Introduction
toPersistent Homology

Žiga Virk

2022
Univerza v Ljubljani
Fakulteta za računalništvo in informatiko

Kataložni zapis o publikaciji (CIP) pripravili v Narodni in univerzitetni knjižnici
v Ljubljani
COBISS.SI-ID=99057667
ISBN 978-961-7059-10-6 (PDF)

Digitalna izdaja je prosto dostopna
This digital publication is freely available
http://zalozba.fri.uni-lj.si/virk2022.pdf
DOI: 10.51939/0002

Recenzenta / Reviewers:
prof. dr. Nežka Mramor Kosta, doc. dr. Boštjan Gabrovšek
Založnik: Založba UL FRI, Ljubljana
Izdajatelj: UL Fakulteta za računalništvo in informatiko, Ljubljana
Urednik: prof. dr. Franc Solina
Copyright © 2022 Založba UL FRI. All rights reserved.

Preface
In the past few decades persistent homology has emerged as oneof the focal points of modern topology. While its beginnings havebeen motivated by practical demand in computational geometry, thescope of persistence soon expanded. Results focused on topologicaland algebraic fundamentals have been complemented by algorithmic
and later statistical development leading to successful applications insciences. Today, persistent homology is a wide ranging topic engaging
a diverse community of mathematicians, computer scientists, data
analysts and scientists in general.
This textbook does not aim to encompass the endless variety of topics related to persistence. The main goal is to provide students with amanageable, geometrically intuitive and self contained introduction topersistent homology. The writing arose from the lecture notes on thecourse that has been taught at the University of Ljubljana for morethan a decade. It is intended for a mixed audience of mathematics andcomputer science students at the masters level. However, any motivated student with scientific or technical inclination should (hopefully)
find it accessible.
As a prerequisite only basic linear algebra (including Gaussian elimination) and basic Euclidean geometry are assumed. The textbook
is structured so that it gradually introduces fundamental structures.
Simplicial complexes and Euler characteristic are first defined in theplane, before generalizing them to the abstract setting. Orientation
is first introduced on surfaces, before utilizing it for homology computations. Required algebra is summarized in Chapter 6 to ensure afamiliar algebraic footing.
Side notes throughout the text are intended to clarify certain ideas,
provide some explanation, or to convey a geometric idea without interrupting the flow of the main text. At the end of each chapter, areader may find a short comment on the background, along with a few
references, keywords, and appendices with further topics or additional
material.

Contents

1

Metric spaces

11

1.1 Definition of metric spaces and basic examples
1.2 Maps and equivalence types
Connectedness

2

11

13

16

Planar triangulations

19

2.1 Definition of planar triangulations
Modifications of triangulations

2.2 Recap on convexity

20

2.3 Euler characteristic

22

19
20

2.4 Constructing planar triangulations with line sweep
2.5 Voronoi diagram and Delaunay triangulation
Local Delaunay condition
Construction of D(S)

2.6 Concluding remarks

25
27

29

Appendix: Proof of Proposition 2.5.6

3

Simplicial complexes

33

3.1 Affine independence

33

3.2 Geometric simplicial complex
3.3 Abstract simplicial complex
Two invariants

40

35
37

30

24

23

6

3.4 Simplicial maps

42

Elementary collapses

43

3.5 Concluding remarks

45

Appendix: Proof of Theorem 3.3.4

4

Surfaces

47

4.1 Surfaces as manifolds

47

Combinatorial manifolds

4.2 Orientability

4950

4.3 Connected sum of surfaces

52

4.4 Classification of surfaces
General surfaces

54

4.5 Concluding remarks

55

53

Appendix: imagining S3

5

45

56

Constructions of simplicial complexes
5.1 Rips complexes

59

5.2 Čech complexes

61

5.3 Nerve complexes

62

Alpha complexes
Mapper59

64

65

5.4 Interleaving properties

67

Rips-Čech correlation

5.5 Concluding remarks

69

69

Appendix: the MiniBall algorithm

70

Appendix: a sketch of a proof of the nerve theorem 5.3.2
Appendix: Dowker duality

6

Fields and vector spaces
6.1 Fields

73

75

75

The fields of remainders Z p

76

71

7

6.2 Vector spaces

78

6.3 Concluding remarks

81

Appendix: A very short introduction to Abelian groups

7

Homology: definition and computation
7.1 Definition

85

Chains85

Boundary86

Homology88

Zero-dimensional homology
Homology of a graph

85

90

91

7.2 Computing homology
Echelon forms

92
92

Smith normal form and representatives

94

Incremental expansion and elementary collapse

7.3 Examples of homology
Disjoint unions
Spheres98

Surfaces

98

96

97
97

Euler characteristic

97

Impact of coefficients: the Klein bottle
Alexander duality

101

7.4 Concluding remarks

102

100

Appendix: Homology with coefficients in Abelian groups
Appendix: cubical homology

8106

Homology: impact and computation by parts
8.1 Impact

109

Functoriality of homology
Brouwer fixed point
Hairy ball

81

110

111

112

Invariance of domain

113

109

103

8

8.2 Homology by parts

114

Exact sequences

114

Mayer-Vietoris exact sequence

8.3 Concluding remarks

116

Appendix: zig-zag lemma

117

Appendix: Relative homology

9

115

119

Persistent homology: definition and computation
9.1 Definition

121

Formal definition

9.2 Visualization

125

Barcodes125

123

Persistence diagrams

126

The fundamental lemma of persistent homology

9.3 Computation

127

127

Matrix reduction

128

Extracting persistence
Representatives
Example

121

128

129

130

Computational tricks

9.4 Concluding remarks

132

133

Appendix: zig-zag persistence and multi-parameter persistence

10

Persistent homology: stability theorem
10.1 Continuous filtrations

137

Interleaving distance for filtrations

10.2 Persistence modules

140

141

Persistence modules
Decomposition

137

141

142

Interleaving distance for persistence modules

143

134

9

10.3 Bottleneck distance and stability theorem
Bottleneck distance

145

Stability theorem

147

10.4 Interpretations and examples

148

1-dimensional persistence of geodesic spaces
Stability demonstrated
Spheres

145

148

150

151

De-noising a function

10.5 Concluding remarks

152

153

Appendix: From the interleaving distance to the bottleneck distance11

Discrete Morse theory
11.1 Motivation

157

157

11.2 Discrete Morse functions and discrete vector fields
Gradient vector fields

11.3 Morse homology

160

162

Morse chain complex
Morse homology

162

163

Generating DMFs and gradient vector fields

11.4 Concluding remarks

166

A proof of Theorem 11.2.6

Index

169

Bibliography

173

166

165

159

154

1
Metric spaces

Topology and geometry study the shapes of spaces. In thisbook we will look at the modelling, computation, and representation ofshapes and their properties. Our starting point will be metric spaces.
These are sets with a meaningful notion of a distance (metric). In thischapter, we will focus on an intuitive understanding of three equivalence types of metric spaces: the isometry type, the homeomorphism
type, and the homotopy type of spaces. These types will play a crucialrole in later sections.

1.1 Definition of metric spaces and basic examples

Definition 1.1.1. A metric space ( X, d) is a pair consisting of a set
X and a function d : X ⇥ X ! [0, •), such that for any x, y, z 2 X
the following hold:• d( x, y) = 0 i↵ x = y,• symmetry: d( x, y) = d(y, x ), and• triangle inequality: d( x, z)  d( x, y) + d(y, z).
Function d is referred to as a distance or a metric.
If X is introduced as a metric space, we implicitly assume d or d X is
the metric on X, unless stated otherwise.
Example 1.1.2. The following are some of metrics d⇤ on R n . For
x = ( x1 , x2 , . . . , xn ) and y = (y1 , y2 , . . . , yn ) we define:• d1 ( x, y) = Âin=1 | xi yi |q
• d2 ( x, y) = Âin=1 ( xi yi )2p• d p ( x, y) = p Âin=1 ( xi yi ) p for p > 1

T Word “i↵ ” stands for “if and only

if ”.

12

introduction to persistent homology• d• ( x, y) = maxi2{1,2,...,n} | xiyi |y = (3, 5)

From now on, R n is always considered to be equipped with the
Euclidean d2 metric, unless stated otherwise.
Example 1.1.3. The underlying space can be di↵erent that R n . Hereare some examples:3• Suppose X is a finite graph with a length associated with each edge.
The geodesic distance d g between two vertices in X is the length of
the shortest path between these vertices in X.
• Suppose X is a surface. We can think of it as a sphere or the surface of the earth. Similarly as above, the geodesic distance d g between two points on X is the length of the shortest path between
these points on X. For example, consider the distance between London and Sydney (see Figure 1.2). The distance usually thought of inthis case is the geodesic distance on Earth, that is, the length of theshortest path between the two cities. The actual Euclidean distancein space between the two cities is shorter, but usually not of interest,since the path that realizes it passes fairly close to the center of the
Earth.
• Suppose A is a finite set which we call alphabet. Let X denote a setof finite sequences (words) consisting of the elements of A (letters).
The Levenshtein distance between two words in X is defined as
the minimum number of edits required to transform one word intoanother, where the allowed edits are:x = (1, 2)
2
Figure 1.1: A few distances:pd1 ( x, y) = 5, d2 ( x, y) = 13,d• ( x, y) = 3.
dg (LON, SY D)
LON
d2 (LON, SY D)

SY D

Figure 1.2: Geodesic vs d2 distance
between London and Sydney.

– an insertion of a letter at any position;– a deletion of a letter anywhere;– a substitution of a letter in any place by another letter.
See Figure 1.3 for example.
• Let X be a finite set and let 2X be the collection of all subsets of X.
The Jaccard distance on 2X is defined asd J ( A, B) =

DOG

DOL

DOLF

W OLF

Figure 1.3: Levenshtein distance
between DOG and WOLF is 3 by thefollowing argument. The sequence
above demonstrates that the distanceis at most 3. As WOLF has threeletters that do not appear in DOG,
the distance is at least 3.

| A [ B| | A \ B|.
| A [ B|For a metric space ( X, d), x 2 X and r > 0 we define the closed1
r-ball around x as

1

As we will only consider closed
balls, the phrase will be simplified tojust “balls”.

Bd ( x, r ) = {y 2 X | d( x, y)  r }.
When the metric is apparent from the context we omit it and use
B( x, r ).

Figure 1.4: Balls in d1 , d2 and d•
metric in the plane.

metric spaces

13

1.2 Maps and equivalence types
When transforming or mapping spaces, we will always be using
continuous maps.
Definition 1.2.1. A map f : X ! Y between metric spaces is continuous if for each x 2 X and for each # > 0 there exists d > 0
so that the following holds for all y 2 X:xd X ( x, y) < d =) dY ( f ( x ), f (y)) < #.
The notion of continuity between metric spaces includes the classical continuity from calculus, i.e., all continuous elementary functions
R ! R are continuous in the sense of Definition 1.2.1 on (R, d1 ).
An equivalent definition of continuity could be stated in terms of
convergent sequences. A sequence of points {zi }i2N in Z converges
to w 2 Z (notation lim zi = w) i↵ d Z (zi , w) converges to zero. Iti!•turns out that a map f : X ! Y between metric spaces is continuous if
the following implication holds: If { xi }i2N is any sequence in X withlim xi = u 2 X, then lim f ( xi ) f (u) 2 Y. A practical interpretation of
i!•

i!•

continuity would be the following: if we improve our measurements xi
in the sense that we get a better approximation for the desired state
w, then the values over a continuous map f ( xi ) also converge to thevalue f (w). For example, suppose we want to estimate the area ofMadagascar from a .bmp image representing a map of the island. We
expect that as the resolution increases, we should get a better estimate
for the total area.
A continuous map g : [0, 1] ! X is called a path from g(0) to g(1).
Next, we give three di↵erent equivalence relations on the class of
metric spaces, each of which preserves a di↵erent level of geometric
information. We start with the strictest equivalence, which preserves
the most structure.

Figure 1.5: A graph (above) and aball around x in the geodesic metricon the graph (below).

Definition 1.2.2. A map f : X ! Y between metric spaces is an isometry, if it is bijective and preserves distances, i.e., for every x1 , x2 2
X, d X ( x1 , x2 ) = dY ( f ( x1 ), f ( x2 )). Two metric spaces are isometric, if there exists an isometry between them.
Isometries of the plane are combinations of translations, rotations
and reflections. In R n , isometries are combinations of a translation
and a linear map. Linear isometries in R n are represented by orthogonal matrices.
It turns out that no patch of a sphere (equipped with the geodesic
metric) is2 isometric to a subset of a plane. A practical consequence of

Figure 1.6: Four isometric planarsets.
2This is a consequence of Gauss’
Theorem Egregium.

14

introduction to persistent homology

this fact is that all topographic maps are distorted.
Although isometries are convenient in many situations, they are essentially a geometric notion that is too rigid for topological treatment.
We next introduce a topological counterpart.
Definition 1.2.3. A map f : X ! Y between metric spaces is a homeomorphism, if it is bijective, continuous, and f 1 is continuous. Two
metric spaces are homeomorphic (or of the same topological type;notation: X ⇠= Y), if there exists a homeomorphism between them.
It is not hard to see that every isometry is a homeomorphism.
While homeomorphisms are much more flexible and preserve a numberof invariants of a space (later we will mention dimension, number ofcomponents and holes, etc.), they do not preserve some of the geometric properties, e.g. diameter (the supremum of pairwise distances in aspace), radii of the smallest enclosing balls, etc.
We will often be referring to the following two spaces:

Figure 1.7: Four homeomorphic setsin the plane.

• For n 2 N an n-sphere Sn is any space homeomorphic to the ndimensional spherenon +1{ x 2 R n+1 | d2 ( x, 0) = 1} = ( x1 , . . . , xn+1 ) 2 R n+1 | Â xi2 = 1 ,i =1where 0 = (0, 0, . . . , n) 2 R n . Observe that S0 consists of two
points, S1 is homeomorphic to a circle3 , S2 to the usual sphere, etc.
• For n 2 N an n-disc D n is any space homeomorphic to the ballB(0, 1) = { x 2 R n | d2 ( x, 0)  1},
where 0 is the n-tuple of zeros4 . Observe that D1 is a closed interval, whose endpoints are S0 . Similarly, D2 can be thought of as theunit disc in the plane. Note that its boundary in the plane is S1 .

Figure 1.8: Four homeomorphic setsin the plane.
A circle is a 1-dimensional subset of
R2 defined by ( x a)2 + (y b)2 = r2 ,i.e., it is “empty inside”.
3

4

Example 1.2.4. Here we provide some examples of homeomorphisms.
• Two finite metric spaces are homeomorphic i↵ they consist of thesame number of points. Each map between finite metric spaces is
continuous.
• Any two closed intervals are homeomorphic. In particular, a homeomorphism f : [0, 1] ! [ a, b] for a < b is given by f (t) = a + t(b a).
• A square [ 1, 1]2 in the plane is homeomorphic to the ball B((0, 0), 1)
in the plane. One of the homeomorphisms is given by the radial
map B((0, 0), 1) ! [ 1, 1]2 mapping:(0, 0) 7! (0, 0) andA clarification on terminology: Aball (a metric concept) in a metricspace is a particular specific subspaceof that metric space. An n-disc (atopological concept) is any spacehomeomorphic to the standard unit
ball in R n , and thus defined up to
homeomorphism. A square in theplane is a 2-disc, but is not a ballin the Euclidean metric. Any unit
ball of radius at least 1 on a circle ofcircumference 1 is the entire circle
and so is not a 1-disc.

metric spaces

15( x, y) 7! r( x, y) · ( x, y) for⇣⌘d2 (0, 0), ( x, y)
x 2 + y2⇣⌘.
r( x, y) ==max{| x |, |y|}d (0, 0), ( x, y)p•• All three balls in Figure 1.4 are homeomorphic via radial maps.
• For each n 2 N, Sn ⇠= Sm i↵ n = m. We will prove this result using
homology in a later chapter.
• For each n 2 N, D n ⇠= D m i↵ n = m. We will prove this result in
Theorem 8.1.7.

Figure 1.9: Two homeomorphic
surfaces.

• No n-disc is homeomorphic to any k-sphere. Each n-sphere can beobtained as a union of two n-discs acting as hemispheres.
While homeomorphism is the focal equivalence in the field of topology, it turns out that many computable invariants are in fact invariant
with respect to a continuous deformation of spaces. These deformations are formalized by the concept of homotopy.
Definition 1.2.5. Continuous maps f , g : X ! Y between metric spaces
X and Y are homotopic [ f ' g] if there exists a continuous deformation of f into g, i.e., if there exists a map H : X ⇥ [0, 1] ! Y,such that
H | X ⇥{0} = f and H | X ⇥{1} = g. Map H is called ahomotopy.

Figure 1.10: Two non-homeomorphic
surfaces.

Figure 1.11: The surface of a cubewith a puncture in each of the six
sides is homeomorphic to a planar set
with five holes.

Another way to think about homotopy between f and g would be
as a continuous collection of paths from f ( x ) to g( x ) in X. Homotopies induce an equivalence relation on continuous maps between
X and Y. Two maps belong to the same homotopy class i↵ they arehomotopic.
Example 1.2.6. Some examples concerning homotopies:
1. For each metric space X, any two maps f , g : X ! R n are homotopic. A homotopy consists of line segments between f ( x ) and g( x ).
In particular,H ( x, t) = (1t) f ( x ) + tg( x ).

2. Let w 2 S1 . Then the identity map id : S1 ! S1 is not homotopic tothe constant map cw : S1 ! S1 , which maps each point to w. Laterwe will be able to prove this fact using homology. Note that by theprevious example both maps are homotopic in R2 , hence the relationof being homotopic depends on the target space of the maps.

Hf, gg(S 1 )S1

f (S 1 )Figure 1.12: Two maps in the plane
are homotopic.

16

introduction to persistent homology

3. Consider the two spaces in Figure 1.13. Space X is a single point,space Y consists of a point, an empty triangle (S1 ), a square (D2 )and a disc with a tail. Observe that there are four homotopy classesof maps from X to Y, one for each component of Y.

XY

We are now ready to introduce homotopy equivalence.
Definition 1.2.7. Metric spaces X and Y are homotopy equivalent[X ' Y] if there exist maps f : X ! Y and g : Y ! X, such thatf g ' idY and g f ' id X . Maps f and g are called homotopy
equivalences.

Figure 1.13: There are four homotopy classes of maps from a single
point space X to Y.

Homeomorphic spaces are homotopy equivalent. A metric space X
is contractible, if it is homotopy equivalent to the one-point space.
Example 1.2.8. Some examples concerning homotopy equivalences:
Figure 1.14: Four contractible spaces.

• Let X = [0, 1] and Y = {0}. Then X ' Y, i.e., [0, 1] is contractible.
Map f : X ! Y is the constant map and map g : Y ! X can bechosen to be any map, say g(0) = 0. Composition f g is identity.
It remains to show that h = g f : [0, 1] ! [0, 1], which is theconstant map at 0, is homotopic to the identity. Such a homotopy
is, for example, the linear homotopy from 1. of Example 1.2.6. In
the same way we can prove that D n is contractible for each n 2 N.
• Convex sets and trees are contractible.
• It turns out that no Sn is contractible. The case n = 1 follows from
2. of Example 1.2.6.

Figure 1.15: Four spaces homotopy
equivalent to S1 : Moebius band (topleft), usual band S1 ⇥ [0, 1] (top right)
and two planar sets below. Only twoof them are homeomorphic.

• R n \ {(0, 0, . . . , 0)} ' Sn 1 . Thinking of Sn 1 as the standard
unit sphere, this equivalence can be proved using the inclusion map
Sn 1 ,! R n \ {(0, 0, . . . , 0)}, the radial map (see Figure 1.18)
R n \ {(0, 0, . . . , 0)} ! Sn 1 defined by x 7! x/|| x ||, and linear
homotopy.
Homotopy equivalence does not preserve all topological properties(for example, dimension), but it does preserve many of those that we
can compute: the number of components, holes, etc.

Figure 1.16: Two more homotopy
equivalent spaces.

Connectedness
The first homotopy invariant we will mention is connectedness.
There are a few versions of it in topology. We will focus on the one
generated by paths.

Figure 1.17: A sequence of steps
deforming O to P. While the figure
demonstrated a continuous deformation (homotopy equivalence), thespaces presented in this case are
actually homeomorphic.

metric spaces

17

Definition 1.2.9. Space X is path connected, if for each x, y 2 X
there exists a path from x to y in X.
Subset A ✓ Y of a metric space Y is a path component, if it
is a maximal path connected subset.
A space is path connected i↵ it is itself a path component. As was
mentioned above, path connectedness is a homotopy invariant: if X
is path connected and Y ' X, then Y is also path connected. Similarly, the number of path components of a metric space is a homotopy
invariant. Space Y on Figure 1.13 has four components.

• Metric spaces;

Figure 1.18: Radial map of a punctured ball B2 ((0, 0), 1) \ {(0, 0)} ⇢ R2
to the standard unit sphere (circle) S in the plane. Using theargument used in last part of Example 1.2.8, the induced homotopy equivalence demonstrates
B2 ((0, 0), 1) \ {(0, 0)} ' S.

• Isometry;adjective “path” and only refer to“connectedness”, and “components”.

1.2 Concluding remarks
Recap (highlights) of this chapter" From now on we will be dropping• Homeomorphism;• Homotopy equivalence;• Connectedness.

Background and applications
Mathematics is the language of science and scientific concepts aremodelled by mathematical objects. These objects can range from
simple to sophisticated: a simple Boolean value (0 or 1, i.e., TRUE
or FALSE), a numeric value (integer, real, complex, etc.), a collectionof numeric values (e.g., a point in R n ), a collection of points in R n , afunction, a vector space, a probability distribution, a graph, a matrix,
a metric space, etc. For most of these notions, there is a useful notionof a metric that transfers the possible outputs into a metric space andthus into the realm of geometry and topology, some of which we have
explored here.
The notions introduced in this chapter are covered in standard
books on topology5 .

5James R. Munkres. Topology.
Prentice Hall, Inc, 2nd ed edition,
2000

2
Planar triangulations

In the previous chapter we learned about metric spaces alongwith the homeomorphism and homotopy type. However, the descriptions we used are not of combinatorial nature, and one would havedifficulties using them for computations. In this chapter we will introduce one of the simplest combinatorial descriptions of planar spaces:
triangulations in the plane. Essentially, we would like to describe aplanar region as a “nice” union of triangles. Triangles are used primarily because they are easy to describe: we only have to provide threepoints. In later sections we will use these triangulations to compute
various invariants of the space: components, homology, etc.
It turns out that not every planar subset can be triangulated. However, finite triangulations (i.e., triangulations with finitely many triangles) can be obtained for most planar subsets of interest to us .

edge xyyxvertex z

z
Figure 2.1: Triangle xyz.

2.1 Definition of planar triangulations
A triangle in the plane has three edges and three vertices.
Definition 2.1.1. A triangulation of a closed region D ⇢ R2 is adecomposition of D into triangles, so that:
1. no triangle is degenerate (i.e., a point or just a line segment),
2. interiors of triangles are disjoint, and
3. intersection of any pair of triangles is either a common edge, a common vertex, or empty.
Geometric description of Definition 2.1.1 is provided by Figure 2.3.
The idea of a triangulation may be generalized in various ways. One
could use di↵erent shapes of pieces to decompose a planar region or
the entire plane. Such decompositions are called tessellations. General-

Figure 2.2: A planar triangulation.
T A planar region admitting a

triangulation is called a polygonal
region.

20

introduction to persistent homology

X

7

X

7

X

7

Figure 2.3: Conditions of Definition
2.1.1.

izing by dimension, one could use “higher dimensional triangles”, suchas tetrahedra, to decompose a higher dimensional space. This idea will
be formalized as simplicial complex in the next chapter.

Modifications of triangulations
Occasionally we will want to modify a triangulation. Here are someof the most used modifications:• add a triangle;• remove a triangle;• flip a common edge;• refine using a subdivision. An example of a subdivision is thebarycentric subdivision: for each edge and each triangle consider
its geometric center (centroid) as a new vertex in our triangulation, and then decompose each triangle as demonstrated by Figure
2.4. This subdivision is convenient when we want to refine a triangulation, i.e., systematically decompose the triangles into smaller
triangles.
We will often focus on triangulations of convex polyhedra, i.e.,
convex hulls of finitely many points in the plane, as defined below.
Given a finite S ⇢ R2 we say a triangulation on S is any triangulationof the convex hull of S, whose vertex set is S.

Figure 2.4: Modifications of tirangulations.

2.2 Recap on convexity
Given points x, y 2 R n , the line segment between them is parameterized asg(t) = tx + (1 t)y, t 2 [0, 1].
Note that g(0) = y, g(1) = x, and g(1/2) corresponds to the
midpoint of the line segment.

(1/2)
x = (1)Figure 2.5: Line segment.

y = (0)planar triangulations

21

Definition 2.2.1. A subset A ⇢ R n is convex, if for each a, b 2 A
the entire line segment between a and b lies in A, i.e., if 8t 2 [0, 1]
we have ta + (1 t)b 2 A. See Figure 2.6.
Given a subset B ⇢ R n , its convex hull Conv( B) is the smallest convex set containing B.
The closed region on Figure 2.2 is not convex, while the ones on
Figure 2.4 are convex.
A triangle is the convex hull of the set of its vertices, which provides a convenient description of a triangle: the triangle with affinely
independent vertices x, y, z 2 R2 can be parameterized by all possible
convex combinations of these vertices:{a1 x + a2 y + a3 z,|

3Â a i = 1}.

8i 2 {1, 2, 3} : ai 2 [0, 1],i =1

The term “convex combination” (as opposed to “linear combination”)
refers to the fact that the coefficients ai are from [0, 1] and add up to
1. These coefficients are called barycentric coordinates in a triangle.
The point with a1 = a2 = a3 = 1/3 is the centroid of the triangle,
while points with two barycentric coordinates1 1/2 are the midpointsof the corresponding edges; all these points are vertices in the barycentric subdivision shown in Figure 2.4.
Convex hull can be constructed by iteratively adding all feasible line
segments. It is important to note that for B ⇢ R n the set obtained byadding all line segments{ a1 x + a2 y|

x, y 2 B,

8i : ai 2 [0, 1],

Figure 2.6: A convex (left) and a
non-convex (right) subset of theplane.

1...and the third coordinate equal to0.

2

Â a i = 1}i =1

is typically not the convex hull. For example, starting with three vertices and adding the line segments between all three pairs we would
obtain the set consisting of the edges but not the interior of the triangle (which constitutes the convex hull of three points). Instead, we
have to add all possible line segments, or alternatively, add all convex
combinations in one step:
Conv B =[m{ Â ai xi

m 2N i =1| 8i :

xi 2 B,ai 2 [0, 1],mÂ a i = 1}.

i =1

By the Carathéodory theorem (see original reference2 or any modern
book treating convexity) we can bound the number of summands3 by
the dimension of the ambient space plus one:n +1

Conv B = { Â ai xi
i =1| 8i :

xi 2 B,ai 2 [0, 1],

Figure 2.7: A collection of points andits convex hull.

n +1

Â a i = 1}.

i =1

In particular: for a finite subset F ⇢ R2 , each point of Conv( F ) is
contained in the convex hull of some triple of points from F.

2Constantin Caratheodory. Ub̈er den
Variabilitätsbereich der Koeffizienten
von Potenzreihen, die gegebene
Werte nicht annehmen. Math.
Ann. 64, no. 1, 95–115, 1907. doi:
10.1007/BF01449883
3...and also the number of iterative
steps in the procedure above...

22

introduction to persistent homology

2.3 Euler characteristic
Along with the number of components of a space, the Euler characteristic is one of the first real topological invariants we come across. In
particular, while there are many triangulations of Conv(S) on a finite
subset S ⇢ R2 , the Euler characteristic is the same for all of them.
For a given triangulation let:• V be the number of its vertices,• E be the number of its edges,=1• F be the number of its triangles.

Definition 2.3.1. The Euler characteristic c of a triangulation isdefined as
c = F E + V.

=0

Theorem 2.3.2 (A simple version of the Euler-Poincaré formula). Assume S ⇢ R2 is finite. For each triangulation of Conv(S) we havec = 1.
Proof. Let us assume our triangulation has no vertical edge: if necessary this can be achieved by a small rotation. Assign to each triangle
value +1 and to each edge value 1. Position each of these values towards the unique rightmost vertex of the corresponding triangle/edge
as suggested by Figure 2.9. Assign to each vertex value +1. The totalsum of all assigned values is c.
For each single vertex add: the value at the vertex and all the values of the triangles and edges, that gathered at that vertex. We cansee that for each vertex the total sum is zero (arising from a sequenceedge-triangle-edge-...-edge-triangle-edge on the left from the vertex
plus the vertex itself) except for the leftmost vertex, where the value
equals one.
Remark 2.3.3. It turns out that c = 1 for each triangulation of acontractible set4 in R2 . In fact, for a triangulation of D ⇢ R2 , cequals the number of components minus the number of holes of D.
More technical details on this fact will be provided in later sections.
Let us just mention that the number of holes of a bounded set D ⇢ R2
equals the number of components of R2 \ D minus one (see Figure2.10).

=

3

Figure 2.8: A few planar triangulations along with their Euler
characteristics.

4

Even more: c is a homotopy invariant.

planar triangulations

+

+

+ +

Figure 2.9: The assignment of values
on triangles (red +), edges (blue )and vertices (green +) from the proofof Theorem 2.3.2. Vertices also holdadditional +1 value. The triangles
are present but not shaded.

+
+

23

+
+
+

+
+
+

+
+

+

2.4 Constructing planar triangulations with line sweep
Let S ⇢ R2 be finite. Perhaps the simplest way to construct a
triangulation on S is using a line sweep, which we now describe. Assume no two points of S have the same horizontal coordinate (this canbe achieved by a small rotation if necessary). Now imagine a verticalline sweeping Conv(S) from left to right. Each time the line reaches
a point of S (a vertex in our triangulation), add all possible edges towards left without creating intersections. Furthermore, for each new
bounded region add the corresponding triangle. As the line sweeps S
we thus obtain a triangulation on S.

Figure 2.10: The top set has 2 holes.
Equivalently, its complement on thebottom has 2 + 1 components.

Figure 2.11: A line sweep using thevertical dashed line. Each time thevertical line reaches a point, we add
all possible edges from that point
to a point with smaller horizontal
coordinate.

The condition that no two points have the same horizontal coordinate was added for reasons of simplicity only. If more points, say
a1 , a2 , . . . , ak have the same horizontal coordinate then, instead of
adding all edges for all points ai at once, proceed point by point: add
all possible edges for a1 , then for a2 , etc. Depending on the order ofpoints ai we typically get di↵erent triangulations.
It should also be obvious that the line sweep does not need to proceed from left to right, but can proceed along any direction by sweep-

24

introduction to persistent homology

ing a line perpendicular to that direction.
While the line sweep is conceptually simple, it does tend to construct triangulations with very thin triangles, which may be undesirable in applications. The triangulation that avoids thin triangles asmuch as possible is the Delaunay triangulation.

2.5 Voronoi diagram and Delaunay triangulation
Throughout this section let S ⇢ R2 be a finite subset satisfying a

Figure 2.12: A line sweep triangulation resulting in thin triangles.

general position property: no four points of S lie on the same circle.
We will first present the Voronoi diagram of S, which is a decomposition of the plane into specific regions.
For each s 2 S define the Voronoi region of s:
Vs = { x 2 R2 | 8u 2 S \ {s} : d( x, s)  d( x, u)}.

Figure 2.13: Circle on left, and a ballon right. The boundary of the ball isthe circle.

Vu

Figure 2.14: An example of a
Voronoi decomposition.

u

If a pair of Voronoi regions Vs1 , Vs2 has a non-empty intersection,then (due to the general position condition above) this intersection is abounded or unbounded line segment called a Voronoi edge and lies on
the bisector between s1 and s2 .
If a triple of Voronoi regions Vs1 , Vs2 , Vs3 has a non-empty intersection, then this intersection is a point called a Voronoi vertex. As thispoint lies on all three pairwise bisectors, it is the center of the circle
containing s1 , s2 and s3 .

Figure 2.15: A Voronoi vertex ⇤ isthe center of the circle containing
the corresponding points • of S.
Voronoi edges lie on the bisector lines
between the corresponding points ofS.

planar triangulations

25

Definition 2.5.1. The Voronoi diagram (or decomposition) of S isthe collection of Voronoi regions, edges and vertices.
T Voronoi regions and Voronoi

A Voronoi region Vs consists of points, whose closest point of S iss. If for some point w there are two such closest points in S, then w
is on the corresponding edge. If for some point w there are three suchclosest points in S, then w is a Voronoi vertex. The general position
criterion above states that for each point in the plane, there can be nofour closest points in S.
Voronoi diagram can be thought of as a result of a uniform expansion from the points of S. Suppose that in the initial stage we startwith a finite set of locations S. Then, as time goes by, each point of
S is being expanded into a region by growing at the same speed in all
directions. At the beginning all these regions are balls centered at the
points of S. As the growing regions collide, the growth towards the regions (edges) of contact stops, but continues along all other directions.
The Voronoi decomposition is the final result of such growth, witheach Voronoi region Vs containing the points that were reached from s
first.

diagrams can be defined for subsets
of R n in the same way. In this casethe general position property is: nocollection of n + 2 points from S lieson the same sphere.

Definition 2.5.2. The Delaunay triangulation on S, denoted by D(S),is the triangulation on S, such that:• its vertices are all points of S,• xy is an edge i↵ Vx \ Vy 6= ∆, and• xyz is a triangle i↵ Vx \ Vy \ Vz 6= ∆.
It turns out that Definition 2.5.2 indeed defines a planar triangulation on S. As a curiosity we mention that an edge xy of a Delaunay
triangulation may partially lie outside of the union Vx [ Vy .
Note that the edge xy of a Delaunay triangulation is a boundaryedge (meaning it is contained in precisely one triangle) i↵ Vx \ Vy is
unbounded. Similarly, x is a boundary vertex of a Delaunay triangulation (meaning it is an endpoint of some boundary edge) i↵ Vx is
unbounded.

Local Delaunay condition
For a triple of non-colinear points x, y, z 2 R2 in the plane defineC( x, y, z) to be the circle containing x, y, z, and let B( x, y, z) be theball whose boundary is C( x, y, z).

Figure 2.16: Voronoi diagram arisingfrom expansion around points.

26

introduction to persistent homology

Figure 2.17: An example of a Delaunay triangulation with its underlying
Voronoi decomposition.

yDefinition 2.5.3. Suppose an edge xy is shared by two di↵erent triangles xyz and xyw of a triangulation. The edge xy is locally Delaunay [abbreviation: LD], if w 2/ B( x, y, z).

Proposition 2.5.4. Suppose edge xy is shared by two di↵erent triangles xyz and xyw from a triangulation.

x

x

x

x

w

2. Each edge in a Delaunay triangulation is LD.

z

The property of being LD is a local property, shared by all edges of
a Delaunay triangulation. It turns out that the converse of Proposition
2.5.4(2) is also true.

z

Figure 2.18: C( x, y, z) on the left andB( x, y, z) on the right.

1. Definition 2.5.3 is symmetric, i.e., w 2/ B( x, y, z) i↵ z 2/ B( x, y, w).

Proof. Part (1) is apparent from Figure 2.19.
(2): Since abc is a triangle in D(S), there exists the corresponding
Voronoi vertex q = Vx \ Vy \ Vz , which is the center of C( x, y, z).
As q 2/ Vw (recall that no four Voronoi regions have a nonempty
interesection by the general position property), d(q, x ) = d(q, y) =d(q, z) < d(q, w) by the definition of Voronoi regions, hence w 2/B( x, y, z).

y

z

w

z
y

y

Figure 2.19: Proof of Proposition
2.5.4 (1).

planar triangulations

27

Theorem 2.5.5. Suppose K is a triangulation on S. Then K is the Delaunay triangulation i↵ each edge is locally Delaunay.
A proof can be found in a textbook5 .

Construction of D(S)Theorem 2.5.5 motivates the edge-flipping construction of Delaunay
triangulations: starting with any triangulation on S (say, one obtainedby a line sweep), keep flipping the non-LD edges. In order to algorithmically implement this construction we have to clarify two issues:

5Mark de Berg, Marc van Kreveld, Mark Overmars, and Otfried
Schwarzkopf. Computational Geometry: Algorithms and Applications.
Springer-Verlag, second edition,
2000. doi: 10.1007/978-3-540-77974-2

1. How do we verify the LD condition?
2. Does the procedure stop?
We address 1. first. It turns out it is not hard to verify the condition of LD using the incircle test.

Figure 2.20: Edge flip.

Proposition 2.5.6. [Incircle test] Suppose x = ( x1 , x2 ), y = (y1 , y2 ), z =(z1 , z2 ) and w = (w1 , w2 ) are four points in R2 . Assume x, y, z are
not collinear and form a positively oriented triple, i.e.:
1
1
1

x1
y1
z1

x2
y2 > 0
y2

x1
y1
z1
w1

x2
y2
z2
w2

x12 + x22
y21 + y22> 0.
z21 + z22
w12 + w22

Then w 2/ B( x, y, z) i↵
1
1
1
1

z

x
A proof and technical details of Proposition 2.5.6 are provided in
the Appendix. While Proposition 2.5.6 provides a convenient way
to verify LD property (and answer 1.), it does not suggest whether
the edge flip algorithm actually stops (2.). In order to address thisquestion we provide a couple more equivalent conditions to LD.
Suppose edge xy is shared by two di↵erent triangles xyz and xyw
from a triangulation K on S. We say that edge xy is a MaxMin edge,
if the minimal angle appearing in triangles xyz and xyw is larger thanthe minimal angle appearing in triangles xzw and yzw.

yFigure 2.21: Positively oriented triple( x, y, z).

28introduction to persistent homology

Proposition 2.5.7. Suppose edge xy is shared by two di↵erent triangles xyz and xyw from a triangulation K on S. Then the following
conditions are equivalent:↵a(i) xy is LD.

p

c

↵

u

(ii) xy is a MaxMin edge.
(iii) ]xzy + ]xwy < p.

v
Proof. Let us prove the equivalence (i) , (iii) first using the inscribed
angle theorem (see Figure 2.22).
definition

Proposition 2.5.7 implies that each edge flip, which makes an edge
in a triangulation LD, increases the minimal angles in the triangulation. Let us explain this statement in more detail. For each triangle
Ti in a triangulation K on S let ti denote the size of its minimal angle.
Construct a lexicographically ordered list of these minimal angles, i.e.,
ti0  ti1  . . .  tim . Proposition 2.5.7 implies that every time we
execute an edge flip making an edge in a triangulation LD, the new
lexicographically ordered list of the minimal angles ti00  ti01  . . .  ti0m
is lexicographical larger than the previous list, i.e., ti j  ti0 , 8 j withjstrict inequality holding for at least one index j . Hence by making
the required edge flips that keep turning edges into LD edges we can’t
return to the initial or any already visited triangulation. Since there
are only finitely many triangulations on S, and therefore finitely many
possible ordered lists of minimal angles, the edge flipping algorithm
terminates, answering 2. above affirmatively.
Conclusion: the edge flipping algorithm terminates with D(S).

↵binscribed angle

xy is LD () w 2/ B( x, y, z)()
p ]xzy > ]xwy ,]xzy + ]xwy < p.
We now turn our attention to (i) , (ii). Let a be the minimal angle appearing in triangles xyz, xyw, xzw and yzw. It is easy to see thata has to lie either along xy or zw as all the other angles get dissected(and hence decreased) by either xy or zw in one of the configurations.
Assume xy is LD. According to the inscribed angle theorem, ]xyz >]xwz, hence ]xyz does not equal a. In the same way we can provethat no angle along xy equals a, hence xy is the MaxMin edge.
Assume now that xy is not LD. Using the identical argument as
in the previous paragraph we can prove that each angle along zw islarger than the corresponding angle along xy. Hence a has to lie along
xy and therefore xy is not a MaxMin edge.

⇡

Figure 2.22: Inscribed angle theorem.
Suppose u, v 2 C( a, b, c), as the figure
demonstrates. Then ]acb = ]aub =p ]avb. This obviously implies]acb > ]apb.

w
x

z
y
Figure 2.23: Proof of Proposition
2.5.7, (i) , (ii).

planar triangulations

29

A triangulation, for which the lexicographically ordered list of theminimal angles is maximal in the lexicographical order, is called a
MaxMin triangulation.
Theorem 2.5.8. A MaxMin triangulation on S coincides with D(S).
In particular, there exists only one MaxMin triangulation on S.

2.6 Concluding remarks
Recap (highlights) of this chapter• Planar triangulations;• Convexity;• Euler characteristic;• Line sweep;• Voronoi diagram and Delaunay triangulation;• Constructing the Delaunay triangulation using the locally Delaunay
condition and the incircle test.

Background and applications
The Euler characteristic was introduced by Leonhard Euler in the
18th century. The line sweep algorithm, Voronoi diagram and Delaunay triangulation are basic notions studied especially in computational
geometry6 . Applications of the Euler characteristic include image
analysis7 , target enumeration8 , etc. The edge flip algorithm we mentioned requires O(n2 ) edge flips, where n is the number of vertices of
S. There are known algorithms to construct the Delaunay triangulation in O(n log n).
The above mentioned properties of the Delaunay triangulation makeit one of the favorite choices for a triangulation on a finite planar set
S. For example, assume you are given a collection of points modellinga geographic profile in a small region. The points consist of coordinates and elevations at these coordinates. The task is to model thesurface approximating the geographic profile. A standard solution
would be to construct the Delaunay triangulation on the set of coordinate points, and then lift these points and triangles according to thegiven elevations. Triangles lifted this way provide a good approximation of the geographic profile on the sampled region.

6

Mark de Berg, Marc van Kreveld, Mark Overmars, and Otfried
Schwarzkopf. Computational Geometry: Algorithms and Applications.
Springer-Verlag, second edition,
2000. doi: 10.1007/978-3-540-77974-2
7
A. Roy, R. A. I. Haque, A. J. Mitra, M. Dutta Choudhury, S. Tarafdar, and T. Dutta. Understanding
flow features in drying droplets
via Euler characteristic surfaces—
a topological tool. Physics of
Fluids, 32(12):123310, 2020. doi:
10.1063/5.0026807
8
Yuliy Baryshnikov and Robert
Ghrist. Target enumeration via
Euler characteristic integrals.
SIAM Journal on Applied Mathematics, 70(3):825–844, 2009. doi:
10.1137/070687293

30

introduction to persistent homology

Appendix: Proof of Proposition 2.5.6
Proof. Let us explain the positively oriented criterion first, see Figure! !
2.24. Points x, y, z form a positively oriented triple i↵ vectors xy,
yz
are positively oriented, meaning that the third component of the crossproduct (y1 x1 , y2 x2 , 0) ⇥ (z1 x1 , z2 x2 ) is positive. This third
component equals the 3 ⇥ 3 determinant
1
1
1

x1
y1
z1

z

x2
y2 .
y2

We now turn our attention to the proof of the proposition. Surprisingly enough, we need to use three-dimensional geometry, see Figure
2.25 throughout the proof. Embed R2 (and points x, y, z, w) into R3
by assigning the third coordinate to be 0, i.e., x = ( x1 , x2 , 0), etc.
Consider the graph of the function f ( x, y) = x2 + y2 . Lift points
x, y, z, w to the graph of f and let x 0 , y0 , z0 , w0 denote the lifted points,
i.e., x 0 = ( x1 , x2 , x12 + x22 ), etc. Let P denote the plane containing
x 0 , y0 , z0 .

xyFigure 2.24: Positively oriented triple
x, y, z.

Figure 2.25: Proof of Proposition
2.5.6.

z = x2 + y 2
z
x0

0

⇧

y0
z

x

y

C(x, y, z)

Let C be the intersection of the graph of f and P. Note that thevertical projection of C onto R2 ⇥ {0} is a circle: substituting z in
z = x2 + y2 by an equation of a plane z = ax + by + c we obtain
an equation of a circle in the plane of the form x2 + y2 ax by
c = 0. As this circle contains x, y, z, it coincides with C( x, y, z). It isplanar triangulations

geometrically apparent that w 2/ B( x, y, z) i↵ w0 lies above P (theregion where the graph of f is below P is B( x, y, z)). Since x, y, z are
positively oriented, a normal of P with a positive third component!
!
!
is ~n = x 0 y0 ⇥ x 0 z0 . Point w0 lies above P i↵ ~n · x 0 w0 is positive. It is!
elementary to verify that ~n · x 0 w0 equals
1
0
0
0

x1
y1
z1
w1

x2
x1
x1
x1

y2
z2
w2

x2
x2
x2

x12 + x222
y1 + y22 ( x12 + x22 )z21 + z22 ( x12 + x22 )w12 + w22 ( x12 + x22 )

1
1
=
1
1

x1
y1
z1
w1

x2
y2
z2
w2

x12 + x22
y21 + y22.
z21 + z22
w12 + w2231

3
Simplicial complexes

Topological and computational treatment of metric spaces
relies on their convenient description. Given a metric space, we would
like to have a finite combinatorial description, that can be used forcomputations. In the previous chapters we introduced planar triangulations as an example of such a description for planar subsets. In thischapter we will introduce simplicial complexes, which will form thebasic structure upon which all our later computations will depend.
Simplicial complexes are higher-dimensional analogues of planar
triangulations. While the latter are collections of triangles that fit
together nicely, simplicial complexes are collections of higher dimensional simplices (generalizations of triangles) that fit together nicely.
Essentially we will be building spaces from simple building blocks(simplices) given a rule describing how these blocks fit together... justlike building a castle from LEGO cubes.

3.1 Affine independence
A point, a line segment, a triangle, a tetrahedron, etc. These aresome of the geometric simplices. They are basic building blocks ofgeometric simplicial complexes. A geometric simplex is a convex hullof a finite collection of points. Before we state their formal definition
we need to clarify a general position property required of a set ofpoints spanning such a simplex. Under this property we want a pair ofpoints to span a line segment, a triple of points to span a triangle (andnot just a line segment), etc.
Choose d, k 2 N and let V = {v0 , v1 , . . . , vk } ⇢ R d be a collection ofpoints. Their affine combination is any sum of the formk

Â ai vi ,

i =0

k

with

Â ai = 1.

i =0The affine hull of V is the collection of all affine combinations of ele-

Figure 3.1: Some geometric simplices: a point, a line segment, atriangle, a tetrahedron.

34

introduction to persistent homology

ments of V. An affine hull is always an affine linear subspace in R d ,meaning it is obtained from a linear subspace of R d by a translation.
Points {v0 , v1 , . . . , vk } are affinely independent if no vi can be expressed as an affine combination of points V \ {vi }. Proposition 3.1.1
explains how to test points for affine independence using linear independence, and why each affine hull is a translated linear subspace.
Proposition 3.1.1. Points of V = {v0 , v1 , . . . , vk } ⇢ R d are affinely
independent i↵ {v1 v0 , v2 v0 , . . . , vk v0 } are linearly independent.
Proof. Assume points of V are not affinely independent. Then, without the loss of generality, v0 = Âik=1 ai vi and Âik=1 ai = 1, which
implies equality Âik=1 ai (vi v0 ) = 0 and not all ai are zero. We
conclude that the points of V are not linearly independent.
On the other hand assume Âik=1 b i (vi v0 ) = 0 with not all b ibeing zero. We define b 0 = Âik=1 b i and observe thatk

Â b i v i + b 0 v0 = 0kÂ bi = 0.

and

i =1

i =0

Choose K 2 {0, 1, . . . , k } so that b K 6= 0. Then
vK =

k

bi
v
bK i

Â

i =0, i 6=K

k

z

y

x
Figure 3.3: The convex hull of threeaffinely independent points is a triangle. The affine hull is the supporting
plane of the triangle.

bi
= 1.
bK

Â

and

Figure 3.2: The affine hull of thetwo points on the left is a line. The
affine hull of the three colinear points
on the right is also a line, implying
these three points are not affinely
independent.

i =0, i 6=K

Hence points of V are not affinely independent.
T A linearly independent collection

Proposition 3.1.2. Suppose points of V = {v0 , v1 , . . . , vk } ⇢ R d
are affinely independent. Then for each point x 2 Conv(V ) there
exist unique coefficients ai 2 [0, 1], i 2 {0, 1, . . . , k }, such thatkx = Â ai vi

and

i =0

k

Â ai = 1.

i =0Coefficients ai in are called barycentric coordinates of point x inConv(V ).
Proof. The existence of such coefficients ai follows from x 2 Conv(V ).
In order to prove the coefficients are unique assume the statement
holds for two di↵erent sets of coefficients ai and ai0 , i.e.,
k

k

i =0

i =0

x = Â ai vi = Â ai0 vi

and

k

k

i =0

i =0

Â ai = Â ai0 = 1.

of vectors in R d can have at most delements. An affinely independent
collection of points in R d can have at
most d + 1 elements.

simplicial complexes

35

At some index i the coefficients ai and ai0 di↵er. Without loss of generality we can assume that index is zero, i.e., a0 a00 6= 0. Thenk( a0a00 )v0 = Â (ai0
i =1

and

k

ai0
a
i =1 0v0 = Â

ai ) vi

ai
v,
a00 i

which contradicts the assumption that the points of V are affinely
independent.

3.2 Geometric simplicial complex
We are now ready to define our basic building blocks.
Definition 3.2.1. Let k, d 2 {0, 1, . . .} with k  d. A geometric ksimplex s in R d is the convex hull of an affinely independent family V = {v0 , v1 , . . . , vk } ⇢ R d , i.e., s = Conv(V ).
The following is some terminology related to a geometric simplex
s = Conv({v0 , v1 , . . . , vk }):• Dimension: dim(s ) = k. We sometimes express it by writing it assuperscript: s = sk .
• Vertices of s: v0 , v1 , . . . , vk .

Figure 3.4: A two-dimensional
simplex has six simplices as faces(three edges and three vertices),
three of which are facets (edges).

• Edges of s: convex hulls of pairs of vertices.
• We say that s is spanned by the set of its vertices.
• If simplex t is spanned by a subset of the vertices of s, we say that:– t is a face of s.
– s is a coface of t.
– t is a facet of s if dim(t ) = dim(s )1.

Note that sk ⇠ D k . By Proposition 3.1.2 each point of s is uniquely=
described by its barycentric coordinates using the vertices of s.
We can now use these building blocks to assemble more complicatedspaces.
Definition 3.2.2. Let d 2 {0, 1, . . .}. A (finite) geometric simplicial complex K ⇢ R d is a (finite) collection of geometric simplices,such that:a: If s 2 K and t is a face of s, then t 2 K.

T All our simplicial complexes

will be finite. For that reason wewill be dropping the word “finite”.
There also exist simplicial complexes
with infinitely many simplices.
However, a proper definition of
infinite simplicial complexes brings
along additional technicalities which
we want to avoid in our context.

36

introduction to persistent homology

b: If s, t 2 K, then s \ t is either empty or a common face of both.
Each planar triangulation has a corresponding simplicial complex
consisting of all triangles, edges and vertices of the triangulation.
Let K be a simplicial complex. We define:• Dimension dim(K ) = maxs2K dim(s ). A one-dimensional simplicial
complex is a graph.
• Vertices of K as the collection of all vertices of all simplices of K.
• Edges of K as the collection of all edges of all simplices of K.
• A geometric simplicial complex L is a subcomplex of K [notation:
L  K], if L ✓ K.

Figure 3.5: The smallest twodimensional simplicial complex
consists of a triangle and all its faces:
three edges and three vertices.

• For n 2 {0, 1, . . .} the n skeleton of K [notation: K (n) ] is the simplicial subcomplex of K consisting of all simplices of K of dimension
at most n. For example, K (0) is the set of vertices of K.
• The body of K [notation: |K |] is the union of all simplices of K.

Formally speaking, a geometric simplicial complex in R d is a collection of simplices and its body is a subset in R d . In practice however
we will be often identifying the two objects in geometric discussions.
From now on we will be visualizing simplicial complexes by drawing
their body and assuming the underlying structure of asimplicial complex.
We are now ready to describe a connection between a metric subspace of a Euclidean space and its combinatorial description.
Definition 3.2.3. Let d 2 {0, 1, . . .}. A triangulation of a subspace
X ⇢ R d is a simplicial complex K in R d , such that |K | ⇠= X.
Not every subspace of R d admits a triangulation. However, all thesubsets that will arise in our context will admit it. Triangulations of
BR2 ((0, 0), 1) include examples in Figure 3.8 and Delaunay triangulations. A geometric simplicial complex is a triangulation of its body.
Occasionally we will want to refine the triangulation of a space,meaning we will want to decrease the size of simplices in order toimprove visualisation, level of details, etc. Such refinements are called
subdivisions. Given a geometric simplicial complex K, a geometricsimplicial complex L is its subdivision, if each simplex of K is theunion of a collection of simplices from L. As an example we alreadymentioned the barycentric subdivision of planar triangulations, which
also exists for simplicial complexes. At this point we will refrain fromintroducing its formal definition. The lower right part of Figure 3.8
depicts a subdivision of a single 2-simplex, see also Figure 3.10.

Figure 3.6: The simplicial complex
from Figure 3.5 (left) and its 1skeleton (right) consisting of threeedges and three vertices.

Figure 3.7: On the left there isa geometric simplicial complex
presented by drawing its body. Eachedge of a sketched triangle and eachvertex of a sketched edge is assumed
to be in the complex. On the rightis the 1-skeleton of the simplicial
complex on the left.

Figure 3.8: A few triangulations ofD2 .

Figure 3.9: A few triangulations ofS1 .

simplicial complexes

37

3.3 Abstract simplicial complex
When buying a commercial object to be assembled, be it a piece offurniture, a toy or a model made of cubes, or a picture made of puzzles, the package usually arrives in a big box. On the box is a pictureof the object, which in our context represents the body of a geometric simplicial complex. On the picture we can often determine pieces,which in our setting would be geometric simplicial simplices. Pieces onthat picture have specific locations and just like geometric simplices,
could be described by specific coordinates. However, the assembly instructions contain no coordinates. There is a good reason for that1 .
In order to assemble the object, the instructions only provide a listof pieces and instructions about how to put them together. That information is sufficient to reconstruct the object. Abstract simplicial
complexes play the role of such instructions.
Assume we want to describe a geometric simplicial complex. Thatmeans we have to provide a list of all simplices. A simplex could be
provided by a list of coordinates of its vertices, but then we also haveto make sure the simplices intersect appropriately. It would be mucheasier to just list the simplices and describe how they fit together in acoordinate free way. Here is a way to do it.
Definition 3.3.1. Let V be a finite set. An abstract simplicial complex L on V is a family of non-empty subsets of V, such that if s 2
L and t ✓ s is non-empty, then t 2 L.
A few more accompanying definitions using the notation of Definition 3.3.1:• An abstract simplex s is an element of L. Its dimension is dim(s) =|s| 1.
• If t ✓ s 2 L, then:– t is a face of s.
– s is a coface of t.
– t is a facet of s if dim(t ) + 1 = dim(s ).
• Dimension dim( L) of L is the maximal dimension of a simplex in L.
• The (closed) star of a vertex v 2 K is StK (v) = St(v) = {s 2 K |s [ {v} 2 K }  K.
• The link of a vertex v 2 K is LkK (v) = Lk(v) = {t 2 St(v) | v 2/t }  St(v).

Figure 3.10: A geometric simplicial
complex and its subdivision.
1

Besides the fact that nobody wouldpurchase such an item.

T In some sources the non-empty

condition in the definition of an
abstract simplicial complex is omitted
and the empty set is always included
as an abstract simplex of dimension1.

38

introduction to persistent homology

A geometric simplex is a subset of an Euclidean space, given as theconvex hull of the collection of its vertices. Each vertex is given as apoint in space, usually in terms of coordinates. A geometric simplicial
complex is a set of such simplices, contains all faces and has to satisfy
the intersection properties of Definition 3.2.2.
An abstract simplex is just a collection of vertices. No coordinates
are needed. An abstract simplicial complex is a set of such collections
which contains all faces (all subsets of its elements). There are nointersections to be checked. It is a complete and convenient combinatorial description.

Figure 3.11: A star (left) and a link(right) of a vertex.

Example 3.3.2. Let K be a geometric simplicial complex provided by
Figure 3.12. As a geometric simplicial complex, K contains specificgeometric simplices described by the coordinates of their vertices. We
can construct a corresponding abstract simplicial complex L. Label thevertices as demonstrated by the figure. Thend

L = {{ a, c, d}, { a, b}, {b, c}, {c, d}, {d, a}, { a, c}, { a}, {b}, {c}, {d}}.
No coordinates are involved. We could also only list the inclusionmaximal simplices, which completely determine the simplicial complex:{{ a, c, d}, { a, b}, {b, c}}.
A simpler structure of an abstract simplicial complex will suffice
for most of our topological analysis of spaces and the corresponding
computations. Indeed, it will simplify them. A geometric simplicial
complex however is still useful when we want to visualise a complex.
For example, outputs of various scans come in the form of geometricsimplicial complexes modelling the scanned shape. While geometricsimplicial complexes describe geometric information about the space(various sizes, lengths, etc.), abstract simplicial complexes contain only
topological information (homeomorphism type).
It is easy to turn a geometric simplicial complex into an abstract
simplicial complex: replace each coordinate given vertex by a uniquelabel. The opposite is a bit harder. Turning an abstract simplicial
complex into a geometric simplicial complex requires us to choose coordinates of vertices in line with the requirements for a geometric simplicial complex. If it can be done, such a geometric simplicial complex
is called a geometric realization (or just realization) of the original
abstract simplicial complex. It turns out that geometric realizations
always exists, although obtaining them in a low-dimensional space istypically hard. The following are two special cases of such realizations.

cabFigure 3.12: A picture accompanying
Example 3.3.2.

Theorem 3.3.3. Every abstract simplicial complex K with n vertices
admits a geometric realization in R n 1 .

simplicial complexes

39

x1

x2

x3

y1

y2

y3

Proof. Simplicial complex K is a subcomplex of the full simplicial
complex L on n vertices, i.e., the simplicial complex, whose simplices
are all subsets of vertices of K. As L admits a realization in R n 1 asan (n 1)-simplex (i.e., the convex hull of a collection of n affinely
independent points), so does K as its subcomplex.
Theorem 3.3.4. Every abstract simplicial complex of dimension d admits a geometric realization in R2d+1 .
A proof of Theorem 3.3.4 is provided in Appendix.
As an example consider graphs, i.e., one-dimensional simplicial
complexes. It is well known that some graphs are planar, which means
they admit a geometric realization in the plane. However, there aregraphs, that are not planar. These graphs can only be realized in
R3 = R2·1+1 (and, of course, in R m for m > 3). See Figure 3.13.
One of the goals of this course is the following: given an abstract
simplicial complex, extract topological properties of its geometricrealization. We will study and analyze spaces by working with their
triangulations.

Figure 3.13: A sketch of a onedimensional abstract simplicial
complex (graph) with no realizationin R2 . The complex consists of alledges between xi and y j .

Remark 3.3.5. It is important to understand the di↵erences between
a metric space, its triangulation and a corresponding abstract simplicial complex. In practice however, we will be frequently vague in our
expression for the sake of simplicity, ofter referring to just simplicial
complex. Given a picture of a space like the one in Figure 3.12, wewill keep in mind the three possible interpretations and use the onethat fits the context at the moment.
Also note that the terminology is essentially the same for the abstract and geometric simplicial complexes. We declare this to be thecase for all further2 definitions as well. For example, an abstract
simplicial complex L is a triangulation of a metric space X if thecorresponding geometric simplicial complex is, i.e., if the body of ageometric realization of L is homeomorphic to X.

Figure 3.14: A metric space(left) and the corresponding
geometric simplicial complex(right). The corresponding abstract simplicial complex is{{ a, b, c}, { a, b}, {b, c}, { a, c}, { a}, {b}, {c}}.

Example 3.3.6. One of our standard examples of a metric space will be
the torus T. It is a two-dimensional metric space, actually a surface,depicted in Figure 3.15. A triangulation of T in terms of an abstract
simplicial complex is provided by Figure 3.17.
Topologically speaking, the torus can be obtained from a square byidentifying the opposite sides along the same direction. This construction is depicted in Figure 3.16.

c

a

b

2

Including the concepts of link andstar of a complex at a point.

Figure 3.15: Torus.

40

introduction to persistent homology

a
a

b

b

b

b

a

Figure 3.16: The torus arising from asquare. Starting with a square (topleft) we identify its pairs of opposite
sides along the same direction as thelabels and arrows suggest. Identifying the sides a we obtain a cylinder(top right). Identifying the otherpair of sides, which represent loops bin the cylinder, we obtain the torus(bottom right).

b

b

b

As a result we can obtain a structure of an abstract simplicial complex by triangulating a square and respecting the mentioned identifications. This provides a convenient topological visualisation of a torus.
Observe that a triangulation of T in terms of a geometric simplicial
complex would be more complicated and not presentable in the plane.
The mentioned abstract simplicial complex is provided by Figure
3.17. We divide the square into 18 triangles and keep in mind theidentifications suggested by the arrows. The two sides of the squarealong the single arrows get identified along the direction of the arrows,
and the same holds for the two sides along the double arrows. For thesake of clarity we also labeled the outer vertices of the triangles as eachlabel appears at least twice due to the identifications.
Example 3.3.7. Choose n 2 {1, 2, . . .}. In this example we provide thesimplest triangulations of discs and spheres.
Let sn be an n-simplex and define K to be the simplicial complex
whose only maximal simplex is sn , i.e., K contains sn and all of its
faces. Simplicial complex K is a triangulation of D n .
To obtain a triangulation of Sn 1 remove from K the maximal simplex, i.e., K 0 = K \ {sn }. Simplicial complex K consists of all faces ofsn but does not contain sn itself. Simplicial complex K 0 is a triangulation of Sn 1 .

Two invariants
Here we provide two invariants (of a space) that can be extracted
from a triangulation. Both are homotopy invariants (and hence also

x

v

w

x

z

z

y

y

x

v

w

x

Figure 3.17: A triangulation ofa torus in terms of an abstract
simplicial complex.

Figure 3.18: A triangle as a triangulation of D2 and its boundary asa triangulation of S1 . In a similar
way a solid tetrahedron is a triangulation of D3 while its boundary is atriangulation of S2 .

simplicial complexes

topological invariants), meaning they coincide for homotopically equivalent spaces. A space typically has infinitely many possible triangulations. Imagine all possible Delaunay triangulations in R2 : they are
all triangulations of D2 . We conclude that the numbers of vertices,
edges, or higher dimensional simplices in a triangulation cannot be
topological invariants.
The first invariant is the number of components. Given a triangulation K, it is easy to extract that number from K (1) (which is a graph)using standard approaches of graph theory. Later we will explain indetail how to obtain this number in terms of homology. For example,
the simplicial complex in Figure 3.19 has two components.
The second invariant is the Euler characteristic, which can be defined for simplicial complexes.

41

Figure 3.19: A complex with twocomponents.

Definition 3.3.8. Suppose K is a simplicial complex and let ni denotethe number of i-simplices in K. The Euler characteristic c(K ) 2
Z is defined as c(K ) = n0 n1 + n2 n3 + . . . .
The Euler characteristic of a metric space is the Euler characteristic of any of its triangulations.
As was mentioned above, the Euler characteristic is homotopy
invariant. Using this fact we can compute the following cases:• Let X be a one-point space. Then c( X ) = 1. Since each Delaunay
triangulation K in R2 is homotopic to a point (meaning that |K | 'X), we also conclude c(K ) = 1, a statement which we have alreadyproved directly. In fact, the homotopy invariance implies that eachtriangulation of a contractible space is of Euler characteristic 1.

Figure 3.20: A series of homotopic
simplicial complexes, each di↵ering from the previous one by asmall modifications. Note that themodifications preserve the Euler
characteristic. In the first step we
add a vertex and an edge; in the second step we add a vertex, two edges
and a triangle; and so on. Each newaddition contributes a total of 0 to
the Euler characteristic.

• The Euler characteristic of a torus is 0. It can be computed directly from the triangulation presented by Figure 3.17, which has 18
triangles, 27 edges and 9 vertices (keep in mind the identifications).
• Let n 2 {0, 1, 2, . . .}. Then c(Sn ) = 1 + ( 1)n .
Proof. Let sn+1 be an (n + 1)-simplex and define K to be thesimplicial complex whose only maximal simplex is sn+1 . As weknow K is a triangulation of D n+1 . As D n+1 is contractible, c(K ) =1. We also mentioned that K 0 = K \ sn+1 is a triangulation of Sn .
As K 0 is obtained from K by removing an (n + 1)-simplex, c(K 0 )is obtained from c(K ) by removing a contribution of that simplex,
which is ( 1)n+1 . Hence c(Sn ) = 1 ( 1)n+1 = 1 + ( 1)n .

T c(Sn ) could also be computed fromthe triangulation K 0 directly using thebinomial formula.

42

introduction to persistent homology

3.4 Simplicial maps
Just as simplicial complexes provide a convenient combinatorialdescription of metric spaces, simplicial maps provide a combinatorialdescription of continuous maps. We first define them in the abstract
setting.
Definition 3.4.1. Suppose K and L are abstract simplicial complexes.
A simplicial map between K and L is an assignment f : K (0) ! L(0)on vertices, such that for each abstract simplex {v0 , v1 , . . . , vk } 2 K
its image { f (v0 ), f (v1 ), . . . , f (vk )} is an abstract simplex in L.
Remark 3.4.2. A simplicial map will be usually denoted by f : K ! L.
However, since K and L are collections of sets, such a notation would
formaly include maps that map, say, a vertex to an edge, a highly
unfavourable occurence. When talking about simplicial maps K ! L
we thus always consider only maps in the sense of Definition 3.4.1,
i.e., maps that map a vertex to a vertex, while images on simplices arealways determined by the values on the vertices:,!
Figure 3.21: An embedding of asimplicial subcomplex L  K into K is
identity on the vertices and always asimplicial map.

• For each vertex v 2 K we define a corresponding vertex f (v).
• For each abstract simplex s = {v0 , v1 , . . . , vk } 2 K its image
f (s) = { f (v0 ), f (v1 ), . . . , f (vk )} is determined by the values on its
vertices.
• Note that f (s ) is a set, meaning there are no repetitions of elements. In particular this means that each vertex appears at mostonce in f (s ), even if it appears multiple times as f (vi ). As a result the image of an n-dimensional simplex can be of dimension lessthan or equal to n, but never more than n.
Example 3.4.3. Let K be the simplicial complex in Figure 3.22. Assignment a 7! a; b 7! c; c 7! c; d 7! d; e 7! b can be verified
to induce a simplicial map K ! K. Note that triangle { a, b, c} gets
mapped to edge { a, c}.
We are now ready to define simplicial maps in the geometric setting.
Definition 3.4.4. Suppose K and L are geometric simplicial complexes.
A map f : K ! L is a simplicial map, if:
1. For each vertex v of K its image f (v) is a vertex of L.
2. The corresponding map between the corresponding abstract simplicial complexes is simplicial, i.e., if {v0 , v1 , . . . , vk } span a geo-

b
e
d

a

c
Figure 3.22: Simplicial complex K of
example 3.4.3.

simplicial complexes

43

metric simplex in K then { f (v0 ), f (v1 ), . . . , f (vk )} span a geometric simplex in L.
3. Map f is linear on simplices (in terms of barycentric coordinates),i.e.,

8ti 2 [0, 1],kÂ ti = 1,i =08 v i 2 K (0) :

k

k

i =0

i =0

f ( Â t i v i ) = Â t i f ( v i ).

Given a simplicial map between geometric simplicial complexes theinduced map (i.e., the restriction to vertices) between abstract simplicial complexes is simplicial. Conversely, each simplicial map between
abstract simplicial complexes corresponds to the unique simplicial
map between the corresponding geometric simplicial complexes: theextension from vertices to geometric simplices is defined using the formula of item 3 of Definition 3.4.4. In accordance with our declarations
simplicial maps will be used to denote maps either in geometric or
abstract setting: in case of a preferred interpretation it will be statedexplicitly or should be obvious from the context.
Simplicial maps between geometric simplicial complexes are continuous maps as they are linear (hence) continuous on each simplex.
Surprisingly enough, each continuous map can be (up to homotopy)
represented by a simplicial map, which means that as long as we areinterested in homotopical properties, we can restrict ourselves to simplicial maps.
Theorem 3.4.5. Suppose f : K ! L is a continuous map between geometric simplicial complexes. Then there exist sufficiently fine subdivisions K 0 of K and L0 of L, and a simplicial map f 0 : K 0 ! L0 ,such that f ' f 0 . We call f 0 a simplicial approximation of f .
The subdivisions above can be taken to be sufficiently fine barycentric subdivision. A continuous map between simplicial complexes isformally a map between the bodies of the simplicial complexes. In thissense both f and f 0 map |K | = |K 0 | to | L| = | L0 | hence f ' f 0 makes
sense.

Elementary collapses
Elementary collapses are minor local modifications of simplicial
complexes, which preserve its homotopy type. Conveniently enough,they can be described in purely combinatorial terms. Their importance stems from the following lemma.
Lemma 3.4.6. Let K be a geometric simplicial complex containing
simplex s = {v0 , v1 , . . . , vk } and let t = {v1 , . . . , vk } be its facet. If

Figure 3.23: The upper part of the
figure presents a simplicial complex K. The middle part consistsof a smooth curve (bold) given bya continuous map f : S1 ! K, superposed on the simplicial complex.
The bottom part consists of a simplicial approximation (bold red)of the curve superimposed on thesimplicial complex and the curve.
The indicated simplicial approximation requires a triangulation of thedomain S1 with at least 12 edges.

44

introduction to persistent homology

s is the only coface3 of t, then the inclusion i : K \ {t, s } ,! K is a
homotopy equivalence.
Proof. The proof is sketched in Figure 3.24.
We first need to subdivide s and t. Choose a point a in the middleof t and connect it to all vertices of s. This induces a subdivision of sand t and in fact of K as no other simplex contains4 s or t.
In order to obtain a continuous deformation5 from K to K \ {t, s },slide a towards v0 . This sliding is a linear homotopy and can be easily
described in the barycentric coordinates of the new subdivision of
K.

⌧

a

3

i.e., a simplex which contains t.

4

If there was another simplex containing s, then that simplex would
have been a coface of t, which contradicts our assumptions. In particular, If s is the only coface, then t isa facet of s. This fact also refers toconditions of Definition 3.4.7.
5
Homotopy equivalence.

v0

Figure 3.24: Elementary collapse
from Lemma 3.4.6 with s being thetriangle and t its left side.

Definition 3.4.7. Let K be a simplicial complex, t (k 1) ⇢ s(k) 2 K,
and assume s is the only coface of t. A removal K ! K \ {t, s }is called an elementary collapse.
By Lemma 3.4.6 each elementary collapse preserves the homotopy
type of a complex. Note that the collapsing map K ! K \ {t, s }is not6 a simplicial map on K. It is, however, a simplicial map on
the subdivision of K employed in Lemma 3.4.6, defined by mappinga 7! v0 and keeping all the other vertices intact. Its homotopy inverseis the inclusion K \ {t, s } ,! K, which is a simplicial map.
Elementary collapses are convenient because they provide us with asimple combinatorial condition that can be used to induce homotopy
equivalence on abstract simplicial complexes. This idea will be further
expanded later within the context of discrete Morse theory.

Figure 3.25: The elementary collapseof Figure 3.24 is usually indicated byan arrow from t into s.
6
Except if t is a vertex.

simplicial complexes

45

Figure 3.26: An example of a simplification of (the homotopy type of) a
simplicial complex using elementarycollapses.

3.5 Concluding remarks
Recap (highlights) of this chapter• Geometric simplex;• Geometric simplicial complex;• Abstract simplicial complex;• Geometric realization;• Simplicial map;
• Simplicial approximation and elementary collapse.

Background and applications
Simplicial complexes model spaces in a wide spectrum of theory andapplications. Great portions of topology and geometry are based onthem due to their simple structure and amenability to combinatorial
treatment. On the applied side simplicial complexes are typically used
to model shapes.
The notions introduced in this chapter are covered in standard
books on topology7 .

Appendix: Proof of Theorem 3.3.4
Before we begin with the proof we clarify a fact that will be used.
A generic8 (random) collection of n + 1 points in R n is affinely independent. Geometrically this is easy to believe:• Generic two points in R will be di↵erent;• Generic three points in R2 will not be colinear;• Generic four points in R3 will not be coplanar.
Similarly, even for k > n + 1 a generic set V of k points in R n has
the same property: each collection of n + 1 points in V is affinely

7

James Munkres. Elements of
Algebraic Topology. Perseus Books,
1984. doi: 10.1201/9780429493911

8

Notion “generic” as used in thisappendix is usually referred to as“general linear position” in theliterature.

46

introduction to persistent homology

independent. For example, in a generic collection of points in theplane no triple of points will be collinear. We will only use the factthat generic collections exist. This fact can be proved using linear
algebra.
Proof of Theorem 3.3.4. Let K be an abstract simplicial complex
of dimension d, whose vertices are v0 , v1 , . . . , vk . Choose a genericcollection of points V = { x0 , x1 , . . . , xk } ⇢ R2d+1 , meaning that eachcollection of 2d + 2 points from V is affinely independent. We will
prove that the correspondence vi $ xi for all i 2 {0, 1, . . . , k} provides
a geometric realization of K.
For each abstract simplex s 2 K spanned by v j0 , v j1 , . . . , v jm , thecorresponding geometric simplex s0 is spanned by the affinely independent collection x j0 , x j1 , . . . , x jm . It remains to prove that if s, t 2 K,then s0 \ t 0 = (s \ t )0 .
As (s \ t )0 ✓ s0 , t 0 by the definition, we have s0 \ t 0 ◆ (s \ t )0 .
To prove the other inclusion we will make use of the dimension
assumption. Let z 2 s0 \ t 0 , which means that z can be expressed as aconvex combination of vertices in s0 and also as a convex combinationof vertices in t 0 . As the total number of vertices in t 0 and s0 is at
most 2n + 2 (by the dimension assumption), the generic condition
implies these are affinely independent, and thus the convex (affine)combinations above coincide as they have to be unique by Proposition
3.1.2. In particular, this means that only the barycentric coordinatescorresponding to the vertices that lie in both simplices (i.e., s \ t)can be non-zero, which implies z 2 (s \ t )0 and hence s0 \ t 0 ✓(s \ t )0 .

Figure 3.27: A generic collection of 5
points in the plane, meaning that nothree are collinear. Only by adding
a point on any grey line does thegeneric condition brake. If we add
any other point to this collection, theobtained collection of 6 points is still
generic.

4
Surfaces

Surfaces are some of the simplest topological spaces appearing
frequently in science and data analysis. From each local perspective
appearing as a part of the plane, the global shape of a surface may
take many forms. Think of the surface of the earth: because it appears
to be “planar” at each point, it had long been believed that Earth isactually a part of a plane or maybe a disc instead of a sphere.
Surprisingly enough, most surfaces of interest can be recognized
up to homeomorphism fairly easily. In this chapter we explain thisrecognition process and the accompanying theory, both of which will
come handy in the chapters to come.

4.1 Surfaces as manifolds
Ever since the ancient times people were wondering about the shapeof the world. They agreed that from the perspective of a human being, the world looked like a plane, a part of a surface. What was muchharder to figure out was the global picture. The first and most obvious
idea was that the world was a flat disc. Later came indications, suchas deviations in the angle of the shadow depending on the latitude,that the world might be curved. Magellan’s first circumnavigation ofEarth does not constitute a rigorous proof that the world is a sphereby modern mathematical standards, but at the time it was a momentous achievement which confirmed that the Earth is indeed round.
While we will not be sailing around the world in this course, we will
be interested in the moral of this story: things that locally look likea plane may globally not be a plane. We will want to determine theglobal structure from local information.
Spaces that locally look like a plane are called surfaces and theirgeneralizations to other dimensions are called manifolds. Here is aformal definition.

Figure 4.1: Some of the surfaces wementioned before: a planar set (topleft), a space homeomorphic to S2(top right), Moebius band (center
left), cylinder (center right), andtorus (bottom).

48

introduction to persistent homology

Definition 4.1.1. Let n 2 {0, 1, 2, . . .}. A metric space X is an nmanifold, if for each x 2 X there exists r > 0, such that BX ( x, r )is homeomorphic to the n-dimensional disc D n .
2-manifolds are called surfaces.
Point x on an n-manifold X is:• a boundary point if a homeomorphism BX ( x, r ) ! D n from Definition 4.1.1 maps x to a point on the boundary of D n .
• an interior point if a homeomorphism BX ( x, r ) ! D n from Definition 4.1.1 maps x to a point in the interior of D n .

Figure 4.2: A neighborhood of apoint on S2 homeomorphic to B2aThese two notions are independent of the choice of homeomorphism
BX ( x, r ) ! D n . Each point of X is either a boundary point or aninterior point. The boundary of X consists of all the boundary points,
and the interior of X consists of all the interior points.
We say that a manifold X is without boundary, if it has no boundary points. For an n-manifold Y its boundary is an (n 1)-manifold
without boundary, as can be seen from the examples below. For our
purposes a closed manifold (closed surface) will be a manifold (surface)
without boundary admitting a (finite) triangulation.
Example 4.1.2. We provide some examples of connected n-manifolds
listed by dimension n.
• n = 0: This one is fairly unimpressive: a single point.

ab

Closely related to the torus are the Klein bottle (see Figure 4.3)
and the projective plane, neither of them has a boundary andneither can be obtained as a subset of R3 . However, they can beobtained as subsets of R4 . While these two spaces are challenging
to imagine geometrically, it is fairy easy to provide their (abstract)
triangulations (see Figure 4.3) and compute some of their topological invariants, such as the Euler characteristic. The Torus and the

b

b

a

b

b

Figure 4.3: Klein bottle obtainedby identifying the edges of a square:
two along the same direction andtwo along the opposite direction.
The resulting space (bottom right) is
not realizable in R3 due to the selfintersection. However, Klein bottle
can be embedded into R4 without
self-intersections.

a• n = 1: Circle and intervals (0, 1), [0, 1], (0, 1]. Each connected 1manifold is homeomorphic to one of these. A circle and an open
interval have no boundary, while the boundary of [0, 1] consists1 of 0
and 1.
• n = 2: We will provide a list of all surfaces by the end of thischapter. Here we list some of the more prominent ones. The alreadymentioned ones are recapped in Figure 4.1: note that the boundaryof the band consists of two copies of S1 , while the Moebius band hasa single boundary component. A closed disc D2 is also a surface,
whose boundary is S1 .

b

b

b

a
Figure 4.4: Projective plane obtained by identifying the edges of asquare: both pairs along the opposite direction. The resulting space
is not realizable in R3 . However, it
can be embedded into R4 without
self-intersections.
1
Also, the boundary of [0, 1) is 0.

v

surfaces

49w

x

Klein bottle have Euler characteristic 0, while the Euler characteristic of the projective plane is 1.

x

The projective space is homeomorphic to the space of all 1-dimensional
subspaces in R3 .

zy

• General n: D n and Sn are both n-manifolds. The boundary of D n is
Sn 1 , while Sn has no boundary. There are many other n-manifolds.

yzCombinatorial manifoldsx

We will mostly be working with triangulated manifolds. A natural
question that arises in this context is how to recognize whether a given
simplicial complex is a triangulation of a manifold. Tackling this task
we first introduce nice combinatorial descriptions of manifolds.

qDefinition 4.1.3. Suppose K is a simplicial complex and n 2 N. We
say that K is a combinatorial n-manifold, if for each vertex v 2
K its link Lk(v) is homeomorphic either to Sn 1 or D n 1 .
Properties and notation:• Each combinatorial n-manifold is a triangulation of an n-manifold.
• For n < 4, each n-manifold admits a triangulation as a combinatorial n-manifold2 .
• Vertices of a combinatorial manifold K satisfying Lk(v) ⇠= D n 1 arecalled boundary vertices.

v

w

w

v

x
x

z

y

y

z

x

v

w

q

Figure 4.5: Triangulations of theKlein bottle (top) and the projective
plane (bottom).
2

Surprisingly enough, this does nothold for n 4• Vertices of a combinatorial manifold K satisfying Lk(v) ⇠= Sn 1 arecalled interior vertices.
• Edges of a combinatorial surface K that are contained in only one
triangle are called boundary edges. The union of the boundaryedges corresponds to the boundary of the manifold.
• Edges of a combinatorial surface K that are contained in two triangles are called interior edges. No edge in a combinatorial surface iscontained in more than two triangles.
Using these properties it is fairly easy to recognize whether a given
simplicial complex K is a combinatorial surface and thus a triangulation of a surface: for each vertex v 2 K we verify whether Lk(v)is homeomorphic to S1 or D1 . It is easy to see that a connected 1dimensional simplicial complex is homeomorphic to:• S1 i↵ each of its vertices is contained in two edges.

Figure 4.6: Triangulation of theKlein bottle (top) and of the Moebius band (bottom). In the top
triangulation each vertex is an interior vertex as each link (bold) is
homeomorphic to S1 . In the bottomcase each vertex is a boundary vertex
as each link (bold) is homeomorphic
to B1 .

50

introduction to persistent homology• D1 , i.e., the line segment, i↵ two of its vertices are contained in one
edge, and all other vertices are contained in two edges.
We will leave the elementary proofs of these two facts to the reader.

4.2 Orientability
Orientability is about defining “up and down”. It is quite easy toagree on the two directions on the surface of the earth. However, ingeneral that may not be the case for all surfaces. Consider the cylinderfrom Figure 4.1. It is orientable because we can define two di↵erent
sides of it. To put it into a more colorful language, we can color one
side of the band in red and the other side in blue, without the colors
ever touching each other. The story is di↵erent on the Moebius band:it has only one side. We could start coloring it in red at some spot
and keep expanding the color along the surface (but not across theboundary): eventually we will color the whole band, i.e., there is no“other” side.
Orientability is an important property of surfaces. It will be required for our classification result. In order to fully understand it wehave to define orientation for simplices first. Besides its applicationin this section orientation on simplices will feature prominently later
within the context of homology computation.
Up to now a simplex was given by a set of its vertices. An oriented
simplex is a simplex with a choice of orientation. For an edge thatmeans direction, for a triangle that means “a normal” (see Figures 4.7
and 4.8). This direction/orientation will be described by a choice of anorder on vertices.
Definition 4.2.1. An oriented simplex on vertices v0 , v1 , . . . , vk isan ordered (k + 1)-tuple s = hv0 , v1 , . . . , vk i. For a permutation p
on {0, 1, . . . , k} we identify:s = ( 1)sgn(p ) hvp (0) , vp (1) , . . . , vp (k) i,
where sgn(p ) is the signature of permutation p, i.e., value 0 if p iseven and value 1 if p is odd.
A 0-dimensional simplex with vertex v can also be oriented in two
ways: as hvi and as hvi.
Figures 4.7 and 4.8 provide examples of descriptions of oriented
edges and triangles, and their geometric interpretations. Here are someproperties that follow from Definition 4.2.1:• Each simplex on vertices v0 , v1 , . . . , vk can be oriented in two di↵erent ways: s = hv0 , v1 , . . . , vk i and s.

y

x

y

x

y

x

Figure 4.7: The edge { x, y} (left) andthe oriented edges h x, yi (center) andhy, x i = h x, yi (right).

z

x

z

y

x

z

y

x

Figure 4.8: The triangle { x, y, z}(left) and the oriented triangles
h x, y, zi = hy, z, x i = hz, x, yi (center)and hy, x, zi = h x, z, yi = hz, y, x i =h x, y, zi (right).

ysurfaces• An oriented simplex has a sign + (usually omitted) or

51

prepended.

• Exchanging two vertices in an oriented simplex t changes the orientation of t by changing the prefixed sign.
An important property of an oriented simplex is that it induces anorientation on each of its facets.
Definition 4.2.2. Suppose s = hv0 , v1 , . . . , vk i is an oriented simplex
and p 2 {0, 1, . . . , k }. Then induced orientation of the facet ofs obtained by dropping v p is( 1) p h v0 , v1 , . . . , v p 1 , v p +1 , . . . , v k i.
Oriented edge h x, yi induces orientations hyi and h x i on its facets(vertices). Oriented triangle h x, y, zi induces orientations hy, zi, h x, zi
and h x, yi on its facets (edges), see Figure 4.9.
Now that we established a way to orient a single simplex, we turn
our attention to orienting the whole surface.

z

x

z

y

x

y

Figure 4.9: Oriented triangle h x, y, zi(left) and induced orientation on theedges (right): h x, yi, hy, zi, and hz, x i.
Note that the edges are oriented
along the direction of the circular
arrow indicating the orientation of
the triangle.

Definition 4.2.3. Suppose oriented 2-simplices s and s0 share a common edge. Simplices s and s0 are oriented consistently, if they induce the opposite orientation on the common edge. (see Figure 4.10)

Definition 4.2.4. Let K be a triangulation of a surface |K |. We saythat |K | is oriented, if all triangles of K are oriented (as simplices)
so that the following holds: each pair of oriented triangles with a common edge is oriented consistently.
A surface is orientable if it can be oriented.

Figure 4.10: Consistent orientation:note that the orientations of thetriangles agree (both directed circular
arrows point counter-clockwise). Thisimplies that the induced orientations
on the common edge are opposite toeach other.

Orientability of a surface does not depend on a triangulation but on
the topological type of the surface only. The following are two basic
examples that demonstrate the underlying geometric idea.
Example 4.2.5. The cylinder S1 ⇥ [0, 1] is orientable as Figure 4.11
demonstrates. To the contrary, the Moebius band is not orientable
as Figure 4.12 demonstrates. Since the Klein bottle and the projective plane both contain a copy of the Moebius band (any of the threehorizontal strips of triangulations in Figure 4.5), neither of them isorientable.
As Example 4.2.5 and Figure 4.12 suggest it is fairly easy to check
whether a connected triangulated surface is orientable. This can bedone directly by orienting one triangle and then inductively orienting
all neighboring triangles with shared edges, while checking that each
newly oriented triangle is oriented consistently with respect to thealready oriented triangles.

y

y

x

x

Figure 4.11: Orientable triangulation of a usual band. The oriented
simplices induce the opposite orientation on all the edges, including theedge { x, y}, along which the glueing
occurs.

52

xintroduction to persistent homology

y

x

y

x

y

x

y

Figure 4.12: A proof that the Moebius band is not orientable. Assume
we want to orient a triangulation of
the Moebius band on the left. We
first choose an orientation of one
triangle (far left) and then inductively induce consistent orientationon the neighboring triangles. In theend (far right) we obtain conflicting
requirements on the orientation on
the last (bold) triangle, which means
there is no consistent way to orient
all the triangles in this triangulation.

!

y

x

y

x

y

x

y

x

4.3 Connected sum of surfaces
One of the ways to make new surfaces (and actually manifolds ingeneral) out of known ones is the connected sum.
Definition 4.3.1. Suppose X and Y are connected surfaces. Choose
topological 2-discs DX ⇢ X and DY ⇢ Y, neither of which contains
any boundary point of the surfaces. The corresponding boundaries ofthese discs are topological 1-spheres (circles) SX ⇢ X and SY ⇢ Y
respectively. The connected sum X#Y is obtained by removing theinteriors of discs DX and DY from X and Y, and gluing the resulting spaces by identifying SX with SY .
See Figure 4.13 for a sketch of this construction. A few technical remarks about connected sums as defined above:• It turns out that the topological type of X#Y does not depend on
the choice of discs DX , DY .
• A connected sum is a surface, whose boundary components correspond to the union of the boundary components of X and theboundary components of Y.

Figure 4.13: Two tori (top) and theirconnected sum (bottom) obtained by
identifying the boundaries of the tworemoved discs (center).

surfaces

53• Surfaces X and Y are both orientable i↵ X#Y is orientable.
• For each surface X, the following holds: X#S2 ⇠= X.
If abstract simplicial complexes K and L are triangulations of surfaces X and Y respectively and X \ Y = ∆, we can obtain a triangulation M of X#Y in the following way:
1. Choose triangles D X and DY in K and L respectively, so that nopoint of these two triangles lies on the boundary of X or Y.
2. DefineM = (K \ {D X }) [ ( L \ {DY })/ ⇠,where ⇠ stands for the identification of each of the boundary edgesof D X with an appropriate boundary edge of DY .
In short, M is obtained by removing D X and DY from the union of K
and L, and then identifying the boundaries of the removed triangles.
This procedure is a discrete version of the one in Definition 4.3.1.
Proposition 4.3.2. c( X#Y ) = c( X ) + c(Y )2.

Proof. Assume abstract simplicial complexes K and L are triangulations of surfaces X and Y respectively and K \ L = ∆. It is obviousthat c(K [ L) = c(K ) + c( L). In order to obtain a triangulation of
X#Y from K [ L, we:• Remove two triangles (change

2 to the Euler characteristic);• Identify three pairs of vertices, meaning we have three vertices less(change 3 to the Euler characteristic);• Identify three pairs of edges, meaning we have three edges less(change +3 to the Euler characteristic);
The total change to the Euler characteristic after these steps is2.
ab

4.4 Classification of surfaces
We can now describe the classification of surfaces. Let T denote thetorus and let P denote the projective plane.

baFigure 4.14: The three closed connected surfaces (the sphere S2 , thetorus T and the projective plane P),
that are used to construct any otherclosed connected surface using theconnected sum operation.

54

introduction to persistent homology

Theorem 4.4.1. [Classification Theorem for closed connected surfaces]
Suppose X is a closed connected surface. Then X is homeomorphic
to one of the following:
1. S2 .
2. n-torus nT = T#T#|
{z. . . #T} for some n 2 N.
n

3. nP = P#P#|
{z. . . #P} for some n 2 N.
n

It turns out that the surfaces appearing in Theorem 4.4.1 can be
distinguished using orientability and the Euler characteristic. From theproperties of the connected sum recall that (for each n 2 N) S2 andnT are orientable while nP are not. Furthermore, using Proposition
4.3.2 and we can inductively deduce3 :• c(nT ) = 2

2n as c(T ) = 0.

• c(nP ) = 2

n as c(P ) = 1.

3

Using Proposition 4.3.2 we candeduce c(2T ) = c(T#T ) = c(T ) +c(T ) 2 = 0 + 0 2 = 2, c(3T ) =c(T ) + c(T#T ) 2 = 0 + 2 2 = 4,
and proceed inductively. The proof
for nP is analogous.

Consequently we obtain the following table of connected closed
surfaces.
Surface

c

S2
T
nT

2

P
nP

2

2
0
2n
1
n

9
=
;

orientable

Table 4.1: A list of closed connectedsurfaces along with their Euler
characteristic and orientability.

not orientable

Theorem 4.4.1 motivates the following classification algorithm for aclosed connected surface given as an abstract simplicial complex4 K:
1. Check for orientability of K.
2. Compute the Euler characteristic.
3. Consult Table 4.1.
Example 4.4.2. Which of the surfaces in Theorem 4.4.1 is the Kleinbottle? We have already discovered that it is not orientable and that
its Euler characteristic is 0. By the Classification Theorem the Kleinbottle is homeomorphic to P#P.

General surfaces
Theorem 4.4.1 can also be used to classify general surfaces admitting a finite triangulation. Suppose X is a surface:

4I.e., we assume the triangulation
K is a connected combinatorial
2-manifold and has no boundary
components.
T A shortcut to computing c: while
the Euler characteristic is formallydefined on a triangulation, it turns
out it can also be obtained from therepresentation of a surface in termsof a polygon with identified sides.
For example, the representationsof the torus in Figure 4.14 andKlein bottle of Figure 4.3 haveone 2-dimensional square, two 1dimensional edges, and one vertex,
yielding c = 2. The representationof the projective plane in Figure
4.4 has one 2-dimensional square,
two 1-dimensional edges, and two
vertices, yielding c = 1. This trick
could assist with Figure 4.15. A
justification will be provided in thechapter on discrete Morse theory.

surfaces55

1. If X is not connected, it is a disjoint union of connected surfaces
and it obviously suffices to recognize each of its components.
2. If X is connected and has a boundary Y, then Y is a 1-manifold
without boundary, meaning Y is a disjoint union of k copies of S1for some k 2 N. By glueing a disc along each component of Y we
obtain a closed connected surface X 0 , which we can recognize 5 . We
conclude that X is homeomorphic to X 0 with k discs removed6 .
Example 4.4.3. It is easy to see that the cylinder S1 ⇥ [0, 1] is obtained
from S2 by removing two discs. It is a bit harder to see how to get the
Moebius band M this way. It is easy to see that M has one boundary
component and has Euler characteristic7 0. Gluing a disc along theboundary component we obtain a closed connected non-orientable surface of Euler characteristic 1, which is P. Hence the Moebius band is
obtained by removing a disc from the projective plane.

5

Such a gluing of discs does notchange the orientation, i.e., X isorientable i↵ X 0 is. However, an
addition of each disc increases theEuler characteristic by 1.
6
It turns out that the homeomorphic
type does not depend on the discs
we remove from X 0 , only on their
number.
7
We could count the simplices in
Figure 4.12.
f

We are now ready to state a classification algorithm for a surface
given as an abstract simplicial complex K:f

1. Partition K into its connected components and classify each of
them.

e

e
a

2. For each component K 0 :
(a)

c

d

Count the number n(K 0 ) of boundary components of K 0 .

(b) Check for orientability of K 0 .
ab(c) Compute the Euler characteristic of K 0 .
(d) Let Y be the surface matching the orientability of K 0 and of
Euler characteristic8 c(K 0 ) + n(K 0 ) by Table 4.1.

With this classification algorithm we can always determine whether
two surfaces are homeomorphic or not.

4.5 Concluding remarks
Recap (highlights) of this chapter• Surfaces, combinatorial surfaces;• Orientation and orientability;• Connected sum of surfaces;• Classification of surfaces;dc(e) Surface K 0 is homeomorphic to Y with n(K 0 ) many discs removed.

b

Figure 4.15: Which surfaces arethese?
c(K 0 ) + n(K 0 ) is the Euler characteristic of a surface obtained from
K by gluing n(K 0 ) discs along theboundary components of K 0 .
856

introduction to persistent homology

Background and applications
For most of the practical purposes, we live in a three-dimensional
space. Objects in our everyday life are often modelled by surfaces
enclosing the objects. Outputs of many 3-D scans are given in terms oftriangulated surfaces (for example, as .stl files).
Surfaces and other higher-dimensional manifolds are also often
assumed to be the underlying spaces in specific settings. A randomly
generated bitmap image will seldom represent something reasonable,
and yet there is a huge number of images that convey an information
to the human eye. A space of “recognizable” images is a huge subspace(perhaps a manifold) in the space of all bitmap images. Manifold
recognition approaches the attempt to detect the underlying manifolds
from sample data.
The third source of surfaces and manifolds are spaces describedwith two or more degrees of freedom: configuration spaces of molecules,
robotic arms, etc. For example, the configuration space of a roboticarm (i.e., the space of all possible positions of the arm) with two independent joins, each of which allows a full rotational motion, is thetorus S1 ⇥ S1 . On a similar note, given two annotated9 points on S1 ,the configuration space of all possible positions of the two points isagain a torus S1 ⇥ S1 , since the degree of freedom of each point is S1 .
It is interesting to observe the di↵erence that appears if the points are
not annotated10 : in such a case all possible configurations actuallyform the projective plane.
On a more theoretical note, the question of whether each manifold
admits a triangulation had been one of the focal points of topology inthe previous century. It turns out that every manifold in dimension
3 or less admits a triangulation. Surprisingly enough, there are manifolds in higher dimensions that do not admit any triangulation. A
detailed proof of the classification theorem of surfaces can be found inmany textbooks11 , 12 .

Appendix: imagining S3
In this appendix we will try to explain two ways of thinking about
the three-dimensional sphere S3 and spheres in general.
1. The first observation has to do with the relationship between discs
and spheres. We have already mentioned that Sn 1 appears as
the boundary of D n . It should also be apparent (see Figure 4.18)
that gluing two copies of a disc D n along their boundaries (copies
of Sn 1 ) results in Sn . In particular, we obtain S3 by taking two
3-discs (solid balls) and gluing them along the boundary.
2. As for the second observation we will refer to Figure 4.19. It turns

Figure 4.16: Spheres S0 (two points),
S1 (a circle), and S2 (a sphere).

Figure 4.17: Discs D1 (a line segment), D2 , and D3 .
9

The annotation refers to the factthat each point has a name in thesense that if the two points lie atdi↵erent positions, then exchanging
them changes the configuration. The
situation is sometimes also describedusing “ordered pairs” of points ( x, y)
for which x, y 2 S1 .
10

In particular, if two non-annotatedpoints lie at di↵erent positions, thenexchanging them does not change theconfiguration as the pair represents
the same collection of points on
S1 . The situation is sometimes
also described using “unorderedpairs” of points ( x, y) by additionallyidentifying ( x, y) ⇠ ( x 0 , y0 ) if x = y0
and y = x 0 .
11

L. Christine Kinsey. Topology ofSurfaces. Springer New York, 1993.
doi: 10.1007/978-1-4612-0899-0
12Jean Gallier and Dianna Xu. A
Guide to the Classification Theorem
for Compact Surfaces. Springer
Berlin Heidelberg, 2013. doi:
10.1007/978-3-642-34364-3

surfaces57

out that Sn can be obtained in the following way: pick two opposite points (the north pole and the south pole) and span an interval’s worth of copies of the sphere Sn 1 between them, so that thespheres are shrinking as they are approaching the poles.
Note that if we only take one point and have the spheres shrinking
only as they approach that point (and have them increase otherwise) we obtain D n (see Figure 4.20 for some low-dimensional
examples).
It may come as a surprise that these points of view can be observedin Dante’s Divine Comedy, written about seven centuries ago. Dante’s
description of the universe coincides with the topology of S3 : on one
extreme are the depths of Hell (part of Inferno), from which Dante
is guided by Beatrice through the spheres of Inferno, Purgatory, andParadise, until he reaches Empyrean, the place which contains theessence of God.
This description coincides with 2. above in terms of spheres. Equivalently, we may consider Inferno and Purgatory together as one 3-discwith center at Hell, and Paradise as another 3-disc with center at
Empyrean: in this setup the Universe consists of both 3-discs thatintersect along the surface of the Earth (which coincides with theboundary S2 ).

Figure 4.18: Gluing two copies of adisc together results in a sphere.

Figure 4.19: Obtaining Sn as acollection of spheres Sn 1 between
two points.

Figure 4.20: Obtaining R2 (left) andR3 (right) as a collection of concentric spheres of all positive radii. In
the same way we can decompose R k
for any k 2 {1, 2, . . .}.

5Constructions of simplicial complexes

Topological methods typically take a simplicial complex as
input. However, objects of interest are often not provided in this form.
The first step in a topological treatment is thus frequently a creationof simplicial complexes. In this chapter we will present various constructions of complexes from a point cloud, i.e., from a finite collectionof points in a metric space. These points may represent a sample ofour shape, a collection of numerical data (a subset of R n ), etc.
Our discussion will include two properties we expect from sucha construction. The first one is a relationship with the underlying
shape, which is often guaranteed by the nerve theorem. The second
one describes stability to perturbations and a way to measure distance
between di↵erent constructions, as formalized by the interleaving
property.

5.1 Rips complexes
Rips complexes represent perhaps the simplest construction of acomplex from a finite collection of points.
Definition 5.1.1. Let X be a metric space and let a sample S ⇢ X
be a finite subset. Choose a scale r 0. The Rips complex Rips(S, r )is an abstract simplicial complex defined by the following rules:
1. The vertex set is S.
2. A subset s ✓ S is a simplex i↵ Diam(s )  r.

T Rips complexes are a special caseof clique complexes. Suppose G isa graph with vertices V and edges
E. The clique complex of G is theabstract simplicial complex with thevertex set V, whose simplices satisfy
the following condition: a subsets ✓ S is a simplex i↵ each pair ofvertices of s is an edge in G. A Rips
complex is the clique complex of its
1-skeleton.
T Diameter of a finite subset A ⇢ X

of a metric space X is defined as

Remark 5.1.2. A few comments:• Rips complexes are sometimes also called Vietoris-Rips complexes.
• Rips(S, r ) represents a combinatorial snapshot of S at scale r.

Diam( A) = max d( x, y).
x,y2 A

Diameter of X is defined as
Diam( X ) = sup d( x, y).
x,y2 X

60

introduction to persistent homology• Diam(s ) is the diameter of s. Condition Diam(s )  r means thatthe distance between any two vertices of s is at most r.
• It is easy to verify that Rips complexes are indeed abstract simplicial
complexes: if s is a simplex then so is each of its subsets.

Figure 5.1: Five points in the plane
and three corresponding Rips complexes Rips(S, r ). Visualisation is assisted by circles of radius r/2 aroundeach point. For much larger scales
the Rips complex is not planar andeventually becomes 4-dimensional.

Remark 5.1.3. Some properties of the Rips complexes:
1. Rips complexes are often the preferred construction in TDA due totheir simplicity in terms of the definition and computation.
2. Rips(S, r ) is an abstract simplicial complex, typically not embeddable in X.
3. For r smaller than the smallest pairwise distance between the points
in S, Rips(S, r ) is a discrete set, i.e., a complex with no edges orhigher-dimensional simplices.
4. For r at least as large as Diam(S), i.e., the largest pairwise distance between the points in S, Rips(S, r ) is the (|S| 1)-simplex,
i.e., the simplicial complex on S containing all subsets of S.

Figure 5.2: Suppose we are given
three points in the plane. These
three points span a triangle in the
Rips complex for r greater or equal
to the maximal pairwise distance
between these points. Equivalently,
the three balls of radius r/2 should
pairwise intersect.

5. If r1  r2 , then Rips(S, r1 ) ✓ Rips(S, r2 ).
Definition 5.1.4. Let X be a metric space and let a sample S ⇢ X
be a finite subset. The Rips filtration on S is the collection of abstract simplicial complexes {Rips(S, r )}r 0 along with inclusions
ir1 ,r2 : Rips(S, r1 ) ,! Rips(S, r2 )for all r1  r2 .

A Rips filtration provides the collection of all Rips complexes on S.
While a single Rips complex depends on the choice of the scale, thefiltration does not. Filtrations will play a fundamental role later in thedefinition of persistent homology.

T More generally, a filtration of asimplicial complex K is a family of
subcomplexes {Kr }r 0 indexed by aparameter r, such that Kr  Kr0 for
all r < r 0 . The Rips filtration ona finite set S is a filtration of the(|S| 1)-simplex.

constructions of simplicial complexes

61

5.2 Čech complexes

Definition 5.2.1. Let X be a metric space and let a sample S ⇢ X
be a finite subset. Choose a scale r 0. The Čech complex Cech(S, r )is an abstract simplicial complex defined by the following rules:
1. The vertex set is S.
2. A subset s ✓ S is a simplex i↵T

T Recall B( x, r ) = {y 2 X | d( x, y) r } is a closed ball.

T Čech complexes are a special casex 2s B ( x, r ) 6 = ∆.

Remark 5.2.2. A few comments about the definition:• Čech complexes are a classical topological construction used in many
contexts throughout topology.

of nerve complexes, a connectionthat will be explained below.

Figure 5.3: Five points in the plane
and three corresponding Čech complexes Cech(S, r ). Visualisation is
assisted by circles of radius r aroundeach point. For much larger scales
the Čech complex is not planar andeventually becomes 4-dimensional.

• Cech(S, r ) represents a combinatorial snapshot of S at scale r.
• It is easy to verify that Čech complexes are indeed abstract simplicial complexes.
Remark 5.2.3. Some properties of the Čech complexes:
1. While harder to compute1 , Čech complexes are attractive due to awell understood geometric interpretation, which will be explained in
the next section within the context of nerve complexes.
2. Cech(S, r ) is an abstract simplicial complex, typically not embeddable in X, although it is often homotopy equivalent2 to a subset ofX.
3. For r smaller than one half of the smallest pairwise distance between the points in S, Cech(S, r ) is a discrete set.
4. For r at least as large as twice the largest pairwise distance between
points in S, Cech(S, r ) is the (|S| 1)-simplex.
5. If r1  r2 , then Cech(S, r1 ) ✓ Cech(S, r2 ).

1

See the MiniBall algorithm in the
Appendix.

2

See the nerve theorem below for
details.

62

introduction to persistent homology

6. It is easy to verify3 that Cech(S, r ) ✓ Rips(S, 2r ).
7. It is also easy to see that Rips(S, r ) ✓ Cech(S, r ). A non-trivialpinclusion Rips(S, r 2) ✓ Cech(S, r ) holds in Euclidean spaces by
Jung’s theorem4 .
Definition 5.2.5. Let X be a metric space and let a sample S ⇢ X
be a finite subset. The Čech filtration of S is the collection of abstract simplicial complexes {Cech(S, r )}r 0 along with inclusions
ir1 ,r2 : Cech(S, r1 ) ,! Cech(S, r2 )for all r1  r2 .

As was the case with Rips filtrations, Čech filtrations also provide ascale-free approximation of the underlying set by simplicial complexes.
The smallest example demonstrating a di↵erence between the twoconstructions is given in Figure 5.4.

If balls of radius r intersect thenthe pairwise distances between thecenters are at most 2r.
3

4

Theorem 5.2.4 (Jung’s theorem). If D is the diameter of afinite subset F ⇢ R n , then F iscontained
q in a ball of radius at
most D 2(nn+1) .
For X = (R n , d2 ) we actually obtain!
r2( n + 1)
Rips S, r✓ Cech(S, r ).
np
The factor r 2 in 7 of Remark
5.2.3 is thus only the smallest upperbound that holds for all n.
Figure 5.4: Suppose we are given
three points in the plane. These
three points span a triangle in theČech complex i↵ the three balls ofradius r intersect. The left complex
consisting of three edges and no
triangle does not appear as a Rips
complex of any triple of points.

5.3 Nerve complexesČech complexes are a special case of a classical topological construction called the nerve.
Definition 5.3.1. For k 2 N let U = {U1 , U2 , . . . , Uk } be a collection of subsets of X. The nerve of U is the abstract simplicial complex N (U ) defined by the following rules:
1. The vertex set is U = {U1 , U2 , . . . , Uk }, consisting of k elements.
2. A subset s ✓ U is a simplex i↵Ti 2s Ui 6 = ∆.

A Čech complex is the nerve of the corresponding collection of rballs, i.e., Cech(S, r ) = N ({ B(s, r )}s2S ). Another example is theDelaunay triangulation, which is the nerve of the Voronoi diagram.

constructions of simplicial complexesU

N (U )

63

Figure 5.5: An example of a nerve.

Figure 5.6: Two examples of Čechcomplexes: balls and the corresponding complex superimposed (left),complex only (center) and the unionof balls homotopy equivalent to thecomplex by the nerve theorem(right).

64

introduction to persistent homology

One of the main advantages of nerve complexes is that their homotopy type in some cases represents the union of the elements of U .
This is formalized within the context of the nerve theorem, of which
we now state a special case.
Theorem 5.3.2. [Nerve theorem] Let n 2 N and assume a collection U = {U1 , U2 , . . . , Uk } consists of closed convex subsets of R n .
Then U1 [ U2 [ . . . [ Uk ' N (U ).
An idea of a proof is given in Appendix. The nerve theorem does
not hold for an arbitrary collection of subsets as Figure 5.5 demonstrates.
For Delaunay triangulations the nerve theorem provides no additional information. As the Voronoi cells are convex, the nerve theorem
implies that the Delaunay triangulation is contractible, a fact we already know as it triangulates a convex (hence contractible by Lemma
5.3.3) planar region.
On the other hand, the nerve theorem provides a homotopical description of the Čech complex. As Euclidean balls are convex, we
obtain[Cech(S, r ) = N ({ B(s, r )}s2S ) 'B(s, r ),T Nerve theorem actually holds

much more generally. For example,
assume each finite intersection ofsets of U (including each member
U 2 U , since it appears as theintersection of {U } ✓ U ) is either
empty or contractible. If U is a finite
collection of closed subsets in R n , oran arbitrary collection of open sets
in a metric space, then[U 2UThis is a stronger statement than
Theorem 5.3.2 by Lemma 5.3.3.
Lemma 5.3.3. Let n 2 N. Eachconvex subset of R n is contractible.
Proof. Assume A ⇢ R n is convex
and fix x0 2 A. We can slide each a 2
A into x0 along the line segment froma to x0 . This results in a homotopy
H ( a, t) = (1 t) a + t x0 between
the identity map on A and theconstant map at x0 , hence A iscontractible.

x0

s2S

i.e., the Čech complex Cech(S, r ) has the homotopy type of the rneighborhood of S. This fact is the foremost reason for the use of Čechcomplexes: while they are harder to compute than Rips complexes, weknow that the obtained homotopy type represents the r-neighborhoodof S. In this spirit we can interpret Figure 5.6. Furthermore, thisobservation can be used to prove reconstruction results: given a closed
connected surface X in an Euclidean space, for each sufficiently small
scale parameter r
0 and for each sufficiently dense finite subset
S ⇢ X we have X ' Cech(S, r ), i.e., the homotopy type of a space X
can be reconstructed using Čech complexes (in Euclidean or geodesic
metric)5 . In the Euclidean metric this holds as6 X ' N ( X, r ) for smallr.

Alpha complexes
Alpha complexes are a fusion between planar Čech complexes andDelaunay triangulations.

U ' N (U ).

Figure 5.7: A sketch of Lemma 5.3.3.

5The same result holds for moregeneral spaces and under appropriate
conditions also for Rips complexes.
However, almost all of the proofs are
based on the application of the nerve
theorem to Čech complexes.
6
Imagine a circle in the plane, a knotin R3 or a surface in R3 : its small
thickening is homotopy equivalent tothe space itself.

constructions of simplicial complexes

65

Definition 5.3.4. Let r
0 and assume S ⇢ R n is a finite collection of points satisfying a general position property: no n + 2 pointsof S lie on the same (n 1)-sphere. For each s 2 S let Vs denotethe corresponding Voronoi⇣cell. The alpha ⌘complex of S at scale r
is the following nerve: N {Vs \ B(s, r )}s2S .
Assume S is as in Definition 5.3.4. While Cech(S, r ) may be of arbitrarily high dimension7 , the nerve theorem guarantees it is homotopy
equivalent to a subset of R n . The alpha complex of S at scale r is acomplex8 , that is homotopy equivalent to Cech(S, r ). To see this notethat by the nerve theorem9 both are homotopy equivalent to[s2S

B(s, r ) =[s2S

Vs \ B(s, r ) .

Thus alpha complexes may be seen as an efficient way of obtaining
the homotopy type of a Čech complex in R n .
Another way of thinking of alpha complexes is as a model formolecules. Each atom in a molecule has a radius10 and touches (rather
than intersects) other atoms within the range.

7

This implies it may be computationally inefficient.
8Note that it is a subcomplex of theDelaunay triangulation on S. It isplanar if S is.
9
Sets Vs \ B(s, r ) are intersections ofclosed convex sets thus closed andconvex themselves.
S

s2S B ( s, r ) =S

s2S Vs \B(s, r ) , take any x 2 s2S B(s, r ) andnote that if s 2 S is a closest point tox in S, then x 2 Vs \ B( x, r ).

T To seeS

10

Assume all the radii are the same.
For di↵erent radii there is a wellstudied concept of a weighted alphacomplex.
Figure 5.8: Alpha complexes corresponding to the situation in Figure5.6. Note that the alpha complexes
are smaller (or equal) yet still homotopy equivalent to the correspondingČech complexes. For larger r theČech complexes become higherdimensional while the alpha complexes of planar subsets maintain thedimensionality bound 2.
A decomposition into regionsof the form Vs \ B(s, r ) (on theleft) mimics the decomposition ofmolecules into atoms.

MapperAnother example of a construction based on the idea of the nerve is
Mapper. In contrast to the constructions above it is typically11 a onedimensional simplicial complex, i.e., a graph. Mapper can be thoughtof as a one-dimensional sketch of a space X as detected through thelens of a single map on X.
We first describe a theoretical setup. Assume:

11

By the definition that will be
provided, a Mapper is a simplicial
complex of arbitrary dimension.
However, our discussion and examples will focus on one-dimensionalcase, as do the practical applications
in which Mapper is used.

66

introduction to persistent homology• X is a metric space;• f : X ! [0, 1] is a (continuous) map12 ;12• U is a collection13 of subsets of [0, 1], whose union is [0, 1].
Definition 5.3.5. For each U let VU denote the collection of all comS
ponents of f 1 (U ) and define V = U 2U VU as the collection of all
subsets of X that appear as a component of a preimage f 1 (U ) forsome U 2 U . Mapper is defined as M( X, f , U ) = N (V ).

We restrict ourselves to the cases
when the target space is [0, 1]. However, there is no theoretical reason
for doing so and the construction iswell defined even if we replace [0, 1]
by some more complicated space.
13
Typically we restrict to cases
when no three subsets of U intersect.
In such cases Mapper is a onedimensional simplicial complex.

An example is provided by Figure 5.9.
f
X

1

(U1 )

U1
f

1

M apper

(U2 )

f
U2

f

1

(U3 )

U3

In practice data is often given as a finite set of points along withcertain measurements. For example, we may have a collection of patients along with their heart rate and blood pressure, or a collectionof basketball players along with their statistics, etc. In this case a
modified practical setup comes into play. Assume:

Figure 5.9: The construction ofMapper: a space (the torus on theleft), a continuous map (projection
f onto the vertical axis), coverU = {U1 , U2 , U3 } of the interval [0, 1],
a decomposition of the preimages
into the four components and theresulting graph (right).

• X is a finite set;
• f : X ! I is a map (measurement);• U is a chosen partition of [0, 1] into intervals, typically of fixed
length # > 0.
• P is a chosen clustering scheme14 on X.

14

This step possibly includes additional choices of clustering algorithms.

constructions of simplicial complexes

67

Definition 5.3.6. For each U 2 U let VU denote the collection of all
Sclusters of f 1 (U ) with respect to P and define V = U 2U VU as
the collection of all subsets of X that appear as a cluster of a preimage f 1 (U ) for some U 2 U . Mapper is defined as M( X, f , U ) =N (V ).
f

1

(U1 )

U1
M apper
f

1

(U2 )

f
U2

While a point cloud and a number of measurements on it are often
given, one has to construct a single function f and a partition U ,and choose other parameters very carefully to extract the desired
information. Mapper is usually not analyzed further with topologicaltools but rather visualized, which is why the one-dimensionality ispreferable.

Figure 5.10: The construction of a
Mapper when X is a point cloud.

5.4 Interleaving properties
Given a finite subset of a metric space we have described how to
associate various complexes with that set. If we think for a moment
about finite abstract complexes we note that these objects are discrete:
it would be hard to define an obvious distance between abstract simplicial complexes. On the other hand, we have a continuous selectionof inputs and input parameters: scale r is typically positive and there
are reasonable notions of a distance between finite subsets of a metricspace. As a result any assignment of a single complex is bound to havediscontinuities15 (instabilities) of some sort, see Example 5.4.1 for ademonstration.
However, it turns out we can define a distance on filtrations, for

Example 5.4.1. Let X = {0, 1} ⇢R. Note that Rips( X, r ) changes
discontinuously at r = 1: while
Rips( X, 1) is a single edge (alongwith the two boundary points), foreach r < 1 the complex Rips( X, 1)
consists of only two vertices.
15

Unless it assigns a constant complex, of course.

68

introduction to persistent homology

which the assignment of a filtration16 becomes a continuous functionof the input set and the scale parameter. 17
Definition 5.4.2. Choose # > 0. Filtrations { Ar }r 0 and { Br }r 0
(obtained by the Rips or the Čech construction) are #-interleaved if
there exist simplicial maps jr : Ar ! Br+# and yr : Br ! Ar+# suchthat jr+# yr : Br ! Br+2# and yr+# jr : Ar ! Ar+2# are equal tothe corresponding inclusions.
Maps of Definition 5.4.2 can be visualised by drawing the following
commutative18 “ladder” diagram.

···

/ Ar

···

/ Br

jr
yr

/ Ar + #=/ Ar+2#;!
/ Br+##/ Br+2#/ ···16

Of course, this eliminates thedependency on the scale parameter r.
17
For the sake of simplicity we will
restrict ourselves to the mentioned
Rips and Čech filtrations although
the concept can be defined moregenerally.

18

Adjective “commutative” refers to
the fact that all maps commute, i.e.,
going from one complex to anotherthrough any viable sequence of maps
gives the same result.

/ ···Definition 5.4.3. Given two filtrations their interleaving distanceis defined as the infimum of all values # > 0, for which the filtrations are #-interleaved.
It turns out that in our context (Rips and Čech filtrations on finite
collections of points) the interleaving distance is a metric on the set offiltrations. In contrast, recall that there seems to be no geometricallymeaningful metric on the set of single finite simplicial complexes.
The concept of interleaving will play an important role later in thecontext of the stability of persistent homology. At this point we canuse it to phrase two proximity results.

Example 5.4.4. Let X = {0, 1} ⇢ R
and Y = {0.1, 1.2} ⇢ R. Rips( X, r )consists of:• two points for r < 1;
• one edge for r

1.

Rips(Y, r ) consists of:• two points for r < 1.1;
• one edge for r

1.1.

The filtrations are 0.1-interleaved.

Theorem 5.4.5 (Stability with respect to spaces). Choose # > 0 andassume X = { x1 , x2 , . . . , xk } and Y = {y1 , y2 , . . . , yk } with d( xi , yi ) #, 8i, i.e., X and Y each consist of k points, such that the corresponding distances are at most #. Then:• The Rips filtrations of X and Y are 2#-interleaved.
• The Čech filtrations of X and Y are #-interleaved.
Proof. It follows directly from the triangle inequality (see Figure 5.11)
that if a subset s ⇢ X is of diameter r, then the corresponding19
subset t ⇢ Y is of diameter at most r + 2#. Hence if s is a simplex
in Rips( X, r ), then t is a simplex in Rips(Y, r + 2#). Consequently we
may deduce that:• maps Rips( X, r ) ! Rips(Y, r + 2#) defined by xi 7! yi are simplicial;

19

Subset t is formed by taking thepoints of Y with the same indices asappear in the points of s.

constructions of simplicial complexes

69• maps Rips(Y, r ) ! Rips( X, r + 2#) defined by yi 7! xi are simplicial;• as the above two maps obviously commute with the inclusions weconclude that the Rips filtrations of X and Y are 2#-interleaved;• in a similar fashion we may conclude that the Čech filtrations of X
and Y are #-interleaved.

y1

y2

"

"

r
x1

x2

Figure 5.11: If d( x1 , x2 )  r andd( xi , yi )  # then it is apparent thatd(y1 , y2 )  r + 2#.

These conclusions tell us that if we perturb our point set slightly,
the resulting filtration does not change much in terms of the interleaving distance, i.e., the construction of a filtration is stable. In asimilar fashion we can express the relationship between Rips and Čechfiltrations.

Rips-Čech correlation
Recall that Cech(S, r ) ✓ Rips(S, 2r ) and Rips(S, r ) ✓ Cech(S, r ) ✓Cech(S, 2r ). This implies that the Rips and Čech filtrations, whenconstructed with logarithmic scales20 , are (log 2)-interleaved, i.e.,{Rips(S, er )}r 0

and{Cech(S, er )}r 0

are (log 2)-interleaved.

Note that Cech(S, er ) ✓Rips(S, 2er ) = Rips(S, elog 2+r ) andsimilarly Rips(S, er ) ✓ Cech(S, er ) ✓Cech(S, elog 2+r ). These inclusions are
the interleaving maps.
20

5.5 Concluding remarks
Recap (highlights) of this chapter• Complexes: Rips, Čech, nerve, alpha, Mapper;• Nerve theorem;• Interleaving;

Background and applications
Constructions of simplicial complexes greatly depend on the intended use. Rips and Čech complexes along with the nerve construction are relatively well understood and have been originally introduced
for theoretical purposes in the first half of the twentieth century. Čechcomplexes and particularly nerves were instrumental in developmentof Čech homology and cohomology theories, which later led to shape
theory. Rips complexes have seen various independent introductions,
including in geometric group theory. Their use has recently been extended to the applied setting. They are the complexes most exposedto the curse of dimensionality. Alpha complexes arose decades later

T The curse of dimensionality: an

inconvenient fact that the number ofsimplices typically grows fast with thedimension of a simplicial complex.
This presents challenges for their
computational applications, which
are partially addressed by alternative
constructions of complexes.

70

introduction to persistent homology

within the realm of computational geometry and are intended for computationally intense applications. For more details and references onhistorical background on these complexes see a textbook21 . There isalso a modern perspective22 on nerve theorem, Dowker duality, andconnections to Rips complexes. Mapper is a more recent construction23 . It is often thought of as a low-dimensional projection method
and has turned out to be a commercial success.
At about the same time the interleaving distance emerged24 as ameasure of stability of filtrations and persistent homology, although
equivalent concepts have been known in pure topology for a long time.

Appendix: the MiniBall algorithm
Given a finite subset s ⇢ X ⇢ R n the MiniBall algorithm25 isa recursive algorithm that returns the miniball of s, i.e., the minimal26 ball in R n containing s. As such, the algorithm provides acomputational verification of the containment of s in a Čech complex: s 2 Cech( X, r ) i↵27 the radius of the miniball is at most r.
As the radius of the ball is also provided, the algorithm actually provides the exact lower bound for the scales r at which s is a simplex in
Cech( X, r ), hence a single execution of the algorithm suffices for theentire filtration.
• Input: disjoint finite sets t, n ⇢ R n .
• Output: the minimal ball with:– t in the ball;– n on the boundary of the ball.

Algorithm 1: Miniball(t, n).
if t = ∆ thencompute miniball B directly;
elsechoose u 2 t;
B =miniball(t {u}, n);
if u 2/ B thenB =miniball(t {u}, n [ {u});21Herbert Edelsbrunner and John
Harer. Computational Topology: An
Introduction. Applied Mathematics.
American Mathematical Society,2010. doi: 10.1090/mbk/069
22Žiga Virk. Rips complexes as
nerves and a functorial Dowker-nerve
diagram. Mediterranean Journal of
Mathematics, 18(2):58, 2021. doi:
10.1007/s00009-021-01699-4
23
Gurjeet Singh, Facundo Memoli,
and Gunnar Carlsson. Topological
Methods for the Analysis of High Dimensional Data Sets and 3D Object
Recognition. In M. Botsch, R. Pajarola, B. Chen, and M. Zwicker,
editors, Eurographics Symposium
on Point-Based Graphics. The Eurographics Association, 2007. doi:
10.2312/SPBG/SPBG07/091-100
24
Steve Y. Oudot. Persistence Theory: From Quiver Representations toData Analysis. Number 209 in Mathematical Surveys and Monographs.
American Mathematical Society,2015. doi: 10.1090/surv/209
25
Emo Welzl. Smallest enclosing disks (balls and ellipsoids). In
Hermann Maurer, editor, New Results and New Trends in Computer
Science, pages 359–370, Berlin,
Heidelberg, 1991. Springer Berlin
Heidelberg
26
Minimality is considered withrespect to the radius. Such a ball is
unique.
T27
z 2 x2s B( x, r ) , s ⇢ B(z, r ).
" Given random finite t, n ⇢R n , there typically exists no ballcontaining t and having n on theboundary. The algorithm is designed
so that only the pairs (t, n), forwhich this condition is satisfied arecalled.

return B

The algorithm is initiated by calling Miniball(s, ∆) 28 and terminates with the miniball B when t = ∆. It inductively scans throughthe points of t. At each step it either removes a point (if removing itfrom the set does not change the miniball of the set) or puts a point

28

I.e., t = s, n = ∆constructions of simplicial complexes

71

into n (if removing the point decreases the miniball). When t = ∆ theset n consists of at most n + 1 points that lie on the boundary of theminiball of s and determine it. In this case we can use the standard
circumsphere and circumradius formulas in terms of determinants to
get the miniball.

Appendix: a sketch of a proof of the nerve theorem 5.3.2
A special case of the proof is illustrated by Figures 5.12, 5.13, and
5.14.
Proof. For the sake of simplicity29 let us assume the nerve is of dimension 1, i.e., all triple intersections of sets of U are empty. Define
X = U1 [ U2 [ . . . [ Uk and Z ⇢ X ⇥ N (U ) as:
⌘
[ ⇣\
Z=
Us ⇥ s .
s 2N (U )A complete general proof is much
more technical but broadly follows
the same steps as are presented here.

s2s

We will prove that Z ' X and Z ' N (U ).
In order to prove Z ' X note that for each x 2 X the section({ x } ⇥ N (U )) \ Z is a simplex30 in the nerve spanned by all s 2 S,
for which x 2 Us . Contracting each such simplex to a point in asynchronized manner for each x 2 X we obtain a deformation of Z to
X, hence Z ' X.
In order to prove Z ' N (U ) note that for each y 2 N (U ) the
section ( X ⇥ {y}) \ Z is a contractible set by assumptions. Contract
first the sections of this form for all non-vertices y, and then conclude
by contracting all the sections for vertices. We obtain a deformation of
Z to N (U ), hence Z ' N (U ).

U

29

N (U )

Z

30

In our case, either an edge or avertex.

Figure 5.12: A collection U of subsets
of a circle X = S1 (left), the corresponding nerve (center) and space
Z constructed in the proof (right).
The sets of U are illustrated as subsets of the plane for greater clarity,
while formally U consists of theirintersections with X.

72

introduction to persistent homology

Z

Z

S1

N (U )

Figure 5.13: Proving Z ' S1 wecontract the sections above points ofx 2 X in Z corresponding to edges
in the nerve complex (contract along
the indicated arrows on the left) toobtain S1 .

Figure 5.14: Proving Z ' N (U )we first contract the sections above
non-vertices y 2 N (U ) in Z (contract
along the indicated arrows on theleft) to obtain the space in thecenter. Conclude by contracting thesections above vertices y 2 N (U )in Z (contract along the indicateddashed arrows in the center) toobtain N (U ) on the right.

constructions of simplicial complexes

73

Appendix: Dowker duality
Nerve complexes are natural complexes arising from a collectionof subsets. There is another similar construction, called the Vietoriscomplexes, that is in a way dual to the nerve construction.
Definition 5.5.1. For k 2 N let U = {U1 , U2 , . . . , Uk } be a collection of subsets of a finite space X, whose union is X. The Vietoriscomplex of U is the abstract simplicial complex V (U ) defined by thefollowing rules:
1. The vertex set is X.
2. A set s ✓ X is a simplex i↵ there exists U 2 U containing s.
We see that maximal simplices of V (U ) are determined by (inclusionwise) maximal sets of U . There is a surprising connection31 between
the nerve complexes and Vietoris complexes.
T Dowker duality actually holds for

Theorem 5.5.2 (Dowker duality). For k 2 N let U = {U1 , U2 , . . . , Uk }be a collection of subsets of a finite space X, whose union is X. Then
N (U ) ' V (U ).
Proof. Consider V (U ) as a subspace of a Euclidean space. Each Ui
determines a simplex DUi spanned by all points of Ui . Note that
{DU1 , DU2 , . . . , DU2 } are closed convex sets whose union is V (U ). By
the nerve theorem 5.3.2 V (U ) is homotopy equivalent to the nerve of{DU1 , DU2 , . . . , DU2 }. This nerve, on the other hand, is actually N (U )via the correspondence DUi 7! Ui , see Figure 5.15 for a visual sketch of
the proof.

U

V(U )

N (U )

While Čech complexes are nerves associated to a collection of ballsof radius r, Rips complexes are Vietoris complexes associated to acollection of sets of diameter at most r.

an arbitrary collection of subsets Uof an arbitrary set X, no additional
structure is necessary. Even in suchgenerality it can be proved with easeusing a general form of the nerve
theorem.
31
Cli↵ord H. Dowker. Homology
groups of relations. Annals of
Mathematics, 56(1):84–95, 1952. doi:
10.2307/1969768

Figure 5.15: A cover of six points byfour colored sets (left), its Vietoriscomplex (center) and nerve (right).
Colored simplices on the central
picture provide a collection of subsets
satisfying the conditions of the nerve
theorem. On the other hand, theirnerve is actually N (U ) on the right.
By the nerve theorem we conclude
the Dowker duality.

6
Fields and vector spaces

The material presented up to this point mostly falls into thepremise of geometric topology and combinatorics: we introduced metric spaces and their combinatorial descriptions, simplicial complexes.
Our eventual goal however is to compute meaningful topological invariants from these combinatorial descriptions. Within mathematics
the field dealing with operations is called algebra and our milestone on
the path to computational implementation is an algebraic formulation
based on simplicial complexes. With that intention in mind we firstreview and introduce some algebraic concepts.
In this lecture we will present fields and vector spaces. Specificcases of the first two notions are probably familiar to the reader: real
numbers and vectors in Euclidean space. We will introduce a few
more fields and vector space constructions, which will provide us withenough structure to introduce homology in the next chapter.

6.1 Fields
Within the context of algebra, a field is a set with two operations
satisfying a number of properties. For our purposes we will deflect aformal introduction and rather introduce specific fields which will be of
our interest.
We will think of a field as our number system. We will want tobe able to add, subtract, multiply and divide (except by zero) in ourfield. The fields a reader is most familiar1 with are probably Q, R, andC. However, there is also a family of finite fields (consisting of finitely
many numbers) which often provides convenient examples: fields ofremainders.

1

N is not a field as it does notcontain all results of subtractions,for example, 3 5 2/ N. Z is is
not a field as it does not contain all
quotients by non-zero numbers, forexample 3/5 2
/ Z.

76

introduction to persistent homology

The fields of remainders Z p
Definition 6.1.1. Let p 2 {2, 3, 5, . . .} be a prime number. Define:(a) pZ = { p · n | n 2 Z } = {. . . ,2p,p, 0, p, 2p, 3p, . . .};(b) Z p = Z/( pZ ) as the quotient consisting of remainders when
dividing by p.
Let us discuss (b) 2 in detail. The quotient Z/( pZ ) consists ofclasses, each of which can be represented by a number from the “numerator” Z. If a 2 Z then the corresponding class is represented by[ a]. Two such numbers represent the same class in the quotient i↵their di↵erence is3 in the “denominator” Z p . To phrase it di↵erently4 ,[ a] = [b],ba 2 pZ.

Example 6.1.2. Let p = 5. In Z5 two numbers represent the same classi↵ their di↵erence is divisible by 5. Classes [0], [1], [2], [3], [4] are all
distinct but5 :• [5] = [0] as 5

0 = 1 · 5.

• [6] = [1] as 6

1 = 1 · 5.

• [ 1] = [4] as1• [126] = [1] as 126

4=2I.e., the fields of remainders and thequotient construction that defines it.

3I.e., i↵ their di↵erence is a multipleof p. In particular this means [ a] =[b] if the remainder after dividing by
p is the same for both a and b.
4

What we just described is a general
construction of an algebraic quotient
structure. We will come across it
again in the context of vector spaces.
5

A few more examples in Z5 :[ 4] = [1], [8] = [3], [17] = [2],[1346134523451] = [1], [3457] = [2],[ 23513252] = [3].

1 · 5.

1 = 25 · 5.

In particular, two positive numbers represent the same class i↵ theirremainder when dividing by 5 is the same.
We draw another conclusion from Example 6.1.2: the most convenient representation6 of Z p is given by p classes [0], [1], . . . , [ p 1].
These classes are all distinct7 and together form Z p .
Example 6.1.3. The structure of Z2 encodes parity: for a 2 Z we
observe that [ a] = 0 i↵ a is even, and [ a] = 1 i↵ a is odd.

T A few examples in Z7 : [8] = [1] =[15], [ 5] = [2] = [72].

6

We will actually be using thisrepresentation almost exclusively
from now on.
7
They form all possible remainders
after division by p.

Defining addition, subtraction and multiplication in Z p These threeoperations are defined in the obvious way:

[ a ] + [ b ] = [ a + b ],[ a][b] = [ ab]and[ a ] · [ b ] = [ a · b ].

It turns out that the operations are well defined8 in the following
sense:[ a] = [ a0 ], [b] = [b0 ] =) [ a + b] = [ a0 + b0 ]and the same holds for subtraction and multiplication.

8

A proof that addition is well defined:[ a] = [ a0 ], [b] = [b0 ] =)9k a , k b 2 Z : a0 = a + k a p, b0 = b + k b p.

Thus

[ a0 + b0 ] = [ a + k a p + b + k b p] == [( a + b) + (k a + k b ) p] = [ a + b].

fields and vector spaces

77

Example 6.1.4. Addition:
In Z5 : [3] + [4] = [2], [3] [4] = [4], [1] + [2] = [3].
In Z7 : [3] + [4] = [0], [3] [4] = [6], [1] + [2] = [3].
Multiplication:
In Z5 : [3] · [4] = [2], [2] · [4] = [3], [2] · [3] = [1].
In Z7 : [3] · [4] = [5], [2] · [4] = [1], [2] · [3] = [6].
Example 6.1.5. Note that in Z2 = {[0], [1]} we have [ a] = [ a] hence
addition is the same as subtraction. In fact, if we identify [0] and [1]with their Boolean values, addition and multiplication encode9 logicaloperations “Exclusive or” (XOR) and “Conjunction” (AND):[ a] + [b] = [ a XOR b],[ a] · [b] = [ a AND b].

Defining division in Z p
Up to this point the described structure of Z p did not require p to
be prime. This assumption, however, is required10 if we want to define
division . From number theory we know that if p is a prime, then foreach number a 2 Z with [ a] 6= [0], the classes [ a], [2a], . . . [ pa] = [0]represent the entire Z p . In particular, we can choose a coefficient k
representing [1] = [kp] and define the inverse of p by [ p] 1 = [k]. We
can consequently define the division by[ a]/[b] = [ a] · [b] 1 ,9Which means, amongst others, thatthese operations in Z2 are fairly natural in computer implementations,exact, and fast. In fact, computations in topological data analysis areoften performed using Z2 .

If q is not a prime then Z q contains divisors of zero, i.e., non-zeroclasses, whose product is the zeroclass. For example, [2] 2 Z4 is a nonzero class, but [2] · [2] = [4] = [0] 2 Z4
is the zero class. If we wanted tofind an inverse of [2] in Z4 we would
need to find an integer k 2 Z, sothat [2k ] = [1] 2 Z4 , an unattainablefeat as 2k is always even. A divisor ofzero has no inverse.
10which turns out to be well defined if p is prime and [b] 6= [0].
Example 6.1.6. In Z5 we have [1 · 3] = [3], [2 · 3] = [1], [3 · 3] =[4], [4 · 3] = [2], [5 · 3] = [0 · 3] = [0]. The products [k · 3] exhaust entire
Z5 and [3] 1 = [2]. Similarly, [2] 1 = [3].
In Z7 we have [2] 1 = [4], [3] 1 = [5], ...

1

2

3

0

We are now able to add, subtract, multiply and divide (except byzero) in Z p , which makes Z p a field.
Remark 6.1.7. Counting and computing in Z p is surprisingly common
in everyday life. It appears whenever we have a periodic behaviour.
• Z2 is a model for true/false in logic, odd/even numbers, and binary
numbers.
• We use Z4 when thinking about seasons of the year.
• We use Z7 when thinking about days of the week (if today is the ath
day of the week then b days from today it will be [ a + b]th day of theweek).
• We use Z10 whenever we are computing in decimal numbers. Givena, b 2 N, the first digit of a + b equals [ a + b] in Z10 and the samegoes for multiplication.

Figure 6.1: Quotient Z p models
rotations by 2p/p. Adding p suchrotations we arrive at the original situation 0 2 Z p . The Figure represents
Z4 . Given any situation the additionof 1 is represented by a rotation by
p/2 in the positive direction.

78

introduction to persistent homology• We use Z10 whenever we are converting units in the metric system11 .
• We use Z24 when thinking about hours in a day. When thinking
about hours coupled with the am/pm prefixes we actually do a combination of Z2 and Z12 .

11

As the reader might imagine,
there is no reasonable algebraic
explanation for the imperial system.

• We use Z60 when thinking about minutes and seconds.
As a summary let us recall all the fields we mentioned: Q, R, C,
and Z p for any prime number p. These are the only fields we will beconsidering.

6.2 Vector spaces
Let F be a field. A prototype of a vector space over field R a reader
is familiar with is R n for any n 2 N. It consists of n-tuples (vectors)of real numbers, which we can add, subtract, and multiply by any
element of our field R. In a similar way F n is a vector field over F:
it consists of n-tuples (vectors) of numbers from F, which we canadd, subtract, and multiply by any element of our field F. While all
our vector spaces will essentially12 be of the form F n , some of ourconstructions will require us to use a more formal definition.
Definition 6.2.1. Let F be a field. A vector space V over field F isa set of elements (vectors) equipped with two operations,
1. addition + : V ⇥ V ! V andsatisfying the following properties:• addition is associative, commutative, contains the identity (zero)
vector 0, and V contains the opposite element of each vector;• scalar multiplication is compatible, distributive, and normalized.
Roughly speaking, if we have a set of vectors we can reasonably
add, subtract, and multiply by elements of some field, then this set
forms a vector space.
Example 6.2.2. Let n 2 N. Given symbols v1 , . . . , vn and a field F, allformal13 sums Âin=1 ai vi where ai 2 F form a vector space. Operations
are defined in the obvious way:
n

n

i =1

i =1

i =1

Â ai vi + Â ai0 vi = Â (ai + ai0 )vi ,andI.e., up to isomorphism, which willdefined later.
T Glossary of algebraic properties

mentioned in Definition 6.2.1:• associativity: (u + v) + w =u + (v + w), 8u, v, w 2 V• commutativity: u + v = v +u, 8u, v 2 V

2. scalar multiplication · : F ⇥ V ! V

n

12

n

n

i =1

i =1

b Â ai vi = Â (bai )vi .

• zero vector: 0 + v = v, 8v 2 V[it should always be clear fromthe context whether 0 denotes anumber in F or the zero vector inV]• the opposite element of v 2 V isdenoted by v 2 V and satisfies
v v=0• compatibility: ( ab)v =a(bv), 8 a, b 2 F, 8v 2 V• distributivity: ( a + b)v = av + bv
and a(v + w) = av + aw, 8 a, b 2
F, 8v, w 2 V• normalization: 1 · v = v, 8v 2 V.
13

A “formal sum” in this settingmeans that vi + v j is not defined asa single element vk (as a result of asummation) in a vector space, but
is rather thought of as an abstractelement in itself. For example, if
we want to shop for an apple anda pear, our shopping list should be
apple + pear, which does not equal
any other single fruit.

fields and vector spaces

When F = Z2 the corresponding vector space models the power setof v1 , . . . , vn . A subset {vi+1 , . . . , vik } corresponds do vi+1 + . . . + vik .
The sum of two formal sums in this setting models the symmetricdi↵erence14 between the corresponding sets.
For a prime number p and n 2 N the vector space (Z p )n = Z np isa finite vector space consisting of pn elements. While this vector space
appears di↵erent from R n , the formal theory, concepts, and proofs are
the same in both cases. We next recast the familiar notions from R n
in the setting of vector spaces over F.
Let V, W be a vector space over field F.
1. A linear combination of vectors in V is any expression of the formk

Â ai vi ,i =1

ai 2 F, vi 2 V

2. A set of vectors {v1 , v2 , . . . , vk } ⇢ V is linearly independent15 if the
only coefficients ai 2 F satisfying Âik=1 ai vi = 0 2 V are the zerocoefficients, i.e., ai = 0, 8i.
3. A basis of V is a maximal16 linearly independent set in V. A avector space typically has many di↵erent bases. However, if V isfinite dimensional17 , then the cardinality of each basis is the same.
This number is called the dimension of V.

79

14

The symmetric di↵erence of sets
A, B equals A [ B \ A \ B.

T Let X be a metric space andm, n 2 N. The following are vector
spaces over F: the set of all m ⇥ nmatrices with entries in F, the set of
all functions X ! F, the set of allcontinuous functions X ! F, the setof all di↵erentiable functions X ! F
if F 2 {Q, R, C }, ... Operations
on functions in these examples aredefined pointwise.

For example, vectors (1, 3) and(2, 1) are linearly independent in
R2 , Q2 , Z213 , but not in Z25 .
15

In particular, each element of V
can be expressed uniquely as a linear
combination of the basis vectors.
17
I.e., if it admits a finite basis.
16

4. A subset U ✓ V is a vector subspace [notation: U  V] of V if it isitself a vector space over F.
5. A map f : V ! W is linear if it is additive18 and multiplicative19 .
A linear map is completely determined20 by the images of its basis.

18
19
20

6. A bijective linear map is called an isomorphism [notation: ⇠=].
Every vector space over F of dimension d 2 N is isomorphic to F d .
7. Let f : V ! W be a linear map.
(a) The kernel of f is defined asker( f ) = {v 2 V; f (v) = 0}  V(b) The image of f is defined asIm( f ) = { f (v); v 2 V }  W.
The dimension of Im( f ) is called the rank of f .
(c) Given bases {v1 , v2 , . . . , vk } of V and {w1 , w2 , . . . , wl } of W,
map f may be represented by an l ⇥ k matrix with entries in F.
j
If f (vi ) = Â j=1 ai,j w j , then the entry at ( j, i ) equals ai,j .

f (v + w) = f (v) + f (w), 8v, w 2 Vf ( av) = a f (v), 8 a 2 F, v 2 VConsequently, a linear map canbe represented by a matrix M withcoefficients in F if we chose basesof V and W, with the matrix-vector
product M · v representing f (v).

80

introduction to persistent homology

8. Given a matrix with coefficients in F, we can still preform Gauss
reduction to, for example, compute the rank of a linear map, solve
systems of linear equations, ... The procedure is the same as in R n .
9. Given U  V, the quotient V/U is defined as the vector space overF consisting of classes [v] for v 2 V under the following identification21 :[u] = [v] , u

v 2 U.

In particular, [v] = [0] i↵ v 2 U.
Our forthcoming descriptions of holes in simplicial complexes will
be expressed in terms of dimensions and bases of vector spaces, forwhich the following proposition will turn out to be very handy.
Proposition 6.2.3. Assume U, V, W are vector spaces over a field F.
1. Let f : V ! W be a linear map. Then Im( f ) ⇠= V/ ker( f ).
2. Let U  V be a subspace. Then dim(V/U ) = dim(U )The operations of addition [u] +[v] = [u + v] and multiplication bya scalar a[u] = [ au] for a 2 F, u, v 2
V are well defined by the same
argument that was provided in theprevious section for the fields.
21

TTake the following system in Z5 :(1)

2x + 3y = 2

(2)

3x

Multiply (1) by 2

(3)

x + 4y = 1.

To match with the leading coefficientof (2) multiply (3) by 3 to obtaindim(V ).

(4)

3x + 2y = 3.

Now subtract (4)Proof. 1. Consider the map g : V/ ker( f ) ! Im( f ) defined by [u] 7!
f (u). The map is:• well defined because [u] = [v] =) u v 2 ker( f ) =) f (u0 =) f (u) = f (v) =) g([u]) = g([v]);y = 1.
1 = 3 to obtain(2) to obtain

3y = 2and thus y = 4 and x = 2.

v) =• surjective by the definitions of Im f and g;• injective as g([u]) = f (u) = 0 implies u 2 ker( f ) and thus
[ u ] = [0].
We conclude that g is an isomorphism.
2. Let {w1 , . . . , wk } be a basis22 if U. Complete it by a set B1 ={v1 , . . . , vl } to a basis23 of V. Observe that B2 = {[v1 ], . . . , [vl ]} is abasis24 of U/V:
• B2 is linearly independent: if a linear combination of B2 was thezero vector in V/U then the corresponding combination of theelements of B1 was in U. This can only happen if the later combination equals 0 by the choice of B1 and thus all the coefficientsequal 0 by the linear independence of B1• B2 spans the whole V/U because25 B1 and U span the whole V.

22

This implies dim(U ) = k.

23This implies dim(V ) = k + l.

This implies dim(V ) = l and thus
proves our claim. Furthermore, it
demonstrates a way to obtain a basisof U/V.
24

Take any v 2 V and express it asv = v0 + v00 , where v0 2 U and v00
is a linear combination of B1 . Then[v] = [v00 ].
25

fields and vector spaces

81

6.3 Concluding remarks
Recap (highlights) of this chapter• fields, vector spaces• quotients and dimension

Background and applications
Fields and abstract vector spaces have a long presence in mathematics going back centuries. Finite fields are attractive for computational implementation due to their simplicity. Computations in them
are typically faster than in real numbers. Furthermore they are resistant to some numerical issues present in reals and floating pointcomputations. On the other hand, there is a potential issue of overflowing with computer stored numbers, say integers. Algebraically itstems from the fact that counting with integers in a computer is typically performed in Z p where log2 p is the number of bits assigned
to a variable. Some of related issues are described in a non-technicalbook26 , including an interesting rule in Swiss train regulations27 .
Algebraic predecessor of fields and vector spaces are (algebraic)groups, another classical subject of algebra, which is now present in
virtually every corner of mathematics. For further algebraic background see a textbook28 .
Homology of the forthcoming section is typically introduced throughgroups in the theoretical setting, while in practice fields are used almost exclusively. A short recap of groups is given in the appendix.
Persistent homology, on the other hand, is almost exclusively introduced through coefficients in a field and the resulting persistence
modules due to the accessible description in terms of a persistence
diagram.

26

Matt Parker. Humble pi: a comedy
of maths errors. Allen Lane, 201927
Apparently trains in Switzerland
are not allowed to have an e↵ective
total number of axles equal to 256.
Note that 256 = 28 .
28David S. Dummit and Richard M.
Foote. Abstract algebra. Wiley, 3rdedition, 2004

Appendix: A very short introduction to Abelian groups
T The term ”Abelian” refers to

Definition 6.3.1. An Abelian group ( G, +) is a set G with an associative commutative operation + : G ⇥ G ! G, such that:
1. there exists the zero element 0 2 G satisfying 0 + g = g + 0, 8 g 2
G;
2. for each g 2 G there exists its converseg + ( g) = 0.

g 2 G satisfying

Examples of Abelian groups include (R, +), (C, +), (Q, +), (Z, +),commutativity. If the commutativity condition is not satisfied, thestructure is called a (non-Abelian)group. These include the groups of
permutations (with the operation being the composition) on n elements,
the groups of isometries of a metricspace (with the operation being thecomposition), the group of invertible
matrices (with the operation being
the product), etc.
T We will typically shorten a + ( b)to a b.

82

introduction to persistent homology(Z q , +) for any q, (R \ {0}, ·), (C \ {0}, ·), (Q \ {0}, ·), (Z q \ {0}, ·) for
any prime p, etc.
Many definitions concerning groups are the same as those of fields
and vector spaces.
Definition 6.3.2. Suppose G, H are Abelian groups. A map f : G !
H is a homomorphism if f ( a + b) = a( a) + b(b), 8 a, b 2 A. A
bijective homomorphism is called an isomorphism [notation: ⇠=].
Suppose G, H are Abelian groups and map f : G ! H is a homomorphism.
1. A subset G 0 ✓ G is a subgroup [notation: G 0  G] of G 0 if it isitself a group for the same operation.
2. The kernel of f is defined asker( f ) = { a 2 G; f ( a) = 0}  G.
3. The image of f is defined asIm( f ) = { f ( a); a 2 G }  H.
4. A set of elements a1 , a2 , . . . , ak 2 G is called a generating set29 ofG, if each element of G can be expressed30 as a sum31 Âik=1 ni ai forsome ni 2 Z.
5. Group G is finitely generated if there exists a finite generating set.
6. If G 0  G, the quotient G/G 0 is defined as the group consisting ofclasses [ a] for a 2 G under the following identification:[ a] = [b] , aGOr just “generators”.

30

As opposed to vector spaces, such
expressions in groups are often notunique, which is why the expression“generating set” is used instead of“basis”.
31
For n 2 N and a 2 G we definen · a = |a · a ·{z. . . · }an timesand ( n) · a =b 2 G0 .

7. The direct sum of groups G and H is the group denoted by G
and defined as

29H

H = {( a, b); a 2 G, b 2 H }and the operation being defined coordinate-wise.
A remarkable fact about finitely generated Abelian groups is thatthey can be classified in a wonderful way.

( n · a ).

fields and vector spaces

Theorem 6.3.3. [Classification theorem for finitely generated Abeliangroups] Let G be a finitely generated Abelian group. Then there exist :• k, r 2 {0, 1, . . .},• q1 , q2 , . . . , qk 2 N, and• prime numbers p1 , p2 , . . . , pk 2 N,such that G is isomorphic toZr|{z}free part of G

Z p q1| 1

Z p q2 . . .
2{ztorsion of G

Number r = rank( G ) is called the rank of G.

Z pqk .
k}

Example 6.3.4. Z12 ⇠= Z3 Z4 , while Z4 6⇠= Z2 Z2 : for each elementa 2 Z2 Z2 we have a + a = 0, while the same does not hold in Z4 .
Proposition 6.3.5. Suppose G, H are Abelian groups, a map f : G !
H is a homomorphism, and G 0  G. Then:
1. Im( f ) ⇠= G/ ker( f ).
2. rank( G/G 0 ) = rank( G )rank( G 0 ).

83

7
Homology: definition and computation

Now that we presented combinatorial and algebraic prerequisites, we are ready to define homology. The notion of homology arosefrom the need to detect the holes in a simplicial complex or a moregeneral space. Its definition is not as straight forward as one mighthope, but nonetheless results in a notion amenable to practical computations and consistent with the geometric intuition we presented in thefirst chapter.
In this chapter we will journey through a geometric introduction
and definition of homology, and study the basic methods of computation. We will provide examples of homologies, which should build
up our understanding and detection of holes of all dimension not only
for subspaces of Euclidean spaces, but also within the combinatorial
context of abstract simplicial complexes.

b

a

d

7.1 Definition
Homology measures holes in a simplicial complex. As the latter isprovided by a collection of simplices, we need to devise a computational framework based on the simplices that will result in a meaningful result. The formal treatment of this section will be provided inparallel to a simple example on the right.
Let K be an abstract simplicial complex of dimension n and choosea field of coefficients F.

ChainsChains are formal sums of simplices along with coefficients from
F. They are an algebraic model of collections of simplices as demonstrated in Figure 7.3.
For each p 2 {0, 1, . . . , n} let n p denote the number of simplices ofdimension p in K.

ecFigure 7.1: Abstract simplicial
complex L.

b

a

b

a
d

c

e

d

c

e

Figure 7.2: Two 1-chains in L: thered chain on the left hc, ai + h a, bi +hc, di + hd, bi + hc, bi coincides withthe blue chain on the right hc, ai +h a, bi + hc, di + hd, bi 2hc, bi =hc, ai + h a, bi + hc, di + hd, bi + 2hb, cii↵ the coefficients are from Z3 .

86introduction to persistent homologynppDefinition 7.1.1. A p-chain is a formal sum Âi=1 li si with li 2 Fpand si being an oriented simplex of dimension p in K for each i.
This formalism incorporates orientation: if s is an oriented simplex then ( 1) · s = s is the simplex s with the opposite orientation.
ppp

We assume that {s1 , s2 , . . . , sn p } is the collection of all p-simplices
of K. The p-simplices that are ”absent” in a p-chain have coefficient 0.
p-chains can be added/subtracted and multiplied by any scalar:
np

np

p

np

p

p

Â li si + Â li0 si = Â (li + li0 )si

i =1

i =1

np

kÂ

i =1

p
li si =

i =1

np

Â

i =1

p
(kli )si ,

b

a

d

8li , li0 2 F.

e

c

b

b

a

8k, li 2 F.

• Working in Z2 (top of Figure 7.3) the 1-chains are merely subsetsof the collection of edges as the orientation does not matter (+1 =1 in Z2 ). Adding the red chain { a, c} + {b, c} and the blue chain{b, c} + {b, d} results in the purple chain { a, c} + {b, d}.

d

e

c

a

Example 7.1.2. Consider the simplicial complex L from Figure 7.1.
Two examples of 1-chains and their additions are presented in Figure
7.3.

b

a

d

d

e

c

e

c

Figure 7.3: Top row: addition ofchains in Z2 . Bottom row: additionof chains in any other field.

• Computing in any other field (bottom of Figure 7.3) the orientation
does matter. Adding the red chain hb, ai + h a, ci + hc, di and the blue
chain hb, ai + hd, ci results in the purple chain h a, ci + 2hb, ai.
As a result the collection of all chains forms a vector space1 .
Definition 7.1.3. The chain group C p (K; F ) is the vector space ofall p-chains.
Thinking of p-simplices of K as an abstract collection of linearly
independent vectors, the resulting linear space (with coefficients in F)
spanned by them is the chain group. If n p is the number of p-simplices
of K then C p (K; F ) ⇠= Fn p .

1

For historical and practical reasons
we will match the established terminology in the literature and callthis vector space a chain group, thereason being that if the coefficientsare in a group (as is standard inclassical theoretical approaches, seealso Appendix), the resulting chainsform only a group. In our case thechains still form a group for addition,
but the overall structure along withmultiplication by a scalar is that of avector space.

Boundary
With the definition of chain groups in place, we can now express
the boundary relation as a linear map. The boundary map encodes theassembly instruction for a simplicial complex.

z

x

z

y

x

Figure 7.4: Oriented triangle h x, y, zi
and its boundary ∂2 (h x, y, zi) =h x, yi + hy, zi + hz, x i.

yhomology: definition and computation

87

Definition 7.1.4. Let p 2 N. The boundary map∂ p : C p (K; F ) ! C p 1 (K; F )is the linear map defined by the following rule on the basis of C p ( G; F ):for each oriented p-simplex s = hv0 , v1 , . . . , v p i the image ∂ p s is thesum of facets of s equipped with the induced orientation from s, i.e.,:p∂ p s = Â ( 1) i h v0 , v1 , . . . , v i 1 , v i +1 , . . . , v p i.
i =0

For technical reasons we additionally define ∂0 : C0 (K; F ) ! 0 to be
the trivial map (actually, the only map) into the trivial vector space(the space only containing the 0 vector).
A crucial fact for the algebraic formulation of a homology is thatthe composition of two boundary maps is the trivial map. In particular, this implies that the image of a boundary map is contained in
the kernel of the subsequent boundary map. See the note on the rightconcerning the notation in the following statement.

T The 0 vector is also called thetrivial vector. The trivial map between vector spaces is the map whose
image is the 0 vector.
" We will typically be dropping

the index of the boundary map ∂whenever it will be evident either
that the statement relating to the useof ∂ refers to all indices p or to a
specific p. For example, when talking
about ∂s p , it is apparent that the
map in question is ∂ p . On the other
hand, notation ∂2 = 0 means that foreach p 2 N, ∂ p ∂ p 1 is the trivial
map whose image is the zero vector.

Theorem 7.1.5. ∂2 = 0.
Proof. It suffices to prove that ∂2 s = 0 for an oriented p-simplex
s = hv0 , v1 , . . . , v p i. Note that ∂2 s is a formal sum of faces of s ofdimension p 2. Choose indices i < j from {0, 1, . . . , p} and consider
how does the face2

2

The following face is obtained froms by dropping vertices vi and v j .

s 0 = h v0 , v1 , . . . , v i 1 , v i +1 , . . . , v j 1 , v j +1 , . . . , v p iz+appear in ∂2 s. Such a face appears from two terms:• By first removing vertex v j from s in the expression of ∂ p and thenremoving vertex vi from the resulting simplez in the expression of∂ p 1 . The indices of removed vertices are j and i hence the sign infrom t of s0 is ( 1)i ( 1) j .
• By first removing vertex vi from s in the expression of ∂ p and thenremoving vertex v j from the resulting simplez in the expression of∂ p 1 . The indices of removed vertices are i and3 ( j 1) hence thesign in from t of s0 is ( 1)i ( 1) j 1 .
As the signs are opposite, the total sum equals zero.
Corollary 7.1.6. Im(∂) ⇢ ker(∂).

+

x

y

+

Figure 7.5: An example of Theorem 7.1.5: Oriented triangle
h x, y, zi on the left, its boundary∂(h x, y, zi) = h x, yi + hy, zi + hz, x i,and ∂2 (h x, y, zi) = h x i h x i + hyi
hyi + hzi hzi = 0 as indicated by thesigns at the vertices on the right.
3
As vertex vi has already been
removed and i < j, the vertex v j in
now on position j 1.

88

introduction to persistent homology

Definition 7.1.7. The collection of chain groups bound together by theboundary maps is called the chain complex:
∂

∂

∂

∂

∂

∂

· · · ! Cn (K; F ) ! Cn 1 (K; F ) ! · · · ! C1 (K; F ) ! C0 (K; F ) ! 0
For computational purposes the boundary maps are typically represented as matrices with entries in F. For each p 2 N a matrix M pcorresponding to ∂ p is obtained as follows:• Columns are enumerated by oriented p-simplices of K.
• Rows are enumerated by oriented ( pba1)-simplices of K.

• Entry at position (i, j) equals +1 or 1 if the i-th row appears withorientation +1 or 1 correspondingly in the boundary of the j-thcolumn. All other entries are zero.

d

Example 7.1.8. In particular, the boundary ∂a of a chain a is obtainedby multiplying the boundary matrix with the natural representation of ain the chosen4 basis.
Labeled boundary matrices5 for complex L of Figure 7.6:
0h a, bihb, ciB
B
B
M2 = h a, ciBB
hb, di@hc, dih a, bi hb, ci h a, ci hb, di hc, dih a, b, ci
1
0
1
h ai
1
1
1
C
hbiB
1
1
1 C
C
B 1
C
C
B
C
1 C, M1 = hciB
1
1
1 C
C
B
C
0 A
hdi@
1
1 A
0
hei

e

c

Figure 7.6: Abstract simplicial
complex L.
4
The same basis that is used toenumerate rows.
5
Only non-zero entries are provided.
Matrix M0 has no formal rows as
it represents the zero-map into the
one-element vector space 0.

Homology
We are now finally ready to define homology as a measure of holes.
Let us first build an intuition on the simplicial complex L from Figure
7.6. This will be followed up by a formal introduction in Definition
7.1.9.
Our task is to compute that L has one hole. In the figure the holeseems to be enclosed by edges cd, db and bc. Following this observation
we decide that holes will be represented by a special kind of chains
called cycles, see Figure 7.7. These are the chains that model closed
simplicial loops in our simplicial complex, just as the one describing
the hole in L above. Formally, we define cycles to be those chains,whose boundary is zero. These are our candidates for the representatives of holes.
However, not all cycles represent loops. For example, the top rightcycle in Figure 7.7 is the boundary of a triangle and thus does notenclose any hole. Such cycles thus do not represent a hole and should

b

a

b

a
d

e

c

e

c

b

a

d

b

a
d

c

e

d

c

e

Figure 7.7: Top row: Two cycles.
Bottom row: a chain that is not acycle (left) and the cycle, that is thesum of the cycles of the top row.

homology: definition and computation

be treated as trivial. Similarly, if a cycle is obtained as the boundaryof a 2-chain, then it should be treated as trivial. Such cycles are calledboundaries6 and the structure formalizing the triviality of boundaries
is the quotient space.
Summing up the idea, the holes are represented by the quotientgroup cycles/boundaries.
Recall that for each p 2 {0, 1, . . .} we have Im ∂ p+1  ker ∂ p .

89

6

At this point, the term “boundary”
can refer to a geometric boundaryof a simplex, a boundary map,
or a chain, that is the image of aboundary map.
b

a

b

a
d

Definition 7.1.9. Let K be an abstract simplicial complex. Choose afield F and q 2 {0, 1, . . .}. We define• the group of q-cycles as Zq (K; F ) = ker ∂q  Cq (K; F ).

dc• the group of q-boundaries as Bq (K; F ) = Im ∂q+1  Zq (K; F ) Cq (K; F ).

c
b

a

b

a
d

d

• q-homology group as the quotient Hq (K; F ) = Zq (K; F )/Bq (K; F ).
The dimension of Hq (K; F ) is called the q-Betti number (of K withcoefficients in F) and is denoted by bq = bq (K; F ).
In particular, each element of a homology group is an equivalenceclass7 of cycles. The homology group of example L from Figure 7.6
will depend on F. Defining a = hb, ci + hc, di + hd, bi as the top
left cycle in Figure 7.7, we see that H1 ( L; F ) is {k[a] | k 2 F }. Even
though we have only one hole, the homology group typically has moreelements. However, the entire H1 ( L; F ) is spanned by [a] and thus
the number of holes should be interpreted8 as the dimension of thehomology group, in this case 1.
More generally, each homology group with coefficients in a field
F is a vector space and thus isomorphic to Fr for some dimension r.
The main goal of our computations is thus to compute r = bq , which
represents the number of q-dimensional holes:• b0 is the same for all fields F and equals the number of componentsof K (0-dimensional holes).
• b1 is the number of holes in the usual geometric sense (1-dimensionalholes), although various fields detect di↵erent9 holes in this setting.
For planar graphs however, b1 is always the number of the holes.
• b2 is the number of caves/enclosures.
These interpretations will be explored, demonstrated and partiallyproved throughout the rest of this chapter. Before we do that let us
mention that homology groups are homotopy invariants even though
cycles and boundaries are not.

ccFigure 7.8: Top left: a simplicial
complex with two holes. Its firsthomology group H1 with coefficientsin Z2 has three non-trivial elements,
depicted as the blue, the red, and the
purple chain. However, that does notmean that the number of holes equals
3. Along with the trivial homology
class, the homology groups consistsof 4 elements. This means that its
dimension over Z2 equals 2, which isthe number of holes. We also observethat any two of the three non-trivial
chains above could form the basis of
H1 . In fact, each of the three nontrivial chains is the sum of the othertwo.
7

Given a cycle b, the correspondingclass in homology will be denoted by[ b ].
8

At this point we observe that it
is crucial to preserve the algebraic
structure (of a vector space) of thehomology group in order to computethe dimension as the number ofholes.
9
See the example of the Klein bottle
later in this section.

90

introduction to persistent homology

Theorem 7.1.10. Assume K and K 0 are simplicial complexes. Thena homotopy equivalence K ' K 0 implies H p (K; F ) ⇠= H p (K 0 ; F ), foreach field F and for each p 2 {0, 1, . . .}.

Zero-dimensional homology
In this subsection we prove that b0 is the number of components10of the underlying simplicial complex. Let K be a simplicial complex
and F any field. The homology group H0 (K; F ) is computed from thefollowing piece of information:∂∂0

1
C1 (K; F ) !
C0 (K; F ) ! 0.

In order to compute H0 (K; F ) we need to determine ker ∂0 and Im ∂1 .
Since ∂0 is trivial we have11 ker ∂0 = C0 (K; F ). In order to determine
Im ∂1 we prove the following proposition.

10

While there are alternative ways
to obtain the number of components
employing a smaller amount ofalgebra, there are no alternatives tohomological constructions when it
comes to 1- and higher-dimensionalholes.

Dimension 0 is the only case wherea single simplex forms a cycle.
11

Proposition 7.1.11. Let K be a simplicial complex, F any field andassume x, y 2 K (0) are vertices. Then hyi h x i 2 Im ∂1 i↵ x andy lie in the same component of K.
Proof. Assume x and y lie in the same component of K. Then there
exists12 a path from x to y tracing edges. Let x = x0 , x1 , . . . , xk = ydenote the sequence of vertices traced by one such path. Then the
chain hyi h x i is the boundary of the 1-chain Âik=01 h xi , xi+1 i. See
Figure 7.9 for an example.
In order to prove the other direction assume hyi h x i = ∂a forsome 1-chain a. Let K 0  K be the component of K containing vertex
x and define a0 to be the part of a contained in K 0 . i.e., s0 contains
all those terms of a whose edge is in K 0 . No vertex of K 0(0) \ { x, y}
appears in ∂a0 as none appears in ∂a either and the terms containingedges with such a vertex as an endpoint are the same in both a and a0 .
Hence ∂a0 is either hyi h x i in case y 2 K 0 or h x i otherwise. Since
the coefficients in front of vertices of any boundary add13 up to zero,
only the first of these two options is possible.
Assume K1 , K2 , . . . , Kn are the components of K with xi 2 Ki , 8i.
We now combine the following information that allows us to describe
H0 (K; F ):1. Equality ker ∂0 = C0 (K; F ) means ker ∂0 = Z0 (K; F ) has a basis{hvi}v2K(0) .
2. For each edge h x, yi 2 K we have ∂h x, yi = hyi h x i, meaning thath x i and hyi get identified in the homology group, i.e., [h x i] = [hyi].

12...by the simplicial approximation
theorem.

b

a

d

c

e

Figure 7.9: The boundary of the depicted chain is hdi hbi, which is also
the boundary of hb, di. As a consequence, the column corresponding to
hb, di in the matrix of ∂1 is the sumof the columns corresponding to theedges of the chain.
13
As ∂(khz, wi) = k hwi k hzi thisproperty holds for boundaries ofsingle terms. By linearity of ∂ thesame also holds for chains.

homology: definition and computation

3. By Proposition 7.1.11 the equivalence classes of two vertices are
identified in homology i↵ the vertices lie in the same components.

91

b

a

4. By 1. {[hvi]}v2K(0) span H0 (K; F ) and by 2. and 3. so do {[h xi i]}in=1 .
5. The collection {[h xi i]}in=1 is linearly independent, the proof of thisclaim being similar to the second part of the proof of Proposition
7.1.11.

d

As a result {[h xi i]}in=1 is a basis of H0 ( X; F ) and thus the dimension of
H0 ( X; F ) equals the number of components of K, i.e., b0 = n. For example see Figure 7.10.

ecHomology of a graph
Let K be a simplicial complex which is a connected planar graph,and let F be any field. In this subsection we prove that b1 is thenumber of holes K generates in the plane.
The homology group H1 (K; F ) is computed from the following pieceof information:∂

Figure 7.10: Abstract simplicial
complex L. H0 ( L; F ) is of dimension
two (representing two components)
with a basis [< a >] = [< b >] = [< c >] = [< d >] and [<e >]. H1 ( L; F ) is of dimension one
representing one hole, with a basis[hc, di + hd, bi + hb, ci].

∂2
1
C2 (K; F ) !
C1 (K; F ) !
C0 (K; F ).

As C2 (K; F ) = 0 we have H1 (K; F ) = ker ∂1 so it suffices to determine
the kernel of ∂1 .
1. Let K 0  K be a maximal tree with edges e1 , e2 , . . . , en .
2. The collection ∂e1 , ∂e2 , . . . , ∂en is linearly independent by the following argument14 . As K 0 ' 0 its first homology is trivial by
Theorem 7.1.10 and as K 0 contains no triangles, H1 (K 0 ; F ) =ker ∂1 |C1 (K0 ;F) . In particular, ∂1 |C1 (K0 ;F) is injective. Its matrix contains ∂e1 , ∂e2 , . . . , ∂en as columns and injectivity implies the columns
are linearly independent.
3. Let W denote the span of ∂e1 , ∂e2 , . . . , ∂en .
4. Let en+1 , en+2 , . . . , em be the edges of K that are not contained in
K 0 , with each e j being the edge from vertex x j to vertex y j .
5. Adding edges en+1 , en+1 , . . . , em to K 0 inductively, each addition ofan edge increases the number of holes generated by the resulting
graph by one.
6. In a parallel fashion, each addition of an edge increases the dimension of the kernel of the first boundary map by 1 as ∂e j 2 W, 8 j 2{n + 1, n + 1, . . . , m} by Proposition 7.1.11.
7. In the end of this process of adding edges we have generated m nholes and the dimension of ker ∂1 (and b1 ) turns out to be m n.

14

For an alternative geometricargument see Figure 7.11.

a
b
c
e

d
f

e

d
f

Figure 7.11: In this figure we demonstrate a geometric reason why thecollection of the boundaries of alledges of a tree is linearly independent. Given a tree (on the left sideof the figure) assume a linear combination of the boundaries of its edges
is the zero vector. Since vertex aonly appears in edge h a, di, the coefficient in front of that edge in thementioned linear combination equals
0. The same argument holds for b
and c and thus the mentioned linear
combination only contains edges fromthe subtree on the right. Repeating
the argument above, now for vertices
d and e, we conclude that the mentioned linear combination is trivial
and thus the claim holds. The same
argument works for any tree.

92

introduction to persistent homology

8. For each j 2 {n + 1, n + 1, . . . , m} let c j denote the (simplicial) pathin K 0 from x j to y j represented as a 1-chain. The following form abasis of H1 (K; F ): [e j c j ] for j 2 {n + 1, n + 1, . . . , m}.
An example is displayed in Figure 7.12.
aFigure 7.12: From left to right, thepictures represent a planar graph, amaximal tree, edges not containedin the chosen maximal tree and twocycles representing a basis of thefirst homology. Note that the graphinduces two holes and thus b1 = 2.

b
c
e

d
f

7.2 Computing homology
A systematic way to compute homology groups is through matrix
reduction which allows us to obtain the rank15 of a linear map. Before
we provide the details on the rank computation, let us explain how to
use it in order to compute the Betti numbers.
Let K be an abstract simplicial complex of dimension n and choosea field of coefficients F. For each p 2 {0, 1, . . . , n} let n p denote thenumber of simplices of dimension p in K.
Proposition 7.2.2. 1. dim ker ∂ p = n p
2. b p = n p

rank ∂ p

15

Given a linear map of vector
spaces, its rank is the dimension of
its image.

Proposition 7.2.1. Let f : A ! B
be a linear map of vector
spaces. Then:

rank ∂ p

1. dim A = dim(ker f ) + rank f

rank ∂ p+1

Proof. Part 1. follows from 1. of Proposition 7.2.1 as n p = dim C p (K; F ).
Part 2. follows from 2. of Proposition 7.2.1 and 1.
Thus the Betti numbers can be expressed only using the numberof simplices of a given dimension and the ranks of the corresponding
boundary maps. Now let us turn our attention to rank computations.
Given a boundary matrix, its rank16 is easily obtained17 from the row
or column echelon form.

Echelon forms
In order to obtain a row echelon form of a matrix we can use thefollowing operations18 :R1: exchange two rows;
R2: multiply a row by a non-zero element of F;

2. dim( B/ Im f )rank f=dim B

Part 1. of Proposition 7.2.1 is astandard statement of linear algebra.
Part 2. was proved in the previous
chapter.

16

And thus also the rank of theboundary map. Equivalent definitions of the rank of a matrix include:
the maximal number of linearly independent columns; the maximal
number of linearly independent rows.
17
When using coefficients in R or Q
the numerical procedure to obtainrank might in some cases result incertain instabilities. When using
coefficients in Z p however such
issues do not arise, at least not forreasonably small p.
18

Operations are considered in F.

homology: definition and computation

93

R3: add a multiple of one row to a di↵erent row.
C1: exchange two columns.
In the end we are aiming for the following transformation19
r

0
B
B
B
B
B
B
@

a1,1

a1,n

am,1

am,n

1
C
C
C
C
C
C
A

0

1
B
B0
B
B
B
B
B
B
@
0

⇤

⇤
1
0

⇤

1

C
C
C
⇤C
C
0C
C
C
A
0

The number of the non-trivial diagonal entries, r, equals the rank of
the matrix. In practice we will sometimes refrain from using C1 andonly reduce to the classical row echelon form that is typically obtained
through Gaussian elimination.
In a similar way we can also compute the column echelon form
using the corresponding column operations C1, C2, C3 and (possibly)
R1.

19

Symbols ⇤ denote arbitrary elements of F. The first r elements ofthe diagonal are declared to be 1 inour case. This is one version of therow echelon form and can always beachieved. However, there is a variantof row echelon form in which these
diagonal entries are non-zero but notnecessarily 1, with the other ⇤ entries
still being arbitrary (possibly zero)elements of F. Using this variant
the rank is still obtained in the same
way and as a benefit, the number ofrow operations required to reach it istypically smaller.

b

a

d

Example 7.2.3. Let us compute the homology of simplicial complex Lfrom Figure 7.13. The boundary matrices areh a, bi hb, ci h a, ci hb, di hc, dih a, b, ci
1
0
1
h ai
1
1
h a, bi
1
C
C
hbiB
1
1
hb, ciB
B 1 C
B 1
C
B
C
C
B
M2 = h a, ciB
1 C, M1 = hciB
1
1
1 C.
B
C
B
C
hb, di@ 0 A
hdi@
1
1 A
hc, di0hei
0Performing only row operations we obtain20
0 1
0
1
1
1
B0C
B
1
1 1
B C
B
B C
B
1
B0C and B
B C
B
@0A
@
0

1

C
C
C
1C
C
A

These are the classical row echelon forms typically obtained throughGaussian reduction21 and the rank of such a matrix is the number ofpivots22 . The corresponding ranks of the matrices are 1 and 3. We
thus have rank ∂2 = 1, rank ∂1 = 3, n2 = 1, n1 = 5, n0 = 5 and we
conclude:• b2 = n 2

rank ∂2 = 0, the complex encloses no “void”.

• b1 = n 1

rank ∂1• b0 = n 0

rank ∂1 = 2, which is the number of components.

rank ∂2 = 1, which is the number of holes.

ceFigure 7.13: Simplicial complex L.

20

The reduced form in this casecoincides for all fields F. Later wewill see, for example with the Kleinbottle, that the reduced forms andranks in general depend on F.

21

In order to obtain pivots only on
the diagonal, as the row echelon form
as we defined it requires, we would
need to exchange columns 3 and 4.
22
Equivalently, the number of nonzero rows.
T Row and column operations

amount to changes in the bases ofthe domain and target vector spaces.
These changes can be encoded intransformation matrices and in fact,
most special forms or reductions ofmatrices are often expressed in termsof matrix factorizations. For ourillustrative purposes though we will
stick with the annotations.

94

introduction to persistent homology

Smith normal form and representatives
While the computation of the echelon forms suffices to compute theBetti numbers, we are often interested in the representing cycles23 ofhomology groups as well. To that end we employ a di↵erent canonicalform of a matrix: the Smith normal form. It is obtained from the row
echelon form by eliminating the ⇤ entries to zero using the row andcolumn operations R1, R2, R3, C1, C2, C3.
r

0
B
B
B
B
B
B
@

a1,1

a1,n

am,1

am,n

1
C
C
C
C
C
C
A

0

1
B
B0
B
B
B
B
B
B
@
0

0

0
1
0

0

1

C
C
C
0C
C
0C
C
C
A
0

23

There are also other ways tocompute the representing cycles although, at the end of the day, mostof them use a similar amount of linear algebra. A high-level approach
would be the following. First compute the basis of Im ∂ p+1 , which isthe column space of the corresponding boundary matrix. Then complete
it to the basis of ker ∂ p . The vectors
forming the completion represent thebasis of p-homology. As mentioned,
there are many ways to practicallyformalize these steps, including thepresented one through the Smith
normal form.

In order to obtain representing cycles though, we need to use annotated rows and columns:• The annotations of columns from index r + 1 on form the basis of
the kernel.
• The boundaries of annotations of columns of index up to r on form
the basis of the image.
ker

0

1
B
B0
B
B
B
B
B
B
@
0

0

0
1 0
0

1

C
C
C
0C
C
0C
C
C
A
0

b

a

d

Example 7.2.4. Let us compute the representatives of the homology
groups of simplicial complex L from Figure 7.14. The annotated boundary matrices areh a, bi hb, ci h a, ci hb, di hc, dih a, b, ci
1
0
1
h ai
1
1
h a, bi
1
C
C
hbiB
1
1
hb, ciB
B 1 C
B 1
C
B
C
B
C
M2 = h a, ciB
1 C, M1 = hciB
1
1
1 C
B
C
B
C
hb, di@ 0 A
hdi@
1
1 A
hc, di0
hei0with the annotated row echelon forms being24ce

Figure 7.14: Abstract simplicial
complex L.

24

Only the column annotations will
be displayed as the row annotations
are not required.

homology: definition and computation95

h a, bi hb, ci hb, di h a, ci hc, dih a, b, ci
1
0
1
1
1
1
B 0 C
B
C
1
1
1
B
C
B
C
B
C
B
C
1
1 C.
B 0 C and B
B
C
B
C
@ 0 A
@
A
0
0

The first of these two matrices is already in the Smith normal form.
The Smith normal form of the second matrix is:0

h a,bi

1

B
B
B
B
B
@

hb,ci

hb,di hb,ci

h a,ci hb,ci h a,bi

hc,di hb,di+hb,ci

b

a
1

C
C
C
C.
C
A

1
1

d

We now construct the homology representatives by dimension:
Dimension 0:c
1. ker ∂0 has a basis h ai, hbi, hci, hdi, hei.
2. Im ∂1 has a basis formed by the images of the first three annotated columns of the Smith normal form, i.e., h a, bi, hb, ci, andhb, di hb, ci. The basis obtained in this way ishbi

h a i, h c ihbi, and hdi

hbi + hcieFigure 7.15: Obtained representativesof bases of the homology groups of
L. Representatives h ai and hei in red
spanning H0 ( L; F ), and representativehc, di hb, di + hb, ci in blue spanning
H0 ( L; F ).

h b i.

3. We may complete the basis from 2. to the basis of ker ∂0 by, forexample, adding h ai and hei and thus h ai and hei represent thetwo 0-holes25 spanning H0 ( L; F ).

25

I.e., components.

Dimension 1:
1. ker ∂1 has a basis h a, ci

hb, cih a, bi and hc, di

hb, di + hb, ci.

2. Im ∂1 has a basis formed by the images (boundaries) of the firstannotated column of the Smith normal form, i.e., h a, b, ci. The
basis obtained in this way is

h a, bi + hb, ci

h a, ci.

3. We may complete26 the basis from 2. to the basis of ker ∂0 by,
for example, adding hc, di hb, di + hb, ci and thus hc, di
hb, di + hb, ci represents a 1-holes spanning H1 ( L; F ).

26

The fact that the basis elementfrom 2. is a member from the basisof 1. helps us to see this completion
immediately. However, such a situation is an exception and a completionof basis typically involves some workwith linear algebra.

96

introduction to persistent homology

Incremental expansion and elementary collapse
We conclude the section by analysing how a minimal change to asimplicial complex, an addition of one or two simplices, a↵ects thehomology.
We first discuss the incremental expansion, or how an addition27 ofa simplex to a simplicial complex changes the homology. Let K be asimplicial complex and let s(n) 2/ K be an n-simplex on vertices of K
such that K [ {s } is28 also a simplicial complex. The addition of s toK has the following e↵ect to the homology computation scheme:
1. The number of n-simplices increases29 by 1.
2. If chain ∂s is already contained in ker ∂n , then the addition ofs to the boundary matrix of ∂n adds a column, which is linearly
dependent on other columns and in e↵ect, the dimension of thekernel is increased by 1.

27

Or a removal, which can be analyized in a similar fashion.

28

In particular, all faces of s should
be present in K.
29

This means that either the dimension of the kernel of ∂n or the rank of∂n increases by 1.

3. If chain ∂s is not in ker ∂n , then the addition of s to the boundary
matrix of ∂n adds a column, which is linearly independent on othercolumns and in e↵ect, the rank of the matrix is increased by 1.
As a result (see Figure 7.16), an incremental expansion either increases bn by 1 (case 2.), or decreases bn 1 by 1 (case 3.).
We next discuss an elementary collapse. We have already mentioned it in the chapter on simplicial complexes. Let K be a simplicial
complex, t (k 1) ⇢ s(k) 2 K, and assume s is the only coface of t.
A removal K ! K \ {t, s} is called an elementary collapse. It is a
modification that does not change the homotopy type, and hence thehomology is preserved.
Let us see how an elementary collapse e↵ects the computation ofhomology.
• The boundary of s is not a linear combination of boundaries ofother k-simplices as s is the only30 coface of t. Hence removing s
decreases rank ∂k by 1.
• The boundary of t is a linear combination of boundaries of other(k 1)-simplices by the following argument. Simplex t is contained31 in the chain ∂s. Since the boundary of this chain equals
zero32 , we can express ∂t as a sum of boundaries of other facets ofs with the appropriate coefficients ±1. Hence t is a linear combination of boundaries of other (k 1)-simplices and thus removing it
decreases ker ∂k 1 by 1.
In total, the dimensions of the homology groups do not33 change.

Figure 7.16: A demonstration ofincremental expansion. Adding anedge to a simplicial complex may
either reduce b0 (the number ofcomponents) by 1 (blue case) or
increase b1 (the number of holes) by
1 (red case).

Figure 7.17: An elementary collapse.
30

Meaning that ∂s is the onlyboundary of a k-simplex containing a term with t.

31...with coefficient +1 or

32∂2 s = 0

33

1.

Recall that the only homology
group that may potentially change is
Hk 1 . It is defined as ker ∂k 1 / Im ∂kand since the dimension of bothker ∂k 1 and Im ∂k decreases by one,the dimension of the quotient ispreserved.

homology: definition and computation

97

7.3 Examples of homology
In this section we present some further aspects of homology thatshould aid our understanding of the concept.

Disjoint unions
Two abstract simplicial complexes are said to be disjoint if theircollections are disjoint34 . Two geometric simplicial complexes are
said to be disjoint if their bodies are disjoint. The union of disjoint
simplicial complexes K, L is called a disjoint union and is denoted by
K ‰ L.
Given two disjoint simplicial complexes K, L, the homology of theirdisjoint union is the cartesian product35 of the individual homologies:
Hi (K ‰ L; G ) ⇠= Hi (K; G ) ⇥ Hi (K; G ). Computationally we can see thisby observing that the boundary map ∂ has block-diagonal matrices:
boundaries of chains from K lie in K and the same holds for L. Since
each simplicial complex is the disjoint union of its components, thetechnical computations and treatments of homology are typically
restricted to connected simplicial complexes.

34

I.e., if there is no intersection between the sets of vertices. Formallyspeaking, if such an intersection
existed it would mean that we aretreating both collection of vertices as
subsets of some larger set.
35

In our setting, the term “directsum” could also be used.

Example 7.3.1. Given a planar graph K and any field F:• b0 is the number of components of K.
• b1 is the number of holes of K induces in the plane.
This is consistent with our obervation for disjoint union in case K is
not connected (as in Figure 7.18).

Euler characteristic
Suppose K is a simplicial complex and let ni denote the numberof i-simplices in K. Recall that the Euler characteristic c(K ) 2 Z isdefined as c(K ) = n0 n1 + n2 n3 + . . . .
This invariant has an interesting interpretation in terms of homology.
Proposition 7.3.2. c(K ) = b0

b 1 + b2

Figure 7.18: A planar graph withfour components: b0 = 4, b1 = 7, c =3.

b3 + . . ..

Proof. By 2. of Proposition 7.2.2 we have b p = n p rank ∂ prank ∂ p+1 . Substituting these equality into b0 b1 + b2 b3 + . . .
we obtain c.
Example 7.3.3. Given a planar graph K and any field F, c(K ) equals
the number of components subtracted by the number of holes K generates in the plane.

Figure 7.19: S0 demonstrates nontrivial H0 , S1 represents a onedimensional hole, and S2 encloses a
two-dimensional hole.

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introduction to persistent homology

Spheres
Holes as measured by homology are represented by cycles and thefundamental examples of holes are provided36 by spheres. In this subsection we prove that given a triangulation of an n-sphere for n
1,
the consistently oriented collection of n-simplices represents an n-hole.
In fact, this is the only hole a sphere has. A convenient triangulationof Sn we will be using will be the one37 consisting of all faces of an(n + 1)-simplex.
Proposition 7.3.4. For each F and n 2 {1, 2, . . .} we have:• H0 (Sn ; F ) ⇠= Hn (Sn ; F ) ⇠= F;• Hi (Sn ; F ) = 0, 8i 2
/ {0, n}.
Proof. The full simplicial complex on n + 2 points is contractible
hence all its homology groups are trivial except for H0 , which is ofrank 1. Removing the only (n + 1)-simplex reduces rank ∂n+1 by
one and hence increases bn by 1, as was explained in the context ofincremental expansions and removals.
Since S0 is a collection of two points it is easy to see38 . that the
only non-trivial homology group of S0 is H0 (S0 ; F ) ⇠= F2 .

36

The homology of a metric spaceis, for our purposes, the homology of
any triangulation of that space.

To be precise: take an (n + 1)simplex, add all of its faces to obtaina simplicial complex called the full
simplex on n + 2 points (sometimes
also called the full (n + 1)-simplex),
and then remove the (n + 1)-simplex
to obtain a trianagulation of Sn .
T An observation: The full simplex
on n points is contractible hence its
Euler characteristic equals 1. On
the other hand, computing its Euler
characteristic by definition we getn points (n2 ) edges + (n3 ) triangles. . . ( 1)n · 1 n-simplex. Summing it
up we get:✓ ◆ ✓ ◆ ✓ ◆
✓ ◆
n
n
n
n
+
. . . ( 1) n
= 1,
1
2
3
n
37

by the binomial formula.
38..by a direct computation or by theargument of Proposition 7.3.4

Surfaces
A beautiful demonstration of the two-dimensional homology isprovided by surfaces.
Proposition 7.3.5. Let K be a triangulation of a closed (i.e., withoutboundary) connected orientable surface. For each group F we have
H2 (K; F ) ⇠= F.
Proof. Recall that K being orientable means there exists a consistentchoice of orientations on all triangles of K. Let us fix such an orientation on them.
1. The structure of a surface implies that each edge of K is a face of at
most two triangles.
2. The structure of a closed surface implies that each edge of K is aface of precisely two triangles.
3. Consistency of orientations on triangles implies that whenever two
triangles intersect in an edge, the induced orientations on the edge
are the opposite.

Figure 7.20: Examples of closed
connected surfaces: they all enclose
one 2-dimensional hole in the form ofa “cave”, which is manifested in thefact that b2 = 1.
" The statement of Proposition
7.3.5 does not hold for connectedsurfaces with boundary. If there
was a non-trivial 2-cycle in sucha case, the same argument as in
the proof of the proposition would
imply that the cycle would be theoriented sum of all triangles (possibly
multiplied by a single non-trivial
factor l 2 F). Since a presence ofa boundary of a manifold implies
the existence of an edge, which is aface of precisely one triangle, such a
triangle (multiplied by l) would thus
appear in the boundary of the cycle,
a contradiction.
The second homology of a connected manifold with a boundary isthus always trivial.

homology: definition and computation

Let us define chain a as the sum of all oriented triangles. By 1.-3.
above each edge appears in ∂a twice, once with each orientation (seeFigure 7.21), and thus ∂a = 0, meaning that a is a chain. As theimage of ∂3 is trivial, a represents a non-trivial homology class.
On the other hand, whenever a 2-cycle b contains a term39 +s,
where s is an oriented triangle, observations 2. and 3. imply that alloriented simplices sharing an edge with s also appear in a with coefficient 1. Inductively expanding this conclusion to further neighbors wereach all triangles as K is connected and thus deduce that b = a. The
proposition is thus proved.
Homology class [a] generating H2 (K; F ) as defined40 in the proof
is called the fundamental class of the surface. In the same way wecan prove that if K is a closed connected orientable manifold of dimension n, then Hn (K; F ) ⇠= F with the generator, which is again
called the fundamental class, being the sum of all consistently oriented
n-simplices of K.
The case of non-orientable surfaces is the first presented situation inwhich the choice of coefficients matters.

99

39

If s appear in the term ls forsome non-zero l 2 F, we repeat thesame argument for the chain divided
by l.

40

Formally speaking, there are two
fundamental classes, one for eachorientation of triangles...except when
F = Z2 . If F = Z2 there is only one
non-trivial homology class which is
its own converse.

Proposition 7.3.6. Let K be a triangulation of a closed connected nonorientable surface. Then H2 (K; Z2 ) ⇠= Z2 and H2 (K; F ) ⇠= 0 for⇠each F 6= Z2 .
Proof. As in the proof of Proposition 7.3.5, the fact that K is a surface means that if a 2-cycle a contains a term +s for some oriented
triangle s, it also contains a term +s0 for each oriented triangle s0
sharing41 an edge with s. Again, as K is connected, this means thata is the sum of all oriented triangles. However, as K is non-orientable,
there is no consistent orientation on triangles and thus42 some edges
appear with coefficient 2 in the boundary, see Figure 7.21. Thus if43
0 6= 2 the boundary is non-trivial and the assumed 2-cycle does nothave an empty boundary, a contradiction. Hence the only 2-cycle isthe trivial cycle.
However, if F = Z2 , the obtained boundary equals zero and thus ais the only non-trivial cycle. As a result, H2 (K; Z2 ) ⇠= Z2 .
We may summarize these two propositions and the corresponding
comments as follows:• A connected surface K is closed i↵ H2 (K; Z2 ) 6= 0.
• Given any field F 6= Z2 , a closed connected surface is orientable if
H2 (K; F ) ⇠= F.

Figure 7.21: Top: The boundaryof a chain consisting of all consistently oriented triangles of a surface
without boundary is zero, as theinduced orientations on edges cancel
out. Bottom: The boundary of achain consisting of a not-consistently
oriented collection of all triangles
of a surface without boundary
contains each edge between two
non-consistently oriented triangles
twice.
41

Sharing in the sense of consistentorientation, meaning that the induced orientation on the shared edge
are the opposite.
42
As each edge appears twice in theboundary of such a chain and not all
such appearances may cancel eachother out by the non-orientability.
43
Equivalently, if F 6= Z2 .
T Proposition 7.3.6 also generalizes
to the n-dimensional homology ofclosed connected non-orientablen-manifolds.

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introduction to persistent homology

Impact of coefficients: the Klein bottle
An example where the choice of coefficients makes a di↵erence in
1-dimensional homology computations is the Klein bottle, which will
be denoted by K in this subsection. It is depicted in Figure 7.22.
Its triangulation is given by the black portion in Figure 7.24. We
already know that b0 = 1 as K is connected. However, the second
Betti number of this closed surface depends on the coefficients due tothe non-orientability:• b2 (K; Z2 ) = 1.
• For F 6⇠
= Z2 , b2 (K; F ) = 0.
From this information and the expression of the Euler characteristic
as the alternative sum of Betti numbers we conclude:• b1 (K; Z2 ) = 2, i.e., H2 (K; Z2 ) ⇠= Z22 .
• For F 6⇠
= Z2 , b1 (K; F ) = 1.
These Betti numbers can also be computed through the matrix reduction. Instead of computing them, we will rather demonstrate thegeometric reason for the di↵erence in b1 depending on the coefficients.
The explanation will be based on Figure 7.24. On each of the fiveparts of the figure a triangulation of K is provided by the black/grey
portion. The black arrows indicate the direction in which the identifications are performed. A single red horizontal directed line represents
a cycle a generating the extra dimension of H2 (K; Z2 ). It is also depicted in Figure 7.22. It turns out that [a] is homologically non-zero i↵coefficients are Z2 . In order to prove this statement we first present a
claim.
We claim that 2[a] = 0 in 1-dimensional homology. In order toprove the claim the leftmost part of Figure 7.24 has two copies of adrawn slightly apart from each other. The corresponding homology
class does not change if we move44 each of the copies of a separately.
So let’s move them as on the Figure:• move the upper copy slightly higher;• move the lower copy to the bottom of the side. Due to the reversedorientation, the chain then appears on the top of the square with(in the plane seemingly) reversed orientation. Moving this representative lower to the first copy of a we see, that the copies cancel
each other out: they consist of the same edges with converse orientations.
As a result, the claim holds, i.e, 2[a] is the trivial homology class.
Depending on the coefficients of our computation this has the following ramifications:Figure 7.22: The Klein bottle.

44“Moving” in this setting can bethought of as a homotopic change.
Formally speaking, a moved chain
represents the same homology class
if the di↵erence between the original
and the new chain is in the boundarygroup, see Figure 7.23.

Figure 7.23: Excerpt from the transformation in Figure 7.24. The blue
and the red chain represent the same
homology class because their di↵erence (bluered) is the boundary of
the 2-chain consisting of the strip of
depicted oriented triangles.

homology: definition and computation

101

Figure 7.24: The Klein bottle.

• if F 6⇠= Z2 then we can divide equation 2[a] = 0 by 2 and obtainthat [a] = 0 2 H1 (K; F ).
• if F ⇠= Z2 then we can’t divide equation 2[a] = 0 by 2 as 2 = 0. It
turns out that [a] 6= 0 2 H1 (K; Z2 ) and thus a provides an extra
dimension to H1 (K; Z2 ).
For an alternative argument proving the claim see Figure 7.25.

Alexander duality
Homology is defined for any abstract simplicial complex. However,if there is an underlying geometric simplicial complex K embedded ina sphere or a Euclidean space, there is a connection between the homology of K and that of its complement. The relationship is formallyknown as Alexander duality.
Before we state the duality we should explain a few technical detailsof the complement construction. Let K ⇢ R2 be a geometric simplicial
complex. In particular, K consists45 of finitely many simplices. The
complement of K, denoted by K C = R2 \ K, is unfortunately not homeomorphic to a (finite46 ) simplicial complex. As a proof of this claim
observe that K is a closed47 subset of the plane, while K C is usually48
not. However, K C is homotopic to a finite simplicial complex. For example see Figure 7.26. At this point we defer from specifying details of
triangulation of K C or its homotopy type and rather conclude with thedeclaration: K C is homotopy equivalent to a finite simplicial complex
K 0 and so whenever we will be talking about the homology of K C , wewill formally be thinking of the homology of K 0 . The same discussion
applies if K is a geometric simplicial complex in any Euclidean spaceor a sphere.
Alexander duality provides a connection between the homologies of
K and its complement.

Figure 7.25: Another proof of thefact that 2[a] is homologically trivial
within the Klein bottle. The chain
2[a] is depicted in red and is theboundary of the 2-chain consisting of
all depicted oriented triangles.

Formally, the body of K is theunion of finitely many simplices.
45

46

Recall that all simplicial complexes
considered here are finite. Within
the context of infinite simplicial complexes though, the complement canbe triangulated and the treatmentof complements presented here isimmaterial.
47
In particular, this means that thelimit of each converging sequence inK lies in K.
48
Except if K is empty.

102

introduction to persistent homology

Theorem 7.3.7 (Alexander duality). Let n 2 N and suppose K ⇢Sn is a geometric simplicial complex. Then for any coefficients F wehave:
1. b0 (K; F )1 = bn 1 ( K C ; F ).

2. bn 1 (K; F ) = b0 (K C ; F )1.

3. bq (K; F ) = bn q 1 (K C ; F ) for all q 2 {1, 2, . . . , n

1}.

For a proof see a textbook49 . From the Alexander duality we may
draw a similar conclusion for complexes in Euclidean spaces by taking into account that removing a point50 from Sn results in a spacehomeomorphic to R n .
Corollary 7.3.8. Let n 2 N and suppose K ⇢ R n is a geometricsimplicial complex. Then for any coefficients F we have:
1. b0 (K; F ) = bn 1 (K C ; F ).
2. bn 1 (K; F ) = b0 (K C ; F )1.

3. bq (K; F ) = bn q 1 (K C ; F ) for all q 2 {1, 2, . . . , nFigure 7.26: Simplicial complex K in
the plane in black, and a simplicial
complex K 0 homotopy equivalent
to its complement in red. Note thenumber of holes of K is one less than
the number of components of K 0 , i.e.,b1 (K ) = b0 (K 0 ) 1. Also, numberof holes of K 0 equals the number ofcomponents of K, i.e., b1 (K 0 ) = b0 (K ).
49

James Munkres. Elements of
Algebraic Topology. Perseus Books,
1984. doi: 10.1201/9780429493911
50
A removal of a point from K C ⇢ Sn
increases bn 1 by one.

1}.

Alexander duality is handy when computing homology groups ofsimplicial complexes in Euclidean spaces or spheres. For example,
instead of computing the one-dimensional homology of a planar simplicial complex, we can51 compute the number of components of its
complement, which is typically much faster.

51

Provided there is an easy description of a complement. Such examples
would include bitmap images.

7.4 Concluding remarks
Recap (highlights) of this chapter• Cycles, boundaries, homology• Detecting components and holes with homology• Computing homology through matrix reduction• Euler characteristic• Alexander duality

Background and applications
Homology is one of the focal invariants in topology and geometry.
Homological conditions and constructions can be found throughout
mathematics. We will present one of them in the appendix (CubicalFigure 7.27: A demonstration of
Alexander duality: given a bounded
subset X of the plane, each component of X corresponds to a hole in
X C , and each hole in X corresponds
to bounded component of X C .

homology: definition and computation

homology). The version presented here is usually called “simplicial
homology” as it arises from the structure of a simplicial complex. For
non-triangulated spaces a version called “singular homology” can be
defined. In general though, any reasonable boundary map ∂ satisfying∂2 induces its own homology structure. Examples52 include cubicalhomology (see appendix) and cohomology. For a general reference wemention a few textbooks53 .
Amenability to algorithmic computations through matrix reductions
and, as we will see later, Discrete Morse Theory makes homology an
obvious tool with which we could determine topological propertiesof data. In practice, though, the usual homology is often supersededby persistent homology, which is a richer, parameterized version ofhomology described in later chapters.

Appendix: Homology with coefficients in Abelian groups
Classical introductions of homology typically consider coefficientsfrom an Abelian group rather than a field. By far the most popular
choice among non-fields is the group of integers Z. In this subsection
we review the construction and properties of homology using coefficients54 in a group Z.
Let K be an abstract simplicial complex of dimension n. For eachq 2 {0, 1, . . . , n} let nq denote the number of simplices of dimension q
in K.
The definition of homology in this case remains the same withthe only di↵erence being that the structure of the resulting algebraic
invariants is that of Abelian groups, and the boundary operator ∂ is ahomomorphism:
nq

q

q

1. A q-chain is a formal sum Âi=1 ai si where ai 2 Z and si is anoriented simplex of dimension q in K.
2. The chain group Cq (K; Z ) ⇠= Z nq is the group of all q-chains. Its
generators are oriented q-simplices of K.
3. For each p 2 N the boundary map∂ p : C p (K; Z ) ! C p 1 (K; Z )is the homomorphism defined byp∂ p h v0 , v1 , . . . , v p i = Â ( 1) i h v0 , v1 , . . . , v i 1 , v i +1 , . . . , v p i.
i =0

As before, ∂2 = 0. Additionally define ∂0 = 0.

4. The collection of chain groups bound together by the boundary
homomorphisms is called the chain complex:
∂

∂

∂

∂

∂

∂

· · · ! Cn (K; Z ) ! Cn 1 (K; Z ) ! · · · ! C1 (K; Z ) ! C0 (K; Z ) ! 0

103

52

Another example is the De Rham
cohomology and exterior derivative.
While the theory itself is quite involved, a snapshot of the fact that
∂2 = 0 can be observed in low dimensions via specific derivatives: gradient, divergence, and curl, are specificboundary maps as the compositionof a consecutive pair amongst themequals zero.
53
James Munkres. Elements of
Algebraic Topology. Perseus Books,
1984. doi: 10.1201/9780429493911;
Allen Hatcher. Algebraic topology.
Cambridge Univ. Press, Cambridge,
2000; and Raoul Bott and Loring W.
Tu. Di↵erential Forms in Algebraic
Topology. Springer New York, New
York, NY, 1982. doi: 10.1007/978-14757-3951-0
54

The presented treatment would bepractically identical for any Abeliangroup as the coefficient group.

104

introduction to persistent homology

5. For each q 2 {0, 1, . . .}. We define groups:• q-cycles as Zq (K; Z ) = ker ∂q  Cq (K; Z ).

• q-boundaries as Bq (K; Z ) = Im ∂q+1  Zq (K; Z )  Cq (K; Z ).
• q-homology group as the quotient group

Proposition 7.4.1. Suppose G, H
are Abelian groups, a map
f : G ! H is a homomorphism,
and G 0  G. Then:

Hq (K; Z ) = Zq (K; Z )/Bq (K; Z ).
Up to this point the introduction has been analogous to the one
where coefficients form a field. However, as Hq (K; Z ) is an Abeliangroup, its rank does not completely determine it. In particular,
Hq (K; Z ) ⇠
=

Zr
|{z}

free part of G

Z p q1| 1

Z p q2 . . .
2{ztorsion of G

2. Im( f )  H.
3. Im( f ) ⇠= G/ ker( f ).
4. rank( G/G 0 )rank( G 0 ).

Z pqk ,k}where the rank of the group r = bq (K; Z ), referred to as the q-Betti
number, only determines55 the free part of the group.
Let rank ∂q be the rank of the image of ∂q . By Proposition 7.4.1,
numbers bq can be deduced56 from the ranks of ∂q , ∂q+1 and nq . However, in order to compute torsion we need to delve deeper into thestructure of the boundary maps.
For example, suppose the ranks of the two maps in the following
diagram are 1:j

1. ker( f )  G.

y

Z ! Z ! Z,
and assume Im j ✓ ker y. Defining H = ker y/ Im j, we know thatrank H = 0. However, depending on maps j, y group H could be
any group of the form Z m . For example, if y(n) = k · s · n for somek, s 2 N and j(n) = k · n, then H ⇠= Zs .
In order to compute homology with coefficients in Z we may reduceeach boundary matrix to its Smith normal form. Given a matrix withentries in Z, its Smith normal form is:
0
1
a1 0
0
B0 a
C
B
C
2
B
C
B
C
B
C
D=B
ar 0
0C
B
C,
B
0
0C
B
C
C
B
@
A
0
0
where each diagonal entry ai divides57 the next one. The diagonal
entries ai are called elementary divisors and r is the rank58 of thematrix.

=rank( G )55

Two cases when the homology
group has no torsion:• For any simplicial complex K wehave H0 (K; Z ) ⇠= Zb0 , where b0 is
the number of components of K.
• If K is a planar graph thenH1 (K; Z ) ⇠= Zb1 , where b1 isthe number of holes K generates
in the plane.
56

I.e., b p = n p

rank ∂ p

rank ∂ p+1 .

T It turns out that amongst all pos-sible choices of coefficients, homology
with coefficients in Z contains themost information. Details of thisstatement are formalized in the universal coefficient theorem, which
explains the connection between coefficients Z and all other coefficients.

57
58

I.e., ai | ai+1 , 8i 2 {1, 2, . . . , r

1}.

The rank of the matrix corresponding to a boundary map coincides with the rank of the boundarymap.

homology: definition and computation

Each matrix with entries in Z has59 a Smith normal form. Some ofits properties are:
1. The form is obtained through a combination60 of row reduction
and the Euclidean algorithm for computing greatest common divisors.
2. The form is unique up to the signs of the elementary divisors.
Elementary divisors generate the torsion part of homology.
We now describe how to obtain homology groups using the Smith
normal form.
• Choose q 2 {0, 1, . . .}.
• Assume matrix D above is the Smith normal form of ∂q+1 with all
diagonal entries positive.
• Compute the rank of ∂q , possibly also through its Smith normal
form.
• Then:r
M

Hq (K; Z ) ⇠= Z nq rank ∂q rank ∂q+1

i =1

105

Formally, every matrix A withentries in Z can be factored as
A = UDV, where D is its Smith
normal form, and U and V are
matrices with entries in Z withdeterminant ±1. In particular, thelast condition means that U and V
are invertible, and that its inverses
have entries in Z.
60
At this point the di↵erence of thestructure of a group as compared tothat of a field becomes prominent.
When coefficients were in a field,
we could always divide a row by anon-zero entry. When working withcoefficients in Z that is not allowed(except for ±1, which doesn’t really
help). As a result, obtaining thedesired form of a matrix requires us
to involve greatest common divisorsand even then not all non-trivial
diagonal entries can be transformed
to 1.
59

Z ai .

Note that this form may potentially be simplified61 further.
We conclude by providing analogues of the examples of homology
with field coefficients:

If s1 , s2 are relatively prime, then
⇠ Zs
Z s1 · s2 =Z s2 . Also, if some1ai = 1, then Z ai is the trivial group,i.e., it can be omitted from theexpression.
61• The formula for disjoint union holds as before: Hi (K ‰ L; Z ) ⇠=Hi (K; Z ) ⇥ Hi (K; Z ).
• The expression for the Euler characteristic with integer Betti numbers is the same: c(K ) = b0 b1 + b2 b3 + . . ..
• For each n 2 {1, 2, . . .} we have:– H0 (Sn ; Z ) ⇠= Hn (Sn ; Z ) ⇠= Z;– Hi (Sn ; Z ) = 0, 8i 2
/ {0, n}.
• For each connected manifold K of positive dimension n we have:– Hn (K; Z ) = 0 if K has boundary.
– Hn (K; Z ) ⇠= Z if K is closed orientable.
– Hn (K; Z ) ⇠= Z2 if K is closed non-orientable.
• If K is the Klein bottle, then H1 (K; Z ) ⇠
=Z

Z2 .

For an extended treatment of these examples see a book62 .

62Allen Hatcher. Algebraic topology.
Cambridge Univ. Press, Cambridge,
2000

106

introduction to persistent homology

Appendix: cubical homology
The homology construction we described above is called simplicial
homology as it is based on the structure of a simplicial complex: aspace assembled using simplices. However, there are settings in whichalternative shapes of basic building blocks appear to be more suitable.
One such setting is image analysis, where we work with an image
or a video consisting of pixels. In this setting it would be natural toconsider pixels as the building blocks.
This leads to a new construction63 of complexes and homology: cubical complexes and cubical homology. We will restrict ourselves to thesetting of two-dimensional images, meaning the pixels are chosen froma fixed grid. The construction could easily be generalized to threedimensional (movies of 2-D images or a 3-D image) four-dimensional(movies of 3-D images) or highere-dimensional images with di↵erent
shapes of grids and cubes, or even without a fixed grid.
Let n 2 N and consider a square grid of size n ⇥ n, where n refers
to the number of squares along each side, see Figure 7.28. Our image
is given by a collection of pixels (grey squares). The first task is todefine the building blocks:

Figure 7.28: A 4 ⇥ 4 image consistingof grey pixels.
63
Actually, we could build complexes and the corresponding theory
from many di↵erent shapes of basic
building blocks.

• 0-dimensional cubes are the vertices appearing on the grid. Thereare (n + 1)2 vertices.
• 1-dimensional cubes are the vertical and horizontal edges between
vertices appearing on the grid. There are 2n(n + 1) edges.
• 2-dimensional cubes are the squares of the grid. There are n2
squares.
A cubical complex K on an n ⇥ n grid is a collection of cubes suchthat if s 2 K and t ✓ s, then t 2 K.
Our next task is to determine a convenient systematic labelling forthe squares, edges and vertices. In the context of simplicial complexes
the labels were just the oriented collections of vertices. While the same
approach could64 be used here, there is a more elegant enumeration of
the cubes.
Instead of thinking about coordinates in terms of the n ⇥ n grid,
we systematically imagine all potential cubes of a complex drawn ina table-like pattern as Figure 7.29 demonstrates. Each cube can be
assigned coordinates ( x, y) where x, y 2 {0, 1, 2, . . . , 2n} according
to this pattern. Drawing the corresponding coordinate axes superimposed over the original n ⇥ n grid (Figure 7.30) we see that a pair of
coordinates ( x, y) represents the cube65 , whose center is ( x, y). We
additionally define the orientations:• Each square is oriented with the ordering of its vertices in thepositive-rotational order.

Figure 7.29: The collection of all
potential cubical simplices.
8
7
6
5
4
3
2
1
0

0

1

2

3

4

5

6

7

8

Figure 7.30: The assignment scheme.
64
Although, the approach would
be cumbersome. We would need 4
vertices to describe a square.
65
A square, and edge, or a vertex.

homology: definition and computation

107• Vertical edges are oriented upwards, horizontal to the right.
The resulting assignment of coordinates/labels has the following properties (see Figure 7.31):
8• If x, y are both odd, then ( x, y) is a square.

7
6• If exactly one of x, y is odd, then ( x, y) is an edge.

square(3, 5)

5edge• If x, y are both even, then ( x, y) is a vertex.

4(7, 4)

3

We are now in a position to define cubical homology. The structureof the definition is the same as for simplicial homology with the only
essential di↵erence in the boundary map.
Let K be a cubical complex and choose66 a a field of coefficients F.
For each q 2 {0, 1, 2} let nq denote the number of q-cubes in K.
nq

q

q

1. A q-chain is a formal sum Âi=1 ai si where ai 2 F and si is anoriented cube of dimension q in K.

vertex2(4, 2)edge

1(0, 1)

0

0

1

2

3

4

5

6

7

8

Figure 7.31: The assignment scheme.
66
We could also choose the coefficients from an Abelian group, theconstruction would be analogous.

2. The chain group Cq (K; F ) ⇠= F nq is the vector space of all q-chains.
Its generators are oriented q-cubes of K.
3. For each p 2 N the boundary map∂ p : C p (K; F ) ! C p 1 (K; F )is the linear map defined67 by the following rules:

67

The map encodes the geometricboundary.

If x and y are both even (vertex): ∂ p ( x, y) = 0.
If x is odd and y is even (horizontal edge):∂ p ( x, y) = ( x + 1, y)(x1, y).

( x, y1).

" The operations between coor-

If x is even and y is odd (vertical edge):∂ p ( x, y) = ( x, y + 1)
If x and y are both odd (square):∂ p ( x, y) = ( x + 1, y)( x, y + 1)(x1, y) + ( x, y

1).

As before, ∂2 = 0.
4. The collection of chain groups bound together by the boundary
homomorphisms is the chain complex:
∂

∂

∂

∂

∂

∂

· · · ! Cn (K; F ) ! Cn 1 (K; F ) ! · · · ! C1 (K; F ) ! C0 (K; F ) ! 0
5. For each q 2 {0, 1, . . .}. We define groups:

dinates in the boundary map areformal summations and subtractions in the chain group and should
not be considered as operations onpairs. The coordinates ( x, y) areonly labels and shouldn’t be added toor subtracted from each other. For
example, (0, 0) (2, 0) is a formal
chain consisting of two vertices withcoefficients 1 and 1, while label( 2, 0) is undefined.

108

introduction to persistent homology• q-cycles as Zq (K; F ) = ker ∂q  Cq (K; F ).

• q-boundaries as Bq (K; F ) = Im ∂q+1  Zq (K; F )  Cq (K; F ).
• cubical q-homology group as the quotient

Hq (K; F ) = Zq (K; Z )/Bq (K; F ).
It turns out that the cubical homology of a cubical complex K is
isomorphic to the homology of the simplicial complex obtained bysubdividing the cubes into simplices. In particular, the homology
detects components, holes, and (in the case of higher dimensional
cubical complexes) higher-dimensional holes as simplicial homology
would.
Figure 7.32: The cubical homology of
the above image is given by H0 ⇠=F(one component) and H1 ⇠= F (onehole).

8
Homology: impact and computation by parts

Homology as defined in the previous chapter is an invariant assigned to a simplicial complex. While its homotopy invariance andcomputational amenability make homology a suitable tool for computational purposes, the structural depth of the underlying theory goes
far beyond the presented material.
In this chapter we present some further properties of homology.
The first one is functoriality and its impact on significant topologicalresults from the beginning of the twentieth century: Brouwer fixed
point theorem, hairy ball theorem, and invariance of domains. The
second property is the ability to combine homology computations of
two parts of a space in order to deduce the homology of the whole
space.

8.1 Impact
One of the fundamental tasks of mathematics is a constructionof new objects (invariants) assigned to known objects. For example,
given a closed surface we can assign to it a triangulation. In turn, we
can assign homology groups to the obtained triangulation.
It turns out to be very beneficial if such an assignment can be extended in a consistent manner to maps between the objects as well.
When this is the case, we say the assignment is functorial1 . It turns
out that homology is functorial as Proposition 8.1.2 demonstrates.

1

Functoriality and its formal consequences are studied within the
category theory.

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introduction to persistent homology

Functoriality of homology
Definition 8.1.1. Suppose f : K ! L is a simplicial map between simplicial complexes, q 2 {0, 1, . . .}, and F is a field. The induced maps
f # and f ⇤ are defined as follows:• f # : Cq (K; F ) ! Cq ( L; F ) is the linear map of chain groups defined as⇣⌘q
f # Â ai si =ai f (si ),ai 2 F, si 2 K.
Âi{i | dim( f (si ))=q}• f ⇤ : Hq (K; F ) ! Hq ( L; F ) is the linear map defined asf ⇤ ([a]) = [ f # (a)].
Comments on Definition 8.1.1 using the notation established in it:
1. Given a simplex s 2 K of dimension q, its image f (s ) is a simplex
of dimension q or less. The condition on dimension in the definitionof f # means that only the images of those simplices si , which are offull dimension q, are taken into account. In particular,8< f ( s );dim( f (s )) = q
f # (s) =
:0;
else.

" We refrain from specifying q and

F in the notation f ⇤ in order not tooverload it with the indices. As suchf ⇤ represents the induced map on
homology in any dimension or withany coefficients. The relevant choiceof the dimension(s) and coefficientsshould always be apparent from thecontext.

T The induced maps in the case ofcoefficients in a group are homomorphisms and are still well defined.
T Identity maps between spaces
induce identity maps on homology.
Constant maps between spaces induce
trivial (i.e., zero) maps on homology.

K
f

,!

2. The induced map f ⇤ turns out to be well defined, i.e., if [a] = [ b]then f ⇤ ([a]) = f ⇤ ([ b]).
3. Homotopic maps induce the same maps on homology.

4. Suppose X and Y are metric spaces with triangulations K and
L. By the simplicial approximation theorem there exists for eachcontinuous map f 1 : X ! Y a simplicial map f 2 between some subdivisions of K and L, such that f 1 ' f 2 . Whenever we mention thehomology of X, we formally think of the homology of K. In a similar manner, whenever we talk about the maps on homology induced
by a continuous map f 1 , we formally think2 of maps induced by thesimplicial map f 2 .
The induced maps are consistent with respect to compositions3 as
the following proposition explains.

L

Figure 8.1: An embedding f : K ! L.
While the first homology groupsof K and L are of dimension 2, theimage f ⇤ ( H1 (K; F )) is of dimension
1, demonstrating that the embedding preserves only one hole. This
interpretation will be significantly
expanded within the context of
persistent homology.
2With this explanation, the notionof a map on homology induced bya continuous map between spaces X
and Y is well defined.
3

Formally speaking we express thisproperty by saying that homology isfunctorial.

homology: impact and computation by parts

111

Proposition 8.1.2. [Functoriality of the induced maps] Suppose maps
f : K ! L and g : L ! M between simplicial complexes are simplicial. Then for each q 2 {0, 1, . . .} and for each F we have(gf ) # = g#

f# ,

and

(g

f ) ⇤ = g⇤f⇤.

The proof follows straight from the definition.
One of the most natural demonstrations of the power of functoriality concerns the existence of retractions. Given a space X and its
subspace4 A ⇢ X, a retraction of X to A is any continuous map
f : X ! A such that f ( a) = a, 8 a 2 A. For example, the radial map
from a two-dimensional Euclidean ball with the center removed to its
boundary sphere is a retraction, see Figure 8.7.
Example 8.1.3. For each n 2 N the standard (n 1)-sphere Sn 1 isthe boundary of the standard n-ball Bn . We claim there is no retraction Bn ! Sn 1 . As a special case, there is no retraction of the unit
interval onto its endpoints.
Proof. Assume such a retraction f : Bn ! Sn 1 = ∂Bn exists. Precompose it with the inclusion g : Sn 1 ,! Bn , see Figure 8.3. Let[a] 6= 0 be a basis (generator) of Hn 1 (Sn 1 ; F ). We combine two
observations:• As fg : Sn 1 ! Sn 1 is identity, ( fg)⇤ ([a]) = [a] 6= 0.

• As Hn 1 ( Bn ; F ) = 0, g⇤ ([a]) = 0 and thus f ⇤ ( g⇤ ([a])) = 0.

Figure 8.2: Geometric intuition
dictates that if we want to retract
B2 onto S1 = ∂B2 , the resulting map
would need to have a discontinuity,i.e., at least one point where we“tear” the disc. A fairly simple proofof this fact is given using homology.
4
A required condition for the existence of a retraction is for A to beclosed in X.
f

g

H1 ⇠
=F

g⇤

H1 = 0

f⇤

H1 ⇠
=F

Figure 8.3: The proof of Example
8.1.3. The composition of maps is
identity on S1 , while the compositionof induced maps can’t be identity as
it factors through 0.

By Proposition 8.1.2 ( f g)⇤ ([a]) = f ⇤ ( g⇤ ([a])), a contradiction.
Hence a retraction f does not exist.

Brouwer fixed point
Brouwer fixed point theorem is probably one of the most famous
early results of topology. It has a surprisingly short proof using thefunctoriality of homology.

g(x)xTheorem 8.1.4. Every continuous map f : B2 ! B2 has a fixed point,i.e., a point x0 2 B2 such that f ( x0 ) = x0 .
Proof. Assume map f has no fixed point. Define map g : B2 ! S1 bydeclaring that for each x 2 B2 , point g( x ) 2 S1 = ∂B2 is the intersection of S1 with the ray starting at f ( x ) containing x, see Figure 8.4.
As f has no fixed point, such a ray always exists. Map g is a continuous retraction, a contradiction according to Example 8.1.3. Hence a
fixed point exists.

f (x)

f (y)
y = g(y)Figure 8.4: Map g from the proof ofthe Brouwer fixed point theorem.

112

introduction to persistent homology

Hairy ball
Another prominent theorem that can be conveniently proved using
homology is the hairy ball theorem. The name comes from a popular adaptation of the result: one can’t comb the hair on a hairy ballwithout creating a hair whorl.
Before we state the theorem we need to clarify a few technical details. Throughout this subsection let S2 denote the unit two-dimensional
sphere in R3 .
1. A tangent vector field on the sphere S2 is a continuous map f : S2 !
R3 such that for each x 2 S2 we have x ? f ( x ). A vector field f is
non-vanishing, if it is non-zero at each point.
2. Given a centrally symmetric5 triangulation K of S2 , let a =Âs(2) 2K s be the cycle defined as the sum of consistently oriented
triangles of K. Without loss of generality we may assume the triangles are oriented so that their “upwards” direction is pointing awayfrom the point (0, 0, 0). Recall that [a] is the fundamental classspanning H2 (K; R ) ⇠= R.
3. For each triangle s 2 K the reflection of s through (0, 0, 0) is again
a simplex s0 of K. However, if s has the chosen orientation fromthe previous point6 , then the reflected triangle has the opposite
orientation7 from the originally chosen orientation on s0 , see the left
portion of Figure 8.6. In particular, [a0 ] = Âs(2) 2K s0 is a non-trivial
homology class representing [a].

Figure 8.5: A tangent vector field
on a sphere induces a flow presented
by the streamlines on this figure.
Theorem 8.1.5 states that the vector
field must have a zero, which canbe demonstrated on our example
by the source of streamlines. To
the contrary, there are non-trivial
tangent vector fields in the plane andon the torus.
5I.e., the triangulation K has thefollowing property: for each simplex
t 2 K its reflection through the point(0, 0, 0) is also a simplex.
6I.e., such that the chosen normal ispointing away from the point (0, 0, 0).
7I.e., such that the chosen normal ispointing towards the point (0, 0, 0).

4. Let r : K ! K be the reflection map and let g : K ! K be theidentity map. By 2. and 3. maps g and r are not homotopic asg⇤ ([a]) = [a] 6= r⇤ ([a]) = [a0 ].

x
c
b

a
a

f ( x)

f (x)
b
c

x

Figure 8.6: Elements of the proof of
Theorem 8.1.5.
On the left side is simplex h a, b, ci
and its (oriented) reflection through(0, 0, 0): h a, b, ci. Observe thatin both cases the normal to thesimplex is in the direction (1, 1, 1).
The same argument and picture work
for any odd dimension, which leads
to Theorem 8.1.6.
On the right side is the construction of homotopy from the proofof Theorem 8.1.5. Point x is connected to x by the geodesic passing
through f ( x ) and vice versa.

homology: impact and computation by parts

113

Theorem 8.1.5. Every tangent vector field on S2 has a zero, i.e., there
is no non-vanishing tangent vector field on S2 .
Proof. Suppose f is a non-vanishing vector field on S2 . Without lossof generality8 we can assume || f ( x )|| = 1, 8 x 2 S2 . Using the notation
leading to this theorem, we will prove that g ' r, which is a contradiction by 4. above. We will construct an explicit homotopy between gand r. Such a homotopy can be thought of as a continuous collection9of paths from x to x for all10 x 2 S2 .
The simplest way to connect two diametrically opposite points on asphere, for the sake of simplicity let us assume we are connecting thenorth pole N to the south pole S, is by drawing a meridian between
them. Such a meridian is completely determined by the point at which
it intersects the equator. We define this intersection point to be11
f ( N ).
In general, connect x to x by a geodesic12 on the sphere passing
through f ( x ). This is a continuous assignment of paths and constitutes the homotopy between g and r, which completes the proof.
The argument of Theorem 8.1.5 works for any even dimension
which leads to a more general result.
Theorem 8.1.6. The sphere Sn admits a non-vanishing tangent fieldi↵ n is even.
When n is even there is an easy construction of a non-vanishingtangent field:

8I.e., by normalizing each vector in
the image of f .

9

A homotopy in question is of theform H : S2 ⇥ [0, 1] ! S2 . For eachy 2 S2 the restriction H |{y}⇥[0,1] isthus a path from y to y. The factthat H is continuous means that thecollection of such paths is continuous.
10
While the homology setup aboveis performed in the simplicial settingof K, the homology here will be
constructed on a “smooth” sphere S2 .
11
Recall that || f ( N )|| = 1 andf ( N ) ? N, since f ( N ) lies on theequator.
T A geodesic on S2 is the shortest
path between two points on S2 .
Geodesics between N and S are
meridians.
12This geodesic traces the trail ofx as translated by the resulting
homotopy. On the other hand, thetrail of x as translated by the
resulting homotopy is given by
the geodesic from x to x passing
through f ( x ). See the right portionof Figure 8.6 for a sketch.

( x1 , y1 , x2 , y2 , . . . , x m , y m ) 7 ! ( y1 , x1 , y2 , x2 , . . . , y m , x m ).

Invariance of domain
The last classical result we mention explains why Euclidean balls ofdi↵erent dimensions are fundamentally di↵erent in the sense that they
can’t be homeomorphic13 .
Theorem 8.1.7. For any pair of natural numbers m 6= n the closed
balls B1 = BRm (0, 1) and B2 = BRn (0, 1) are not homeomorphic.

13

While homology itself is a homotopy invariant, the trick we will use
will allow us to use it to di↵erentiate
homeomorphic types of spaces.
T As a consequence of Theorem

Proof. Assume there exists a homeomorphism f : B1 ! B2 . Thenf | B1 \{0} : B1 \ {0} ! B2 \ { f (0)} is also a homeomorphism. Recall
that B1 \ {0} ' Sn 1 via the radial projection (see Figure 8.7), whichmeans Hn 1 ( B1 \ {0}; F ) is non-trivial for any F. On the other hand,
B2 \ { f (0)} is either:

8.1.7, D n 6⇠
= D m if m 6= n. The same
argument gives R n 6⇠
= R m if m 6= n.

114

introduction to persistent homology• homotopy equivalent to Sm 1 if f (0) 2/ ∂B2 , or• contractible if f (0) 2 ∂B2 .
In both cases Hn 1 ( B2 \ { f (0)}; F ) = 0, a contradiction.

8.2 Homology by parts
Given a decomposition of a simplicial complex K = A [ B as theunion of subcomplexes A and B, can14 we compute the homology of X
from the homology of A and B?
The answer to this question is unfortunately negative, for example:• As the sidenote on the right on a similar question demonstrates, thecumulative zero-dimensional homology of K and L may be too large
and should possibly be decreased by the zero-dimensional union of
the intersection.
• On the other hand, a circle is the union of two 1-discs, i.e., a spacewith a one-dimensional hole is the union of two subspaces withoutholes.
These two examples show that A and B can have cumulatively “toomuch” or “too little” homology to deduce the homology of the union
X and that one should probably take into account the homology of
the intersections as well. The algebraic structure through which theconnection between the homologies of X, A, and B is expressed is thatof exact sequences.

Exact sequences
Definition 8.2.1. A sequence of vector spaces V0 , V1 , . . . and linear maps
jn : Vn ! Vn 1 is exact, if for each n we have Im jn+1 = ker f n .
It turns out that in an exact sequence, the dimension of each vector
space (except for the last one) can be deduced from the ranks of theneighboring maps.
Proposition 8.2.2. Suppose the following sequence is exact:
j n +1jn

j2

j1· · · ! Vn+1 ! Vn ! Vn 1 ! · · · ! V2 ! V1 ! V0 .
Then for each n > 0, dim Vn = rank jn+1 + rank jn .
Proof. We know that dim Vn = dim ker jn + rank jn . Now use
exactness: Im jn+1 = ker jn .

Figure 8.7: Radial projection of adisc with the center removed to theboundary of the disc. The induced
homotopy equivalence demonstrates
B1 \ {0} ' Sn 1 in the proof of
Theorem 8.1.7.
14
A similar question: Given a finite
set Y = C [ D, can we determine
the cardinality |Y | from |C | and | D |?
The answer |Y | = |C | + | D | |C \ D |,which also includes the intersection, is not unlike the answer to ourquestion about homology...especiallysince, for discrete sets, the cardinality represents the zero-dimensional
homology.
A

A

A

B

B

B

Figure 8.8: Two contractible complexes, whose union is not contractible.

T Recall that homology is definedfrom a sequence of chain groups
called the chain complex; it is defined as the quotient ker ∂/ Im ∂. In
particular, the homology of a chain
complex is zero at all dimensions i↵
the chain complex forms an exact sequence. Or, to put it locally, Hq = 0i↵ the chain complex is “exact at Cq ”
in the sense that Im ∂q+1 = ker ∂q .
Homology thus measures the extent
to which a chain complex is notexact.

homology: impact and computation by parts

115

Mayer-Vietoris exact sequence
We are now able to express15 the connection between the homology
of X and the homology of its two parts A and B, a connection that
also includes the homology of the intersection A \ B.

15Standard proofs use zig-zag lemma
given in an appendix.

Theorem 8.2.3. Suppose A, B  X are subcomplexes of a simplicial
complex K such that A [ B = X. Then for each choice of coefficientsthe following sequence of homology groups is exact:
dn+1· · · ! Hn+1 ( X ) ! Hn ( A \ B)
· · · ! H0 ( A \ B)(in ,jn )! Hn ( A)(i0 ,j0 )! H0 ( A)µn

Hn ( B) ! Hn ( X ) ! · · ·µ0H0 ( B) ! H0 ( X ) ! 0,with the involved maps defined as follows:• i⇤ , j⇤ are inclusion induced maps, i.e., i⇤ [a] = [a] and j⇤ [a] = [a].
• µ is the subtraction map, i.e., µ⇤ ([a], [ b]) = [ab ].

• d is a variant of a boundary map defined as follows. Given an ncycle a in X, decompose it as a = a A + a B where a A is an n-chain
in A and a B is an n-chain in B. Define d[a] = [∂a A ] as the homology class corresponding to the boundary of the chain a A .
Example 8.2.4. We will compute the homology of S1 with coefficientsin a field F. Express S1 as the union of two 1-discs A and B as Figure
8.9 suggests. The only non-trivial part of the corresponding MayerVietrois sequence is the following:
H1 ( A)

H1 ( B) ! H1 ( X ) ! H0 ( A \ B) ! H0 ( A)

A

A\B

B

Figure 8.9: A decomposition of S1
into two 1-discs.

H0 ( B) ! H0 ( X ) ! 0,

which is of the formd1
0 ! H1 ( X ) !
F2(i0 ,j0 )µ0d0! F2 ! H0 ( X ) !
0,since A \ B has two components. We proceed by the following sequenceof deductions:
1. rank d0 = 0 as it is the trivial map.
2. Map µ0 is of rank16 1.
3. By Proposition 8.2.2 we get dim H0 (S1 ) = 1.
4. By exactness and observation 2. we have dim Im(i0 , j0 ) = dim ker µ0 =1, hence rank(i0 , j0 ) = 1.

Recall that µ(u, v) = u v. Its
rank is either 0, 1, or 2. It can’t be
0 as the map is nontrivial. It can’tbe 2, as it has a non-trivial kernel
generated by (u, u) since A and B are
in the same component of X.
16

5. By exactness and the previous item we have dim Im d1 = dim ker(i0 , j0 ) =1, hence rank d1 = 1.

116

introduction to persistent homology

6. By Proposition 8.2.2 we get dim H1 (S1 ) = 1.
7. All higher homotopy groups (for n > 1) are trivial as they appear as· · · 0 ! Hn ( X ) ! 0 · · · in the exact sequence which, by Proposition
8.2.2, means they are trivial.
Remark 8.2.5. In the same manner we could compute the homology
groups of Sm for each m by observing that it can be decomposed as theunion of two hemispheres (m-discs) whose intersection is homotopy
equivalent to Sm 1 , see Figure 8.10.
Example 8.2.6. In a similar manner we can compute the homologyof the torus X with coefficients in any field F. We will only mention
how to compute its first homology as the homology groups of otherdimensions are already known.
We will use the decomposition of Figure 8.11. The relevant part of
the Mayer-Vietoris sequence is
H1 ( A)

H1 ( B) ! H1 ( X ) ! H0 ( A \ B) ! H0 ( A)

H0 ( B) ! H0 ( X ) ! 0

Figure 8.10: A decomposition of S2
into two discs, whose intersection is
S1 .

A

A\B

B

↵

Figure 8.11: A decomposition of
the torus into two parts, whose
intersection is the disjoint union of
two copies of S1 .

which is of the form
µ1d1
F2 ! H1 ( X ) !
F2(i0 ,j0 )! F2 ! F ! 0.

We proceed by the following sequence of deductions:
1. By the same argument as in Example 8.2.4 we have rank d1 = 1.
2. The generators of H1 ( A) and H1 ( B) are cycles/loops a and b
respectively. Note that a ' b in X thus [a] = [ b] 2 H1 ( X ).
Furthermore, as17 0 6= [a] 2 H1 ( X ), we have18 rank µ1 = 1.
3. By Proposition 8.2.2 we get dim H1 ( X ) = 2.

8.3 Concluding remarks
Recap (highlights) of this chapter• Induced maps on homology and functoriality;• Brouwer fixed point theorem and hairy ball theorem;• Exact sequences;• Mayer-Vietoris exact sequence.

17

An algebraic way to see that
0 6= [a] 2 H1 ( X ) is through theMayer-Vietrois sequence:
Proof. If [a] was trivial in H1 ( X )then ([a], 0) would be in ker µ1 . By
exactness, this would mean that([a], 0) 2 Im(i1 , j1 ). However, Im(i1 , j1 )is generated by the images of thetwo obvious cycles in A \ B, each ofwhich maps into (±[a], ⌥[ b]). SpaceIm(i1 , j1 ) is thus one-dimensionaland generated by (±[a], ⌥[ b]), hence([a], 0) 2
/ Im(i1 , j1 ) as [a] 6= 0 in H1 ( A)and [ b] 6= 0 in H1 ( B).
18

The map µ1 is defined asµ1 (u, v) = u v in the basis([a], 0), (0, [ b]) of H1 ( A) H1 ( B).
Its rank is either 0, 1, or 2. Its can’tbe 0 as the map is nontrivial, since
0 6= [a] 2 H1 ( X ). It can’t be 2, as it
has a non-trivial kernel generated by(u, u) since [a] = [ b] 2 H1 ( X ).

homology: impact and computation by parts

117

Background and applications
Invariants of homological nature appear throughout topology, geometry and other fields of mathematics. The examples of theoreticalapplications presented here barely scratch the surface. Some of thesettings in which such constructions contributed to significant development include knot theory (Khovanov homology), di↵erential geometry(De Rham cohomology, Floer homology), etc. For further backgroundon theoretical foundations see Hatcher’s book19 .
The Mayer-Vietoris sequence arises from a decomposition of a space
into two pieces. A natural question about a similar result in the context of decompositions into more pieces is treated within the context
of spectral sequences, an algebraic formalism far above the reach of
our presentation. These theoretical developments allow for a certain
level of distributed computation of homology.

19

Allen Hatcher. Algebraic topology.
Cambridge Univ. Press, Cambridge,
2000

Appendix: zig-zag lemma
Lemma 8.3.1. [Zig-zag lemma] Let F be a field of coefficients. Assume
the following diagram of vector spaces over F and linear maps20 iscommutative21 :
..
.

..
.
∂

..
.
∂

✏

✏

b

/ Cq+1

∂

∂

a

✏
/ Bq

b

✏
/ Cq

a

✏

b

✏

a

/ Bq+1

∂

0

✏
/ Aq

0

✏

0

/ A q +1

∂

✏

∂

∂

/ Aq 1✏∂..
.

/ Bq 1/0

For the sake of simplicity theindices of maps will be omitted. For
example, maps aq : Aq ! Bq are
all denoted by a even though they
depend on q . For the same reason
we will refrain from mentioning F
again.
21
I.e., ∂ a = a ∂ and ∂ b = b ∂.
T An exact sequence of the form

0!A!B!C!0/0∂/ Cq 1

✏∂
..
.

20

/0

✏∂
..
.

If each row is a short exact sequence, and each columns is a chain
complex22 , then there exists a long exact sequence of homology groups23

is called a short exact sequence.
In such a situation, map A ! B is
injective as its kernel is the trivial
image of the map 0 ! A. On asimilar note, B ! C is surjective as
its image is the kernel of the map
C ! 0, which is C. As Im( B ! C ) ⇠=B/ ker( B ! C ) we conclude C ⇠= B/Asince equality ker( B ! C ) = Im( A !
B) holds by exactness.

I.e., ∂2 = 0.
I.e., the homology groups arisingfrom the vertical chain complexes. In
particular, Hq ( A) is the quotient
22
23

a⇤

···
Hq ( A)/ Hq+1 ( B)b⇤

/ Hq+1 (C )

d
a⇤

t

/ Hq ( B)b⇤ker( Aq ! Aq 1 )/ Im( Aq+1 ! Aq ).

/ Hq (C )d

Hq 1 ( A)ta⇤/ Hq 1 ( B)

b⇤

/ ···

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introduction to persistent homology

The idea of a proof. The proof is performed using the “diagram chasing” technique. We will only prove the existence of the d map.
In order to define d let us choose a non-trivial cycle c 2 Cq+1 .
Charted by the diagrams on the right, the chase after d([c]) begins:

T Diagram 1:∂

Diagram 1: ∂(c) = 0 as c is a cycle.

✏

0

Diagram 2: By the exactness of the row map b is surjective, thus
there exists b1 2 b 1 (c). Define b2 = ∂(b1 ). By the commutativityb(b2 ) = 0.
Diagram 3: By the exactness of the row map there exists a1 2
a 1 (b2 ). Define d([c]) = [ a1 ].

T Diagram 2:

b1

_

b2

a1

• By the commutativity a( a2 ) = 0.

Remark 8.3.2. The construction and proof of the Mayer-Vietoris sequence follows from the zig-zag lemma using the following commutative
diagram (using notation of Theorem 8.2.3)
..
.
∂

0

✏
/ Cq ( A \ B)0✏
/ Cq 1 ( A \ B)

/ Cq+1 ( A)✏

/ Cq ( A)✏/ Cq 1 ( A)✏

Hq+1 ( B)∂

Hq ( B)

∂

✏∂
..
.

_

/c
_

✏
/ b2

✏
/0

/0

Hq 1 ( B)

✏∂
..
.

/c
_

a1

✏
/ b2
_

✏
/0

✏

✏
/0

/0

∂
b

/ Cq+1 ( X )b✏/ Cq ( X )

b

✏

/0

∂

∂
a

a

b1

_

a2

✏

∂
a

✏
/0

b1

_

..
.
∂

a

b

T Diagram 4:• By the exactness of the row a2 = 0, hence a1 is a cycle.

0

/0

/c
_

T Diagram 3:• Define a2 = ∂( a1 ) observe ∂(b2 ) = 0 as ∂2 = 0.

✏
/ Cq+1 ( A \ B)

b

∂

✏

The rest of the proof goes along the same lines. For example, inorder to prove a1 is a cycle we use diagram 4:

..
.

/0

_c

/0

∂

/ Cq 1 ( X )

/0

✏∂
..
.

with maps a being induced by inclusion, and maps b being the subtraction maps24 . Observe that the horizontal maps are short exact
sequences.
Zig-zag lemma provides a useful template for constructions of exact sequences. Another setting in which it applies is that of relativehomology.

24I.e., b([g1 ], [g2 ]) = [g1g2 ].

homology: impact and computation by parts

119

Appendix: Relative homology
Let us fix a field F, a simplicial complex K, and L  K. Homology
construction on K is based on cycles: chains whose boundaries are
trivial. The concept of relative homology expands this construction in
the following way: given L  K, the relative homology construction isbased on relative cycles, i.e., chains, whose boundaries are contained inL.
Algebraic specifics of the definition. From the chain complexes of K
and L we can construct the quotient chain complex:
∂

∂

∂

∂

∂

· · · ! Cq (K )/Cq ( L) ! Cq 1 (K )/Cq 1 ( L) ! · · · ! C0 (K )/C0 ( L) ! 0
The relative homology groups Hq (K, L) are the homology groups
arising from this chain complex. In particular:∂

Hq (K, L; F ) =ker Cq+1 (K )/Cq+1 ( L) ! Cq (K )/Cq ( L)∂Im Cq (K )/Cq ( L) ! Cq 1 (K )/Cq 1 ( L).
T Observe that Hq (K, ∆) = Hq (K ).

Combining Lemma 8.3.1 and the commutative diagram
..
.

..
.
∂

∂

✏

0

/ Cq+1 ( L)0✏
/ Cq ( L) 

0

✏



✏

/ Cq+1 (K )∂∂✏/ Cq (K )∂/ Cq 1 ( L)
✏∂
..
.

..
.

∂



∂

✏
/ Cq+1 (K )/Cq+1 ( L)

/0

∂

✏
/ Cq (K )/Cq ( L)

/0

∂

✏

/ Cq 1 (K )✏/ Cq 1 (K )/Cq 1 ( L)

✏∂
..
.

✏∂
..
.

/0

we conclude that relative homology groups fit into the following exact
sequence:· · · ! Hq+1 (K, L) ! Hq ( L) ! Hq (K ) ! Hq (K, L) ! · · ·· · · ! H0 ( L) ! H0 (K ) ! H0 (K, L) ! 0,
Relative homology has a geometric meaning, which expands thatof the usual homology. Table 8.1 summarizes the relative homology ofthe pair (K, L) of simplicial complexes from Figure 8.12.
Let us geometrically interpret Table 8.1:

120

introduction to persistent homology

h

j

i

k

f
g

e

a

b

c

q

dim Hq (K )dim Hq (K, L)

0

3

1

1

2

2

Figure 8.12: Simplicial complex K.
Its subcomplex L  K contains
vertices a, c, b, d, e, j, k and all edges
between these vertices. It is depictedby bold red edges.

dTable 8.1: The comparison of thehomology of K and the relativehomology of the pair (K, L) from
Figure 8.12.

Dimension 0: K has three components. However, the relative homology detects only component [ f ]. Homology class [e] is contained in
L and is thus trivial by the definition. Homology class [ h] is homologous to [e] and thus trivial as well.
Dimension 1: A convenient basis for H1 (K ) would consist of [h a, bi +hb, ei + he, ai] and [he, i i + hi, hi + hh, ei]. A basis for H1 (K, L)
however would consist of [he, i i + hi, hi + hh, ei] and [h j, ci]. Note
that:

h

f

• [h a, bi + hb, ei + he, ai] is a trivial in H1 (K, L) as it is contained inL.
• h j, ci is a cycle in the relative homology chain complex as its
boundary is contained in L and thus trivial.
Geometrically we can think of the relative homology H⇤ (K, L) as
the homology of the space obtained from K when the subcomplex L
is contracted to a point, see Figure 8.13 for an example. The only
exception to this rule is H0 (K, L), whose dimension is one less25 than
the number of the components of the resulting space26 .

igFigure 8.13: The space obtained from
simplicial complex K from Figure
8.12 by contracting the subcomplex L
to a point. The space has two holes
but is not a simplicial complex ingeneral.
25

In the literature this exception
is usually encoded in the phrase“reduced homology”.
26
Note that the resulting spacedoes not inherit the structure of asimplicial complex from K. However,it can be triangulated.

9
Persistent homology: definition and computation

The concept of persistent homology along with its variations is at the forefront of topological data analysis. Mathematicallyspeaking, persistent homology is an obvious extension of homology:
the functoriality of homology is applied to a sequence of inclusions.
The resulting structure is, somewhat surprisingly, not harder to compute than ordinary homology. When coupled with the standard constructions of complexes, persistent homology contains not only topological but also geometric information.
We will start this chapter by explaining geometric intuition on
persistent homology. We will continue by presenting formal and convenient visualisation techniques. We will conclude with a fairly simple
algorithm for computation one could call a “labelled matrix reduction”.

9.1 Definition
We first describe the geometric intuition of persistent homology.
Given a “growing” simplicial complex, persistent homology describes
the evolution of its holes. As an illustrative example we consider four
simplicial complexes K1  K2  K3  K4 of Figures 9.1 and 9.2.
Here is how we interpret the corresponding zero-dimensional barcode1 described by Figure 9.1:
K1 : There are two components of K1 . This fact is visualised by thefact that there are two bars (blue and red) starting at that time.
The corresponding homology generators2 (points) are colored accordingly.
K2 : There are three components of K1 : this fact is visualised by thefact that there are three bars (blue, purple and green) passing fromthat time on. The corresponding homology generators of the newcomponents are colored accordingly. However, the two componentsof K1 merge, which we interpret as one of the components of K1

1I.e., the evolution of the components.

2

Note that the generator of the redcomponent is unique. On the other
hand, we could have chosen any
vertex of the other component as a
generator and color it blue.

122

introduction to persistent homology

K1

K2

K3

K4

disappearing. We declare3 that the component disappearing is the
red component, which is visualised by the fact that the red bar
terminates just before K2 . The edge making the connection between
the two components is colored in red.
K3 : The purple component terminates by connecting to the blue
component via two edges, one of which is indicated by the purple
color.
K4 : There is no change in components as compared to K3 , both bars
are passing through to infinity.

K1

K2

K3

K4

In a similar fashion we interpret the corresponding one-dimensional
barcode4 described by Figure 9.2:
K1 : There are no holes and hence no bars passing on.
K2 : A blue hole appears inducing a blue bar.
K3 : The blue hole becomes trivial by the blue triangle and hence theblue bar terminates. However, two new holes appear, the red one

Figure 9.1: Nested simplicial complexes K1  K2  K3  K4 are
divided by vertical lines. The horizontal arrows below are called “bars”
and form a barcode. They indicate
the persistence of zero-dimensional
homology classes: components. The
left endpoint of each bar corresponds
to the birth complex of a component.
The right endpoint of each bar corresponds to the terminal complex
of a component. The color of eachbar also appears on one vertex (therepresentative of the component) andpotentially on one edge (the edge,that terminates the component).
3

As the components appeared at
the same time, we might as wellhave chosen to have the blue bar
terminated and keep the red bar
going. The uncolored barcode would
have remained the same. However,whenever there is a merger of components with di↵erent birth times we
act according to the elder rule: theolder component survives. This will
be apparent at K3 . The reader may
rest assured this is not a product of
discrimination but rather a rule thatis consistent with the mathematicalstructure of persistence (especiallythe interleaving and stability) thatwill be described later.
Figure 9.2: Nested simplicial complexes K1  K2  K3  K4 andthe corresponding one-dimensional
homology barcode.

4I.e., the indicated evolution of theholes.

persistent homology: definition and computation123

and the green one. Consequently, there are two bars passing fromK3 on.
K4 : The red hole becomes trivial while the green hole lives on forever, just as the corresponding bar.
The goal of this chapter is to present the theoretical backgroundformalizing the presented geometric idea of persistent homology, andto introduce the computational procedure to obtain the barcodes.

Formal definition
We first formally introduce a filtration: a nested sequence of everlarger simplicial complexes modelling a growing simplicial complex.
Definition 9.1.1. Let K be a simplicial complex. A (discrete) filtration of K is a sequence of subcomplexes
K1  K2  . . .  Km = K.
An example of a filtration is given in Figures 9.1 and 9.2.
Persistent homology measures how homology elements5 persist6
through steps of a filtration. A filtration of a simplicial complex K in
Definition 9.1.1 can be expressed as a sequence of natural inclusionmaps denoted by map7 i , :
i1,2

i2,3

5
6

7

I.e., components, holes, etc.
I.e., remain non-trivial

For example, is,t : Ks ,! Kt .

im 1,m

K1 ,! K2 ,! . . . , ! Km = K.
Given a field F and q 2 {0, 1, 2, . . .} we can apply homology
Hq ( ; F ) to obtain a sequence8 of homology groups connected bylinear maps:
Hq (K1 ; F )(i1,2 )⇤! Hq (K2 ; F )(i2,3 )⇤

! ...

8

By the functoriality of the homology we have (iu,t )⇤ (is,u )⇤ = (is,t )⇤ .

(im 1,m )⇤! Hq (Km ; F ) = Hq (K; F )Definition 9.1.2. Assume K is a simplicial complex, F is a field, andq 2 {0, 1, 2, . . .}. Given a filtration
K1  K2  . . .  K m = K
of K, the corresponding q-dimensional persistent homology groups
with coefficients in F are images of the maps

T In each step of a filtration we

add simplices. The addition of asingle d-dimensional simplex in
one step may either “terminate” anon-trivial homological element ofdimension d 1, create a non-trivial
homological element of dimension d,or have no e↵ect on homology.

(is,t )⇤ : Hq (Ks ; F ) ! Hq (Kt ; F )q

for all 0  s  t  m. The corresponding ranks b s,t = rank(is,t )⇤are called persistent Betti numbers.

q

T Note that b s,t is a non-increasing

function in t and a non-decreasing
function in s.

124

introduction to persistent homology

As is the case with ordinary homology with coefficients in a field,
each persistent homology group is determined9 up to isomorphism
by its Betti number. A single filtration results in a table of persistentBetti numbers.
Example 9.1.3. Given any field F the following are the tables of thezero-dimensional and one-dimensional persistent Betti numbers of thefiltration of Figure 9.3:

t

1

2

3

4

1
2
3
4

2
/
/
/

1
3
/
/

1
2
2
/

1
2
2
2

s

b0s,t !

t

1

2

3

4

1
2
3
4

0
/
/
/

0
1
/
/

0
0
2
/

0
0
1
1

s

b1s,t !

While the rank of (is,t )⇤ determines
the image of the map is,t up to isomorphism, it does not determine a
specific b s,t -dimensional subspace ofHq (Kt ; F ). In this aspect persistenthomology as a specific subgroup of
Hq (Kt ; F ) contains more informationthat persistent Betti numbers, i.e.,
its basis consists of homology representatives spanning the persistenthomology group.
9

Table 9.1: The table of persistentBetti numbers corresponding to
the filtration of Figure 9.3. The
diagonal entries coincide with the
Betti numbers of the corresponding
stages of the filtration. The subdiagonal entries are undefined.

Let us demonstrate how to interpret these numbers geometrically:• b02,3 = 2 means that two of the di↵erent components of K2 are still
disconnected from each other in K3 .
• b13,4 = 1 roughly means that only one homologically non-trivial loopof K3 is still10 homologically non-trivial in K4 .
• b12,3 = 0 means all one-dimensional homology elements in H1 (K2 ; F )are homologically trivial in K3 .

K1

K2

K3

10

A mathematically correct statement would be: the space of onedimensional homology elements in
H1 (K4 ; F ) which have represenatives
in C1 (K3 ; F ) is of dimension one.

Figure 9.3: A filtration K1  K2 
K3  K4 along with the corresponding
Betti numbers of each of the stage
and the zero-dimensional barcode.

K4

0 =2

0 =3

0 =2

0 =2

1 =0

1 =1

1 =2

1 =1

While the tables of persistent Betti numbers are useful, there areother ways to visualize the evolution of homology groups through afiltration. One such visualization we have already presented is thebarcode.

persistent homology: definition and computation

125

9.2 Visualization
Throughout this section we fix a field F, q 2 {0, 1, . . .}, a filtration
K1  K2  . . .  Km = K,
and 1  s < t  m.

BarcodesBarcodes have been geometrically introduced above. In this subsection we will provide their formal definition.
q
Persistent Betti number b s,t represents the dimension of the subspace of homology elements in Kt that have a representative in Ks .
q
Putting it di↵erently, b s,t indicates the dimension of the collection11
of homology elements in Ks that are still non-trivial in Kt in the senseqthat b s,t = dim Hq (Ks ; F )/ ker(is,t )⇤ . Barcodes as indicated above,
however, have more specific information: a bar [s, t) represents a homology element that is born precisely at s and terminates precisely att. Let us phrase this formally:
1. The number of bars containing s and passing through t equals12q
b s,t .
2. Homology born at s is defined as13 Hq (Ks ; F )/(Im is 1,s )⇤ . Its
dimension is b s,s b s 1,s and represents the number of bars startingat s.
3. Homology terminating at t is defined as ker(it 1,t )⇤ . Its dimension14 is b t 1,t 1 b t 1,t and represents the number of bars terminating at t.
4. The quantity b s,t b s 1,t represents15 the dimension of homology
born at s which is still alive at t. It represents the number of bars
starting at s which are passing through t.
5. Quantity ns,t = b s,t 1 b s 1,t 1 ( b s,t b s 1,t ) represents16
the dimension of homology born at s which terminates at t. It
represents the number of bars starting at s and terminating at t.
6. We additionally define ns,• = b s,m b s 1,m , which represents thedimension of homology born at s which is still alive at the end of
the filtration.
The q-dimensional barcode17 consists of intervals18 of the formi. [s, t) for 1  s < t  m, andii. [s, •) for 1  s < m.

11This collection is formally not a
linear subspace. In a formal setting
we represent it as the quotient linear
subspace appearing at the end of thesentence.

12

Through the rest of the section wewill drop the supscript q indicating
the fixed dimension.
For formal reasons we define (i0,t )⇤to be the trivial map.
13

Using the fact that ker(it 1,t )⇤ ⇠=Hq (Kt 1 ; F )/ Im(it 1,t )⇤ .

14

15

Compare to the interpretation of
persistent Betti numbers above.
Also note that b s,t b s 1,t =dim((Im is,t )⇤ / Im(is 1,t )⇤ ), i.e., thedimension of the homology elementsin Hq (Kt ; F ) that have a representative in Ks module the ones that havea representative in Ks 1 .
16

Observation 4. interprets this formula as [the dimension of homology
born at s which is alive at t 1] [the dimension of homology born at s
which is still alive at t] .
17...of the chosen filtration withcoefficients in F...
18
In the setting of a barcode theseintervals will be called bars.

126

introduction to persistent homology

A barcode can have multiple19 copies20 of each interval. Fixing
1  s < t  m:
• The number of the intervals [s, t) is denoted by ns,t .
• The number of the intervals [s, •) is denoted by ns,• .
Example 9.2.1. We again turn our attention the familiar filtrationin Figure 9.3. From the table on the right we can deduce that n2,3 =
1 1 (2 3) = 1 and as a result there is 1 bar of the form [2, 3), as
displayed in the figure. In a similar fashion we compute n1,2 = n1,• =n2,• = 1 and n1,3 = n1,4 = n2,4 = n3,4 = n3,• = n4,• = 0.
A barcode represents the persistences of homology elements. The
longer a bar, the longer the corresponding homology element persists.
In most settings the longer persistence of a homology element alsomeans higher importance21 . However, there are also settings in which
information is contained in shorter bars, especially when the bars are
numerous and specifically distributed.

19...or none...
Alternatively, we could thinkof the barcode as the collection of
all possible intervals of the forms(i) and (ii), each with an assigned
multiplicity from {0, 1, 2, . . .}.
20

t

1

2

3

4

1
2
3
4

2
/
/
/

1
3+
/
/

1+
2
2
/

1
2
2
2

s

b0s,t !

Computing n2,3 of Figure 9.3: thecoloring corresponds to the defining
formula and the subscripts to thesigns within.

21

I.e., a more prominent topologicalfeature.

Persistence diagrams
Another established method of visualisation of persistent homology are persistence diagrams defined as follows. Given a barcode as
defined above we can think of an interval [s, t) as a pair of numbers
and visualize22 it as a point (s, t) 2 R2 . A point of the form (s, •)obviously can’t be drawn in a plane so we choose a y-coordinate abovek, perhaps most conveniently as k + 1, to act as a representative of •,i.e., a bar [s, •) corresponds to a point (s, k + 1). Each point (s, t) ofa persistence diagram has an assigned multiplicity ns,t , which represents the number of bars of the form [s, t). In the case of (s, k + 1), themultiplicity is ns,• .
The result is a collection of weighted points in the plane called apersistence diagram. An example is provided in Figure 9.4.
A barcode encodes precisely the same information as a persistence
diagram. While the persistence of a bar is measured by its length,
the persistence of a point on a persistence diagram is measured by its
distance from the diagonal D = {( x, x ) | x 2 R }. All points of apersistence diagram lie above D.
Persistence diagrams are often the chosen method of visualization
when it comes to representation of persistent homology. Especiallywhen the number of points and bars is large, their distribution seems
to be well represented by persistence diagrams. On the other hand,when the number of points and bars is low, a barcode is often more
descriptive.

22
Just as there can be more bars
with the same endpoints in a barcode, there can be more copies of thesame point visualized at the same
location in a persistence diagram.
While multiple such intervals canbe visualized in a vertical stack, thesame can not be done with points.
For this reason we always consider apoint (s, t) in a persistence diagram
as a weighted point with weight(multiplicity) ns,t .

T Theoretically speaking, if there

existed bars [s, s) of length zero, thenthese would have been the shortest
bars. They would have corresponded
to diagonal points (s, s). This pointof view will come handy in the next
chapter in the context of stability.

persistent homology: definition and computation

K1

K2

K3

127

1

K4

4
3
2
1

1

The fundamental lemma of persistent homology
Numbers ns,t are defined using persistent Betti numbers b s,t . It
turns out that the reverse expression also exists.
Lemma 9.2.2. [Fundamental lemma of persistent homology]
b s,t =

Âns,ts0 s, t0 >tThe formula in the lemma can be verified explicitly. However, thestatement is apparent from the definitions, as• b s,t represents the homology born at s or before and terminating
after t;• ns,t represents the homology born precisely at s and terminating
precisely at t.
Lemma 9.2.2 implies that the information encoded in a barcode
or in a persistence diagram is precisely the same as the information
encoded by persistent Betti numbers.

2

3

4

Figure 9.4: A filtration along withthe corresponding zero-dimensional
barcode and persistence diagram.
The colors of bars match the colorsof the corresponding points in thepersistence diagram.
" In this setting, condition t0 > t

implies that t0 as an index in n could
also take the value •.
T Lemma 9.2.2 has a geometricinterpretation in the context of
persistence diagrams, see Figure 9.5.
It essentially states that b s,t is thesum of all multiplicities of points ofa persistence diagram, which lie inthe upper-left quadrant [0, s] ⇥ (t, •]with the apex at (s, t). In the context
of this interpretation, the formula for
multiplicity ns,t = b s,t 1 b s 1,t 1
b s,t + b s 1,t is the expression of thesquare (s 1, s] ⇥ [t 1, t) in terms ofsuch quadrants.

14

2,3 = 23

9.3 Computation
While the multiplicities ns,t of points of persistence diagrams areformally expressed by persistent Betti numbers, there is an algorithm
to obtain them directly without referring to the Betti numbers andthe corresponding k (k + 1)/2 ranks of maps. In this section we will
present perhaps the simplest23 version of the algorithm, which is also
the most illustrative. We will proceed in two steps:• compute the matrix reduction, and• extract the persistent homology.

2
1

1

2

3

4

Figure 9.5: The sum of multiplicitiesof points in the blue quadrant withapex (2, 3) is b 2,3 by Lemma 9.2.2.
It equals 2 as the point (2, 3) is notcontained in it, see also the table in
Example 9.1.3.
23
There exist many improvementsof this algorithm which may significantly improve the computing
time.

128

introduction to persistent homology

We will conclude the section with an example.
Throughout this section we fix a field F and filtration
K1  K2  . . .  Km = K.
Parameter q 2 {0, 1, . . .} will denote the dimension of a consideredobject.

Matrix reduction
This part could be called an annotated matrix reduction using only
column operation from the left.
1. Order simplices consistently with the filtration. For each q order
all q-simplices in an order in which they appear in the filtration.
If more simplices appear at the same time their internal order isimmaterial24 .
2. For each q construct the boundary matrix Mq using the orderchosen in 1 to label columns and rows.
3. For each q reduce Mq from the left using a single type of column
operations: the addition of an F-multiple of any of the previous25
columns to a treated column. Specifically, starting with the leftmost
column go through all the columns by passing to the right and foreach column:(a) Determine the pivot26 .
(b) If any of the previous columns on the left has a pivot in thesame row, subtract the appropriate multiple of that column sothat the pivot of the current column either disappears or its
location is moved up.

24

It would eventually e↵ect theobtained critical simplices and representatives, but not the persistenthomology.

25

In the chosen order from 1.

26

The lowest non-trivial entry in thecolumn(c) Repeat as long as there are matching pivots on the left.
For each q the resulting matrix is denoted by Mq0 . Each of its
columns is either trivial or has a pivot, whose row is unique amongst
all pivots.

Extracting persistence
At this point we have sufficient information to extract homologyof Km from the number of pivots27 . However, we can also use thelocations of pivots to extract numbers ns,k required to construct thebarcode and persistence diagram. In order to explain the extraction
process we first recall the incremental expansion.
Given a simplicial complex, an addition of a single q-simplex canchange the homology in two ways:

27
Note that the rank of a matrix isthe number of its pivots in a reduced
form, and the ranks themselves
suffice to compute the Betti numbers.

persistent homology: definition and computation• If its boundary is a linear combination of boundaries of otherq-simplices28 , then the simplex gives birth to a non-trivial qdimensional homology element. In this case we call the simplex abirth simplex.
• If its boundary is not a linear combination of boundaries of otherq-simplices29 , then the simplex terminates a non-trivial (q 1)dimensional homology element. In this case we call the simplex aterminal simplex.
A filtration can be considered to be a sequence30 of incremental
expansions. At each stage of the filtration we may assume we firstadd all vertices according to the ordering in 1., then all edges, etc.
Combining such an ordering through all stages we get a sequence ofincremental expansions inducing boundary matrices Mq and their
reduced forms Mq0 .
Based on such an ordering each simplex of K is either a terminal
simplex or a birth simplex. We are now in a position to extract persistence diagram and barcode:• For each terminal q-simplex t there exists a paired birth (q 1)simplex s, which is the label of the pivot in the column t. Such apair induces a bar [s, t) in the corresponding barcode or, equivalently, a point (s, t) in the corresponding persistence diagram, wheres, t are the stages of the filtration at which s and t appear31 .
• Each birth simplex which is not paired to a terminal simplex induces a bar [s, •) in the corresponding barcode or, equivalently, apoint (s, m + 1) in the corresponding persistence diagram, where s is
the stage of the filtration at which s appears.
As a result we obtain a barcode and a persistence diagram as
demonstrated in the example in the last subsection.

Representatives
Occasionally we are also interested in homology representatives ofthe bars and points of persistence diagrams. These can be extracted
from the reduction process. In this subsection we present the mostdirect way of generating representatives. Given a bar with the birth
simplex s and the terminal simplex t we define:• The birth representative of s as the chain formulated32 by the reduction of the column corresponding to s to the zero column in thecolumn reduction scheme. In particular, if the linear combination
turning column s into the zero column in our column reduction
scheme is encoded in terms of columns as ∂s Âi li ∂si = 0, then

129

28

I.e., if, after adding the simplex
to the boundary matrix as therightmost column, its column gets
reduced to the trivial column by theabove reduction.
29

I.e., if, after adding the simplex
to the boundary matrix as therightmost column, its column does
not reduce to the trivial column.
30

A specific sequence should respect
the ordering given by a filtration as
in 1. above, and also the structure ofa simplicial complex, i.e., a simplex
cannot be added before all of its
faces are present.

T As each row contains at most one

pivot, each birth simplex is paired
to at most one terminal simplex. A
terminal simplex cannot appear as
the label of a pivot column.
Note that if s = t we obtainan empty interval in the barcodeand a point on the diagonal in thepersistence diagram, both of which
we ignore in the visualization as they
represent elements of persistence
zero. This is consistent with ourinterpretation of persistent homology,
which measures only holes thatpersist through at least one stage ofthe filtration.
T It turns out that the presented
definition and computation of thebarcode respects the elder rulementioned at the beginning of thechapter.
31

32

For example, in the next subsection we provide an example inwhich the column hc, di is reduced
to the zero column by subtracting the column hb, di and adding
the column hb, ci. This means∂hc, di ∂hb, di + ∂hb, ci = 0 andhence hc, di hb, di + hb, ci is thechain that is our birth representative.

130

introduction to persistent homology

the birth representative is a = s Âi li si . The birth representative
gives a homology class [a] that is born33 by the addition of s.
• The terminal representative is encoded by the column corresponding34 to t in the reduced matrix.
The birth representative and the terminal representative typicallydo not represent the same homology class. The birth representative
may not even be a good representative of the corresponding bar in
the sense that it may remain homologically non-trivial beyond35 theappearance of the corresponding terminal simplex. On the other hand,
infinite intervals do not have a terminal representative. As a result wedefine the representative of a bar as follows:
1. If the bar is a finite interval, the representative of the bar is theterminal representative.
2. If the bar is an infinite interval, the representative of the bar is thebirth representative.
This choice of representatives is algebraically sound in the sensethat the representatives form a basis of the elementary intervals of thedecomposition described in the structure theorem for persistent homology, a result we discuss in details in the next chapter. This statement
includes the fact that the lifespan of each representative matches thelifespan of the corresponding bar, and that the representatives arelinearly independent36 at all times.
In practice we sometimes deviate from the algebraically orthodox
choice when declaring the representatives of 0-dimensional bars: wechoose the birth representative as a bar representative even if the bar
is bounded. Let us explain this geometrically motivated exception on
the example of the next subsection, where pair (hbi, h a, bi) induces a0-dimensional bar. Sometimes we would geometrically like to think ofthis bar as a representation of the component containing b merging
with a larger component, hence the choice of the birth representativehbi which fits into this geometric intuition. However, we should beaware that the homological element [hbi] does not become trivial37
after adding h a, bi. In terms of homology the appearance of h a, bi
identifies38 [h ai] = [hbi] rather than sets [hbi] = 0.

ExampleLet us compute persistent homology of our standard example, see
Figure 9.4. The annotation of simplices we will be using is provided in
Figure 9.6. The chosen order is apparent from the following boundary
matrices, in which vertical and horizontal lines divide simplices fromdi↵erent stages of the filtration.

" Homology class [a] is not the
only homology class born by theaddition of s. If [ b] is another homology class of the same dimension thathas existed before the addition of s,then [a + b] is also a homology classborn by adding s.
34
For example, in the next subsection we provide an example inwhich the column corresponding tohb, c, di in M20 encodes the terminal
representative hb, ci + hd, bi + hc, di.
35
See the discussion on the representatives of 0-dimensional bars below
for an example.
33

U There is no guarantee that these

representatives are geometricallythe most convenient. There are
more involved ways of obtaining
representatives that optimize given
criterion function. For example,
we may want to obtain the shortest
1-dimensional representatives, etc..
T Let us prove that the lifespan of
the homology class of the terminal
representative b corresponds to thelifespan of the corresponding bar:
• [ b] appears by the time s appears
by construction;• if a representative b0 of [ b] appeared before s, then the column
corresponding to t could havebeen reduced further to b0 thus
eliminating the pivot labeled as s,a contradiction.
• [ b] becomes trivial by the time temerges by definition;• if [ b] became trivial sooner, its
expression as a boundary could
be used to reduce the t column to
the zero column, a contradiction.
36...or trivial beyond their lifespans
The terminal representative hbi
h ai does become trivial. In fact, [hbi]never becomes trivial.
38
In this sense, the terminal representative tells us which two components merge.
37persistent homology: definition and computation

0hb, ci hb, dih ai
BhbiB 1
1
B
hciB
1
B
M1 =
hdiB
1
B
B
hei@
hfi

h a, bi hc, di
1
1
1
1

hb, c, dihb, ci1
hb, diB
1
B
B
B
h a, biB
M2 = M20 = hc, diB
B 1
h a, ciBB
B
h a, ei@hb, ei0h a, ci h a, ei hb, ei1
1
1
C
1 C
C
C
1
C
C
C
C
1
1 A
h a, b, ei

1

C
C
C
1 C
C
C
C
C
C
C
1 A
1

We now perform the labelled matrix reduction as described above.

131

d
c
b
f
a

e

Figure 9.6: The annotation of simplices of K.

U Green entries are the pivots.

T In the matrices below a blue col-umn is modified using red columns.

0hb, ci hb, dih ai
BhbiB 1
1
B
hciB
1
B
M1 =
hdiB
1
B
B
hei@
hfi
0hb, cih ai
BhbiB 1
B
hciB
B 1
hdiB
B
B
hei@
hfi

hb, di1h a, bi hc, di1
1
1
1

h a, bi hc, di11

10hb, ci hb, dih ai
BhbiB 1
1
B
hciB
1
B
hdiB
1
B
B
hei@
hfi

h a, bi hc, di1
1

h a, ci h a, ei hb, ei1
1
1
C
1 C
C
C
1
C
C
C
C
1
1 A

U ∂hc, di = ∂hb, di∂hb, ci.

h a, ci h a, ei hb, ei1
1
1
C
1 C
C
C
1
C
C
C
C
1
1 A

U ∂h a, ci = ∂h a, bi + ∂hb, ci.

h a, ci h a, ei hb, ei1
1
C
1 C
C
C
C
C
C
C
1
1 A

U ∂hb, ei = ∂h a, ei∂h a, bi.

132

introduction to persistent homology

0hb, ci hb, dih ai
BhbiB 1
1
B
B
h
c
i
1
B
M10 =
hdiB
1
B
B
hei@
hfi

h a, bi hc, di1
1

h a, ci h a, ei hb, ei1
1
C
C
C
C
C
C
C
C
1
A

We can now extract the barcode from the birth-terminal pairs andunpaired birth simplices. We start by extracting the zero-dimensional
barcode from the pivots of M10 and unpaired vertices.
• Pairs (hci, hb, ci) and (hdi, hb, di) provide no contribution39 .
• Pair (hbi, h a, bi) induces a 0-dimensional (component) bar40 [1, 2)
representedby [hbi].
• Pair (hei, h a, ei) induces a 0-dimensional (component) bar [2, 3)
represented by [hei].

39

Formally, they contribute theempty interval [1, 1) as all involved
simplices appear at K1 .
40
Recall that hbi appears at K1 while
h a, bi appears at K2 , hence the valuesof the endpoints.

• Vertices a and f are unpaired and thus induce 0-dimensional bars
[1, •) (generated by [h ai]) and [2, •) (generated by [h f i]).
We next extract the one-dimensional barcode from the pivots of M20
and unpaired edges.
• Pair (hc, di, hb, c, di) induces a 1-dimensional bar [2, 3) represented41 by [hc, di hb, di + hb, ci].
• Pair (hb, ei, h a, b, ei) induces a 1-dimensional bar [3, 4) represented
by [hb, ei h a, ei + h a, bi].
• Edge h a, ci is unpaired and thus induces the 1-dimensional bar[3, •) generated by [h a, ci h a, bi hb, ci].

Computational tricks
We conclude by mentioning a trick that speeds up the computationof persistent homology. It is based on an observation that boundary
matrices Mi that are being reduced in the reduction process are notcompletely independent of each other.
If the reduction process of Mq reduces the column corresponding
to q-simplex t to a non-trivial column, we can extract the following
information
1. t is a terminal simplex and hence the row corresponding to t in
Mq+1 will have been reduced to the zero-row, which means we canset it to zero immediately.

U In this example the 1-dimensional

birth and terminal representatives
coincide. This is not generally thecase.
41
Recall that hc, di appears at K2
while hb, c, di appears at K3 , hence
the values of the endpoints. The
linear combination that made thecolumn corresponding to hc, di trivial
in M10 was ∂hc, di ∂hb, di + ∂hb, ci
and hence the representative.

persistent homology: definition and computation

133

2. The pivot location reveals the corresponding birth simplex s. As
a result the column corresponding to s in Mq 1 will have been reduced to the zero-column and can hence be set to zero immediately.
T Overview: reducing a t-column toHence a single reduction of a column in Mq corresponding to aterminal simplex also reveals a zero-row in Mq+1 and a zero-column
in Mq 1 . Of course, this information can’t be of much help if it has
already been extracted from previous reductions. For this reasons thematrices Mq can be reduced in the order of decreasing dimension: thisway no column in Mq 1 has been reduced by the time Mq has been
reduced and as a result we avoid reducing almost half42 the columns
resulting in a significant speedup.

a non-zero column reveals:• t is a terminal simplex;• t-row is trivial;• pivot label s is a birth simplex;• s-column is trivial.

42

This estimate depends on thefiltration but seems to hold for mostof the practical cases.

9.4 Concluding remarks
Recap (highlights) of this chapter• Persistent homology;• Barcode;• Persistence diagram;• Computing persistent homology.

Background and applications
Persistent homology43 , 44 , 45 is perhaps the most popular and fruitful construction of topological data analysis. For the past two decades
it has been an inspiration to extensive theoretical and practical treatments, spanning from purely mathematical theoretical foundations46
to computable aspects and applications in numerous fields of scienceand engineering. When coupled with standard constructions of complexes, persistent homology contains information about geometry ofdata. As such, the method is applied whenever the geometric shape ofdata is thought to contain significant information.
Applications include de-noising schemes, dimension reduction
schemes, feature extraction methods, and specific data analysis ofmaterials, molecular structures, medical images, weather patterns, etc.
The combinatorial treatment of this chapter will be followed byfurther properties in the following chapter. While the definition of
persistent homology could have been expressed using coefficients inan Abelian group, the visualizations47 and efficient implementations48
crucially depend on the structure of a field.

43

Herbert Edelsbrunner, David
Letscher, and Afra Zomorodian.
Topological persistence and simplification. Discrete & Computational
Geometry, 28(4):511–533, 2002. doi:
10.1007/s00454-002-2885-2
44
Herbert Edelsbrunner and John
Harer. Computational Topology: An
Introduction. Applied Mathematics.
American Mathematical Society,2010. doi: 10.1090/mbk/069
45
Tamal K. Dey and Yusu Wang.
Computational Topology for Data
Analysis. Cambridge: Cambridge
University Press, 2022. doi:
10.1017/9781009099950
46
We will mention two ideas ofgeneralizations of the standard
persistent homology in the appendix.
47

I.e., barcodes and persistence
diagrams.
48
I.e., matrix reductions.

134

introduction to persistent homology

Appendix: zig-zag persistence and multi-parameter persistence
In this appendix we will sketch the ideas of two generalizations of
the standard persistent homology as presented throughout the chapter.
In both cases the generalization refers to the type of filtration used.
The first generalization is based on zig-zag filtration. While a standard filtration models a growing simplicial complex, a zig-zag filtration
models a changing simplicial complex, in which the simplices may be
appearing or disappearing.
Definition 9.4.1. Let K be a simplicial complex. A zig-zag filtrationof K is a sequence of subcomplexes K1 , K2 , . . . , Km of K, such that foreach i 2 {1, 2, . . . , m 1} either Ki  Ki+1 or Ki+1  Ki .
For example, a zig-zag filtration may be of the following sort:
K1 ,! K2

- K3

- K4 ,! K5

K2

K3

-

,!

K1

K2

K4

-

K3

complexes of a simplicial complex
L not satisfying the condition of a
zig-zag filtration, and we still wantto compute a meaningful zig-zag homology, a standard way to constructa corresponding zig-zag filtration is
to connect them either by unions orintersections:
K1 ,! K1 [ K2
K1

- K2 ,! . . .

- K1 \ K2 ,! K2- Km ,- . . . ,! Km .

Interestingly enough, while the two
options induce generally di↵erent
barcodes, they encode precisely thesame information.

- K6 ,! K7 .

It turns out that even in this setting there exists an algorithm basedon matrix reductions which will produce a well defined barcode49
describing what is called zig-zag persistence. An example is displayedin Figure 9.7.
K1

T In case L1 , L2 , . . . , Lm are sub-K4

The second generalization is based on a multi-parameter filtration. While standard filtration models a one-parameter50 growth ofa simplicial complex, multi-parameter filtration models growth withmore degrees of freedom. For our demonstrative purposes it suffices toformally introduce only a 2-parameter filtration.

49

Or, equivalently, a persistence
diagram.

Figure 9.7: A zig-zag filtration andthe corresponding zero-dimensional
barcode, visualized as a table fortechnical reasons (i.e., the absence ofa designated direction of all arrows
as endpoints of bars). In the same
way we could have presented thebarcodes of ordinary persistenthomology as well.

50

I.e., a sequential.

persistent homology: definition and computation

135

Definition 9.4.2. Let K be a simplicial complex. A 2-parameter filtration of K is a collection of subcomplexes K j,k  K parameterizedwith j, k 2 {1, 2, . . . , m}, such that for each j 2 {1, 2, . . . , m 1}
and for each k the following containments hold (see Figure 9.8):
,!
,!

,!

..
.

,!

,!

..
.

,!

,!

..
.

,!

,!

K1,2 ,! K2,2 ,! K3,2 ,! · · · ,! Km,2,!

There are theoretical and practical settings in which multi-parameter
filtrations arise naturally. A multi-parameter persistent homology is
the object obtained by applying the homology to spaces and maps ofsuch a filtration. Unfortunately, there exists no convenient51 visualization52 in this setting. As a result, theoretical treatments of multiparameter persistent homology typically deal with a multi-dimensional
grid of interconnected homology groups, while practical applicationsof the same object use incomplete information about it such as multiparameter tables of Betti numbers, restrictions to a 1-parameter filtrations yielding a standard barcode, etc. For more details see a book53 .

..
.
,!

• Kk,j  Kk,j+1 .

K1,m ,! K2,m ,! K3,m ,! · · · ,! Km,m,!

• K j,k  K j+1,k , andK1,1 ,! K2,1 ,! K3,1 ,! · · · ,! Km,1

Figure 9.8: A scheme of a 2parameter filtration.
51

While a 1-parameter persistenthomology “decomposes” into simple
pieces called bars (we will explain
this statement in detail in the next
chapter), the pieces of a multiparameter persistent homology canbe quite complicated and not easily
visualized or encoded.
52...such as multi-dimensional barcode or persistence diagram.
53
Tamal K. Dey and Yusu Wang.
Computational Topology for Data
Analysis. Cambridge: Cambridge
University Press, 2022. doi:
10.1017/9781009099950

10
Persistent homology: stability theorem

In the previous chapter we introduced persistent homology andits basic method of computation in the discrete setting. However, itturns out that the concept of persistent homology can be treated on amuch deeper theoretical level, through which many of its advantages
become apparent.
In this chapter we will delve further into the theoretical machinery of persistent homology. We will introduce continuous filtrations
and the underlying algebraic structure of persistence modules. These
structures will be crucial in the formulation of the stability theorem,
which states that, unlike homology, persistent homology behaves continuously with respect to the underlying filtration. We conclude bymentioning a series of interpretations and examples of our expanded
scope of persistence.

10.1 Continuous filtrations
Recall that a discrete filtration of a simplicial complex K is a sequence of subcomplexes
K1  K2  . . .  Km = K.
An example of a filtration is given in Figure 10.1.
K1

K2

K3

K4

Discrete filtrations1 formalize finite nested sequences of complexes.

Figure 10.1: A discrete filtration.

1I.e., filtrations given by finitely
many nested simplicial complexes

138

introduction to persistent homology

While this approach is geometrically intuitive, there is an alternative
shorter description of a filtration.
Rather than storing a sequence of separate subcomplexes, we annotate each simplex s 2 K by the index f (s ) at which s first appears.
Given such an annotation it is easy to reconstruct Ki = {s 2 K |f (s )  i }. See Figure 10.2 for an example. This observation motivates
us to expand the scope of filtrations in two ways: by considering continuous filtrations with infinitely many subcomplexes; and by defining
filtrations from an appropriate annotation function.
1. A continuous filtration of a finite simplicial complex K is a collection of subcomplexes2 {Kr }r 0 of K such that

8r < q : Kr  Kq  K.
2. Given a simplicial complex K, let f be a filtration function, i.e., anannotation of each of the simplices of K by a non-negative number
such that s  t =) f (s )  f (t ). The sublevel filtration associated to f is a continuous filtration consisting of sublevel complexes3
Kr = {s 2 K | f (s )  r }  K for r 0.
There are two motivating reasons for the introduction of continuous
filtrations:• Most of the standard constructions of filtrations actually yieldcontinuous filtrations: Cech filtration4 , Rips filtration5 , filtration byalpha complexes, sublevel filtration, etc.
• The interleaving structure and the resulting stability theorem depend6 on the continuous choice of parameter.
The definition of persistent homology groups for continuous filtrations is the same, with the only di↵erence being the continuous rangeof the indices 0  s  t, which results in nominally infinitely many
persistent homology groups. The matrix-reduction based computationof persistent homology is unhindered by the expansion to continuous
filtrations as the computations depend only on filtration function defined on (finitely many) simplices, and do not require all (infinitely
many) complexes of filtration separately. In a nutshell, we can compute the same barcodes using the same procedure for continuous ordiscrete filtrations.
We next discuss the interpaly between discrete and continuous
filtrations:
1. Given a discrete filtration, there is an obvious extension of it as thesublevel filtration of the annotation function.

1
2
1

1

3
1

3

1
2

1

4

3
2

3
2

Figure 10.2: The annotation of
simplices encodes the filtration of
Figures 10.1.
2

Throughout the book we will additionally require that for each simplex
s 2 K the minimum argminr {s 2 Kr }exists, i.e., there exists the smallest
scale r at which s appears. An equivalent condition is the following: foreach r there exists r 0 > r such thatKr = Kr0 , i.e., if a simplex is absent
at scale r, it is also absent at slightly
larger scales.
Under this condition each continuous filtration is the sublevel filtrationof its associated annotation function. In particular, each sublevel
filtration is a continuous filtrationand vice versa. The Rips and Cech
filtrations as defined in Chapter 5 arecontinuous filtrations of this sort.
In this setting parameter r is often
referred to as the scale, a notion arising from Rips and Cech filtrations,or the level, a notion arising from thefiltration function.
3

4

The corresponding filtration function on simplices being the radiusof the smallest enclosing ball of thevertices.
5
The corresponding filtration function on simplices being the diameterof the set of vertices.
6

To be discussed in detail later. In anutshell, the continuous choice of theparameter eventually results in thecontinuity of persistent homology.

persistent homology: stability theorem

2. Given a continuous sublevel filtration {Kr }r 0 associated to a filtration function f there are two ways of generating a discrete filtration:

Figure 10.3: An excerpt from theRips filtration on the five points on
the left.

(a) By restriction to K1  K2  . . .  Kdmax f e . While mathematically convenient, this approach has many drawbacks7 and ismostly avoided.

7(b) A more beneficial way of thinking about the index i of a discrete filtration is not as the scale parameter8 but rather as theindex of the critical scale9 of the continuous filtration. Formallyspeaking we define critical scales r1 < r2 < . . . < rk as theenumeration10 of the image of f and define

8

The corresponding continuous
filtration as defined by 1. may besignificantly di↵erent from {Kr }r 0 .
The information about the sequenceof changes between each pair of
integer scales is lost.
An interpretation prevalent in thecontext of continuous filtrations.
9
A scale r of a continuous filtrationis critical, if at least one simplex
appears at r.
10{r1 , r2 , . . . , rk } = Im f .

Ki = { s 2 K | f ( s )  r i }.
The corresponding finite filtration contains information about allchanges in the original continuous filtration.

0

1/2
1/

r=0

r = 1/2pr = 1/ 3

0

p

1/2

T From this point on, whenever wemention an unspecified filtration, or
consider a transition from a finite tocontinuous filtration or vice versa,
the underlying interplays we havein mind are 1. and 2. (b). For an
example see Figure 10.4.

Continuous filtrations conveniently model the geometric setup of
the standard filtrations. On the other hand, discrete filtrations area convenient finite description on which we may develop algorithmic
approaches.

1/2

139

3

0

Example 10.1.1. [Topology of o↵sets] Given a finite collection of points
S ⇢ R n we have already mentioned that the nerve theorem implies thatfor each r > 0 the Cech complex Cech(S, r ) is homotopy equivalent to
the r-neighborhood11 N (S, r ) of S:

Figure 10.4: The Cech filtration on
three vertices forming an equilateraltriangle of side length 1 nominallyconsists of infinitely many simplicial
complexes. However,
only at scalesp0, 1/2 and 1/ 3 do the changes
occur and hence the corresponding
discrete filtration (according to 2(b) above) consists of simplices at
those scales, depicted by the first
three complexes in the figure. The
annotation function is provided on
the right, its image consists of thementioned scales.
11

Also called the r-o↵set of S.

140

introduction to persistent homology

Cech(S, r ) = N ({ B(s, r )}s2S ) '[B(s, r ) = N (S, r ).

s2S

It turns out that the conclusion of the nerve theorem behaves consistently12 with the maps, which results in the following fact: if for
r1 < r2 the inclusion N (S, r1 ) ,! N (S, r2 ) is a homotopy equivalence,
then13 so is the inclusion Cech(S, r1 ) ,! Cech(S, r2 ).
The Cech filtration thus models the homotopy type of growing o↵sets: if on some interval the growth of r results in homotopy equivalent
growth of o↵sets, then it also results in a homotopy equivalent growthof the Cech complexes.

12

The formal term corresponding tothis consistency is “functoriality” andthe relevant extension of the nerve
theorem is referred to as “persistentnerve theorem” or “functorial nerve
theorem”.
13
At this point we crucially use thefact the Euclidean balls always forma good cover as required by the nerve
theorem.

Interleaving distance for filtrations
We conclude this section by recalling the interleaving distance between filtrations. The concepts has already been defined in Chapter 5
for Rips and Cech filtrations. With the established general notions we
can use the same definition for filtrations in general.
Definition 10.1.2. Choose # > 0. Continuous filtrations {Kr }r 0 and{ Lr }r 0 are #-interleaved if there exist simplicial maps jr : Kr !
Lr+# and yr : Lr ! Kr+# such that jr+# yr : Lr ! Lr+2# and yr+#
jr : Kr ! Kr+2# are equal to the corresponding inclusions.

···

/ Kr

···

/ Lr

jr
yr

/ Kr + #=/ Kr+2#;!
/ Lr + ##/ Lr+2#/ ···T Suppose filtrations {Kr }r 0 and{ Lr }r 0 consist of subcomplexes of asimplicial complex K and let #0.
It is easy to see that if for each r wehave Kr  Lr+# and Lr  Kr+# , thenthe filtrations are #-interleaved. An
argument of this sort is used in the
proof of Proposition 10.1.3.

/ ···Given two filtrations their interleaving distance is defined as theminimum14 of all values # > 0, for which the filtrations are #-interleaved.
It turns out that the interleaving distance is a metric15 .
In Chapter 5 we proved that Rips and Cech filtrations equippedwith the interleaving distance are continuous (stable) with respect
to perturbations of the underlying points. Generalizing this result
we now prove the sublevel filtrations are continuous with respect toperturbations of the filtration function in the max metric16 .
Proposition 10.1.3. Let K be a simplicial complex. Assume f , g : K !
[0, •) are filtration functions. Then the sublevel filtrations of K corresponding to f and g are || f g||• interleaved.
Proof. In order to align our notation with the diagram above for# = || f g||• define Kr = {s 2 K | f (s )  r }  K and Lr = {s 2 K |g(s )  r }  K. The interleaving maps j, y are defined to be identities
on vertices. The maps are well defined by the following argument:

14

It is not hard to prove that theminimum exists due to the additional requirement imposed on ourfiltrations.
15
In order to maintain this view wedeclare two filtrations to be isomorphic if they are 0-interleaved. Theinterleaving distance is a metric on
the isomorphy classes of filtrations.
16
Given two functions f , g : K ! R
defined on all simplices of a finite
simplicial complex K, the max distance between them is|| fg||• = max | f (s)s2K

g(s)|.

persistent homology: stability theorem

141• For each vertex v 2 K: if v 2 Kr then v 2 Lr+|| f g||• by thedefinition of the max distance. In a similar fashion, if v 2 Lr thenv 2 Kr+|| f g||• . Hence maps j, y are well defined on vertices.
• The same argument for simplices17 implies maps j, y are simplicial.

For example, if a simplex s 2 K is
contained in Kr , it is also containedin Lr+|| f g||• .
17

10.2 Persistence modules
Persistent homology is obtained by applying homology to a filtration. In this section we present the properties of the resulting algebraic
objects (persistence modules) which model persistent homology. Just
as filtrations model the growth of simplicial complexes, persistence
modules model the evolution18 of vector spaces19 .
For the rest of this chapter we fix a field F, which will provide
coefficients to all mentioned vector spaces, including homology groups.

18

I.e., not only growth.
Which we interpret as holes in thecontext of persistent homology
19

Persistence modules
Definition 10.2.1. A persistence module is a collection of (finite
dimensional) vector spaces {Vr }r 0 along with linear maps
hr,q : Vr ! Vq ,8r  qsatisfying hr,q = hr,s hs,q and hr,r = idVr for all r  q  s.
Scale r0 is said to be regular if there exists # > 0 such thatmaps h p,q are isomorphisms for all p, q 2 (r #, r + #) or (in thecase r = 0) for all p, q 2 [0, #), i.e., the maps h are isomorphisms
close to r. Scale r is critical if it is not regular.
Our interest in persistence modules stems from the fact that they
are the underlying algebraic structure of persistent homology of continuous filtrations. In order to simplify our treatment we thus restrict
to persistence modules that appear as persistent homology of continuous filtrations as defined above. In particular, each persistence module
treated here will be assumed to have the following properties:
1. There exists R > 0 such that for each R  r < q maps hr,q areisomorphisms20 , i.e., eventually all maps h are isomorphisms.
2. For each r > 0 there exists r 0 > r such that for all q 2 [r, r 0 ) themaps hr,q are isomorphisms21 .
3. There exist finitely22 many critical scales.

T In general literature persistence

modules may consist of infinite
dimensional vector spaces.

T Critical scales of a continuous

filtration are a supset of criticalscales of its persistent homology as
any change in homology requires achange of the underlying complex,
but not vice versa.

T Properties 2. and 3. imply [0, •)can be decomposed into finitely many
intervals of the form [⇤1 , ⇤2 ) on
which all maps h are isomorphisms.
20
An analogous property holds forcontinuous filtrations as they filter afinite simplicial complex, i.e., given
a filtration function f , all sublevelcomplexes Kr for r > max | f | coincide.
21

This corresponds to the analogous
property assumed for our continuous
filtrations.
22This property corresponds to thefact that continuous filtrations filtera finite simplicial complex.

142

introduction to persistent homology

Definition 10.2.2. Persistence modules {Vr }r 0 and {Wr }r 0 are isomorphic if for each r 0 there exist isomorphisms Vr ! Wr suchthat for each 0  r1 < r2 < . . . the following diagram commutes

···

/ Vr

j

/ Vr

j +1

⇠
=

···

✏
/ Wr
j

/ Vr

j +2

⇠
=

✏

/ Wr

⇠
=

✏

j +1

/ ···

/ Wr

j +2

/ ···

Decomposition
It is often advantageous to decompose23 mathematical objects
into simple pieces and thus obtain a canonical form. In the previous
chapter we decomposed persistent homology into pieces represented
by bars. In this subsection we will formalize such a decomposition for
persistence modules.
We first explain what we mean by a “decomposition”.
Definition 10.2.3. The direct sum of persistence modules {Vr }r 00 , is a perand {Vr0 }r 0 along with respective linear maps hr,q and hr,qsistence module consisting of:• spaces Wr = Vr

23

Functions are decomposed intomonomials (Taylor series) or trigonometric functions (Fourier series).
Closed connected surfaces otherthan the sphere can be decomposed
as a direct sum of tori or projective planes. Every n-dimensional
vector space is of a form F n and in
one of the previous appendices wementioned how finitely generated
Abelian groups can be decomposed
into smallest indecomposable groups:
groups of the form Z p and Z.

Vr0 and0 )• maps h̃r,q = (hr,q , hr,q

for all 0  r < q.
We next present interval modules, which are the pieces represented
by bars.
Definition 10.2.4. Let 0  p < q. An interval module F p,q corresponding to the pair ( p, q) is a persistence module {Vr }r 0 defined
as follows:• Vr = F for r 2 [ p, q) and Vr = 0 else.
• Maps hs,s0 are isomorphisms whenever possible.

T Rewriting the condition on maps

in Definition 10.2.4:• for p  s  s0 < q map hs,s0 is theidentity on F.
• else hs,s0 is the zero map.
T It is easy to verify that interval

Theorem 10.2.5. [Structure theorem for persistent homology] Eachpersistence module is isomorphic to a direct sum of interval modules.
The decomposition is unique up to the permutation of the intervals.

modules F p,q and F p0 ,q0 are isomorphic i↵ p = p0 and q = q0 .

persistent homology: stability theorem

Barcodes and bars introduced in the previous chapter correspond to
this decomposition and interval modules. Theorem 10.2.5 is an algebraic expression of the existence of barcodes. It states that the persistence module can be decomposed into the intervals and is completely
determined24 by the interval modules of its decomposition.

Interleaving distance for persistence modules

143

24

And as a result, barcodes andpersistence diagrams are complete
descriptions of persistence modules.

The interleaving distance has already been defined for filtrations.
Conceptually the same definition applies to persistence modules.
Definition 10.2.6. Choose # > 0. Persistence modules {Vr }r 0 and
0 are #{Wr }r 0 along with their respective linear maps hr,q and hr,qinterleaved if there exist linear maps jr : Vr ! Wr+# and yr : Wr !
Vr+# such that jr+# yr : Wr ! Wr+2# and yr+# jr : Vr ! Vr+2#0
are equal hr,r+# and hr,r +# correspondingly.
Given two persistence modules their interleaving distance d I isdefined as the minimum of all values # > 0, for which the filtrations
are #-interleaved.

···

/ Vr

···

/ Wr

jr
yr

/ Vr+#</ Vr+2#;"/ Wr+##/ Wr+2#

/ ···
/ ···

It is not hard to prove that the minimum in the definition of theinterleaving distance exists due to the additional requirement imposedon persistence modules. It is easy to verify that the interleaving distance is a metric on the isometry classes of persistence modules. As
such the interleaving distance is the metric25 of choice on persistenthomologies.
The functoriality of homology implies that #-interleaved filtrations26
induce #-interleaved persistence modules. Another setting in which#-interleaved persistence modules (but not necessarily #-interleaved
filtrations) are obtained is that of spaces, which are “close” to eachother. Let us first define closeness.
Definition 10.2.7. Let ( X, d) be a metric space and assume A, B ⇢X are finite subsets. The Hausdor↵ distance d H ( A, B) is definedasd H ( A, B) = max max min d( a, b), max min d( a, b) .
b2 B a2 A

a2 A b2 B

The Hausdor↵ distance is a metric on all finite subspaces of a metric space X. It has a natural geometric meaning. Given the setting ofDefinition 10.2.7 find:25

At this point it should be clearthat continuity and small perturbations of persistent homology dependon the ability to perform continuousand small steps in the index set.
An interleaving distance defined on
persistent homology of discrete filtrations or a single complex would havebeen, in the best of cases, restricted
to the integer values, that do notaccommodate the idea of continuity.
26
We have already discussed how
these appear by perturbing points
when using Rips or Cech complexes,
and by perturbing the filtration
function when using the sublevel
filtration.

144

introduction to persistent homology• The minimal r A such that N ( A, r A )of A contains B.

B, i.e., the r A -neighborhood• The minimal r B such that N ( B, r B )A.

A

We conclude that d H ( A, B) = min{r A , r B }. Note that for each a 2
A there exists b 2 B such that d( a, b)  d H ( a, b), and vice versa.
An example is given in Figure 10.5, where a black set A and a red
set B are displayed on top, while their respective neighborhoods are
displayed in the middle and on the bottom.
Hausdor↵ distance measures the distances between finite subspacesof a metric space and heavily depends on a way in which these subspaces are embedded. For example, di↵erent isometric subspaces will
be at a positive Hausdor↵ distance.
Similar to Hausdor↵ distance is the Gromov-Hausdor↵ distance.

B

N (A, rA )N (B, rB )Definition 10.2.8. Suppose A, B are finite metric spaces. The GromovHausdor↵ distance dGH ( A, B) is defined as

Figure 10.5: d H ( A, B) = r B > r A .

dGH ( A, B) = inf{d H (µ( A), n( B))},µ,nwhere the infimum is over all isometric embeddings µ : A ! X andn : B ! X into a metric space X.
It turns out that the infimum in Definition 10.2.8 is always attained
and that dGH is a metric on the isometry classes27 of finite metricspaces.
Proposition 10.2.9. Let A, B be finite metric spaces with # = dGH ( A, B).
Then for each q 2 {0, 1, . . .}:1. { Hq (Rips( A, r ))}r 0 and { Hq (Rips( B, r ))}r 0 are 2#-interleaved.

T Observe that d GH ( A, B)  d H ( A, B)

for finite subspaces of a metric space
X. As a result Proposition 10.2.9
also holds for d H . However, GromovHausdor↵ distance is typically harder
to compute and thus it is occasionally more convenient to use d H as
the easily computable parameter ofinterleaving.
27
In particular, dGH ( A, B) = 0 i↵ thespaces are isometric.

2. { Hq (Cech( A, r ))}r 0 and { Hq (Cech( B, r ))}r 0 are #-interleaved.
Proof. We will only sketch the proof for q = 1 and Rips filtrations.
The proof of other cases follows the same idea but requires some technical diligence. Without loss of generality we may assume A and B aresubspaces of a metric space X and # = d H ( A, B).
We aim to define maps j and y that constitute a commutative
diagram:···/ H1 (Rips( A, r ))···/ H1 (Rips( B, r ))jryr

/ H1 (Rips( A, r + 2#))
5

/ H1 (Rips( A, r + 4#))5)
/ H1 (Rips( B, r + 2#)))
/ H1 (Rips( B, r + 4#))

/ ···
/ ···

persistent homology: stability theorem

145

We first define maps on the vertices of the Rips complexes:• For each a 2 A choose ba 2 B such that d( a, ba )  # and define
jr ( a) = ba , 8r.
• For each b 2 B choose ab 2 A such that d(b, ab )  # and define28
yr (b) = ab , 8r.
Given a 1-cycle a = Âi h ai , ai+1 i in Rips( A, r ) define jr ([a]) =[Âi hbai , bai+1 i]. This gives well defined maps jr (and also yr ) by thefollowing arguments:

28

As defined, maps jr and yr do not
define an interleaving between the
Rips filtrations as in general aba 6= a.

• Âi hbai , bai+1 i is a cycle in Rips( B, r + 2#) as d( ai , ai+1 )  r impliesd(bai , bai+1 )  r + 2#.
• If [Âi h ai , ai+1 i] = [Âi h ai0 , ai0+1 i] holds29 , then j([Âi h ai , ai+1 i]) =j([Âi h ai0 , ai0+1 i]) as well30 .
At last we need to show that
h
i h
i
00 00
h
a
,
a
i
=
h
a
,
a
i
i
i
+
1
Â
Â i i +1iThis means Âi h ai , ai+1 i
Âi h ai0 , ai0+1 i = ∂ Â j h x j , y j , z j i.

29This holds as Âi hbai , bai+1 i
Â i h b a 0 , b a 0 i = ∂ Â j h b x j , by j , bz j iii +1
and hbx j , by j , bz j i are triangles inRips( B, r + 2#).
30iin H1 (Rips( A, r + 4#)) where ai00 = aba . First note that d( ai , ai00 ) i2#, 8i. The di↵erence Âi h ai , ai+1 i Âi h ai00 , ai00+1 i is a boundary as
demonstrated by the blue 2-chain in Figure 10.6.

a00i
a00i 1ai 1

10.3 Bottleneck distance and stability theorem
The many versions of the stability theorem for persistent homology
state that persistent homology is continuous with respect to continuous change of the input parameters31 . We have already seen examplesof this sort: through Propositions 10.2.9 and 10.1.3 we can conclude
that persistent homology behaves “continuously” in the interleaving
distance. One of the main advantages of persistent homology is its
visualization and so the final step towards a geometrically convenient
form of the stability theorem is to interpret32 the interleaving distance in geometric terms as a distance on persistence diagrams33 . The
resulting distance on persistence diagrams is called the bottleneck
distance.

Bottleneck distance
We start by explaining notions and setting needed to define the bottleneck distance. Suppose A = ( a1 , a2 , . . . , am ) and B = (b1 , b2 , . . . , bn )are persistence diagrams, i.e.:a00i+1ai
ai+1Figure 10.6: An excerpt from theproof of Proposition 10.2.9. Eachedge connects points at distance at
most r + 2#.
31

With various versions discussing
various forms of input.

32

A brief idea about a transition
from the interleaving distance to thebottleneck distance is provided inappendix.
33
For this setting, the visualization
with persistence diagrams is muchpreferred to the visualization withthe barcodes.

146

introduction to persistent homology• each ai and bi is a point above the diagonal in the first quadrant in
the plane, and• each point may appear multiple times in any of the diagrams.
For a point v = ( x, y) 2 R2 let34 v̄ = (( x + y)/2, ( x + y)/2) 2 R2 .
A partial matching between A and B is a bijective map j : A0 ! B 0
where35 A0 ✓ A and B 0 ✓ B . The matching distance of such a j isdefined asnod M ( j) = max max{d• (v, j(v))}, max {d• (v, v̄)}, max {d• (v, v̄)} .
v2A0

v2A\A0v2B\B 0Let µ(A, B) denote the collection of all partial matchings between A
and B .

v̄ represents the point on the
diagonal D = {(z, z) | z 2 R } which isthe closest to v in d• (and also in d2 )metric.
35
Again, a point can appear in A0 or
0
B multiple times but not more times
than in A or B respectively.
T Recall that the max distanced• (( x1 , y1 ), ( x2 , y2 )) between points inthe plane is defined as34

max{| x1x2 |, | y1y2 |}.

Definition 10.3.1. The bottleneck distance between persistence diagrams A and B is the minimal matching distance between them, i.e.,d B (A, B) =min

j2µ(A,B)d M ( j ).

Examples of partial matchings are given in Figure 10.7. In orderto demonstrate the additional pairs used in the definition of the bottleneck distance, the unmatched points are connected to the closest
point on the diagonal. The matching with the smallest matching distance is the second from the left, a fact that can be verified in Figure
10.8, which illustrates the matching distances for matchings of three
diagrams of Figure 10.7. The d• ( a, b) distance between points a andb can be thought to represent one half of the side-length of the squarecentered at a which has b on its boundary. The maximal length ofsuch sides is the smallest in the second case and the resulting quantity
is the bottleneck distance d B .

Figure 10.7: Examples of partial
matchings between the red and theblue persistence diagrams with points
unmatched by j being matched tothe closest diagonal point.

T At this point it should become

apparent why it is geometricallyconvenient to consider points on
the diagonal represent the trivial
persistence module. A side e↵ect of
this approach is that any two points
on the diagonal represent the same
trivial persistence module. In a way,the entire diagonal should thus betreated as a single point.

persistent homology: stability theorem

dB

147

Figure 10.8: The distances between
the matched pairs are demonstrated
by the squares arising as the ballsof the d• metric. The diagramwith the smallest maximal squareamongst all matchings (even the ones
not displayed here) is the middle
one. Hence the resulting bottleneck
distance d B arises from the middle
diagram.

2"

Theorem 10.3.2 (Isometry theorem). The interleaving distance between persistence modules equals the bottleneck distance between thecorresponding persistence diagrams.

"

Example 10.3.3. Let A be the persistence diagram presented by thefour blue points in Figure 10.9. If persistence diagram B satisfies
d B (A, B)  #, then B consists of the following:• For each blue point there exists one designated36 red point within
the grey square (i.e., the #-ball in d• ) around it.
• Arbitrarily many points within the grey #-band37 along the diagonal.

Figure 10.9: A schematic representation of the #-neighborhood of the
diagram consisting of blue points as
discusses in Example 10.3.3.
36

Stability theorem
Theorem 10.3.4. [Stability theorem] Assume persistence diagrams A
and B represent persistent homologies of filtrations V and W obtainedby one of the following procedures:
1. As the sublevel filtrations of filtration functions f and g satisfying condition || f g||•  #, see Proposition 10.1.3.
2. As the Rips filtrations of metric spaces X and Y satisfying condition dGH ( X, Y )  #/2, see 1. of Proposition 10.2.9.
3. As the Cech filtrations of metric spaces X and Y satisfying condition dGH ( X, Y )  #, see 2. of Proposition 10.2.9.
Then d B (A, B)  #.
Figure 10.10 is a schematic representation of the discussion leading
to the stability theorem as presented here.
The moral of the theorem is that small perturbations of the input
lead to small changes in persistence diagrams. On the other hand,critical simplices and homology representatives may be unstable.

If some of the squares had nonempty intersection, then within thatintersection there might be morepoints of B , so a single square might
contain more red points. However,
one of them, potentially a diagonal
point, has to be the designated one,i.e., the point to which the blue point
in question is matched in an optimal
matching.
37This band is actually the #neighborhood of the trivial (empty)
persistence diagram. Again, this
implies that the squares intersecting
the band may contain more points.
T The stated version combines sev-

eral separate version of the stability
theorem found throughout the literature by stating several di↵erent
initial assumptions.
T While the presented results explain
stability in terms of the bottleneck
distance, there is another family of
distances on persistence diagrams
called the Wasserstein distances.
For example, the 1-Wasserstein
distance is obtained by defining thematching distance as the sum (rather
than max) of individual terms.
Under appropriate assumptions thepersistence diagrams are also stable
when using Wasserstein distances.

148

"

introduction to persistent homology

interleaving offiltrations"
functoriality

V and W

interleaving ofisometry
theorem

persistence
modules

Filtration

Rips

Cech

functionsfiltration

filtration||fdGH (X, Y )  "/2

dGH (X, Y )  "g||1  "10.4 Interpretations and examples

dB (A, B)  "Figure 10.10: The diagram summarizing the stability theorem andstrategy of its proof that have been
discussed.

With the stability theorem, persistent homology may be thoughtof as a stable description of a geometric shape. Stability itself justifies
the following observations:• Given a geometric shape, ever better approximating point-clouds
induce persistent diagrams ever closer38 (converging) the the persistence diagram of the shape.
• The points of higher persistence (i.e., the longer bars in the barcode) represent more stable39 features and are thus typicallydeemed to be of higher importance, leading to simplification schemes
on data, such as denoising.

Given a closed manifold X, asufficiently small scale r, and a pointcloud S sufficiently close to X in dGH ,the corresponding Rips and Čechcomplexes are homotopy equivalent
to X. As a result, the persistenthomology around scale r of a good
approximation reveals homology of
X.
38

39

In this section we will present several examples40 of persistent homology arising via Rips complexes and comment on their structure.

I.e., they remain non-trivial underlarger perturbations.
40

Generated by Ripserer.jl, withcoefficients in Z2 .

1-dimensional persistence of geodesic spaces
Let X be a closed41 geodesic manifold or, more generally, the bodyof a finite simplicial complex equipped with a geodesic metric. Assume
Sn is a sequence42 of finite metric spaces converging towards X in the
Gromov-Hausdor↵ metric. Let An denote the 1-dimensional persistence diagram obtained from Sn via Rips filtration and coefficients in
F. It turns out that the limiting diagram43 A = limn!• An encodesa shortest base of H1 ( X; F ): for each member44 a of a shortest homology base of X we obtain a bar [0, |a|/3), where |a| is the lengthof a, see Figure 10.11. Without going through all the details let us
demonstrate the situation through examples.

41

This implies it admits a finite
triangulation.
42

Such a sequence may be, roughly
speaking, obtained by constructing
ever finer finite approximations of X
and inducing an approximation of ageodesic metric on them.
43
A can be obtained as persistence
diagram of the Rips filtration of X, aconstruction which involves infinite
simplicial complexes and is formallybeyond the scope of this book.
44
Members are formally cycles whose
length in this case is the length ofthe corresponding loop in X. One
can choose a triangulation for whichthese simplicial loops are the shortest
possible.

persistent homology: stability theorem

Figures 10.12 and 10.13 represent three surfaces in R3 approximated by a finite collection of points. An approximation of a geodesic
metric is induced on the points and used to compute 1-dimensional
persistent homology. The right part of the figures represents thelongest one or two bars obtained from each of the computations.
Starting with a discrete set of points, a multitude of short bars is
also generated, but are the artefact of a finite approximation rather
than topologically significant features. By the stability theorem theirlifespans decrease (although their numbers increase) as we improve theapproximation density.
Let us interpret the results:
1. Our chosen samples are dense enough for the longest bars to detect the shortest 1-dimensional homology bases, which in this casecoincide for all coefficients.
2. The bars would ideally be born at 0 and run until45 one-third of
the lengths of the corresponding homology generators. With increased density the resulting barcode would approach this scenario.
3. The visualizations of approximating points also contain a loop or
two: these are obtained by connecting the vertices of the criticaltriangles by the shortest paths through our points. For bars corresponding to the basis, this gives an approximation of the shortest
homology basis. In the spirit of the stability theorem, the finer theapproximation by points, the closer approximation of the loops we
obtain.

149

↵

|↵|/3

| |/3

Figure 10.11: The 1-dimensional
persistent homology of a torus
detects its shortest homology basis.
45
Žiga Virk. 1-dimensional intrinsic persistence of geodesic spaces.
Journal of Topology and Analysis, 12(01):169–207, 2020. doi:
10.1142/S1793525319500444

Figure 10.12: The longest bar of
1-dimensional persistent homology.

4. Going beyond the basis, we see that the next bar in Figure 10.13
detected a hole in our approximating points. The lifespan of thisbar would decrease towards zero with ever better approximations.

150

introduction to persistent homology

Figure 10.13: The longest two bars of
1-dimensional persistent homology.

Stability demonstrated
Figure 10.14 contains four approximations of a circle by discretesets and the corresponding persistence diagrams arising from the Rips
filtration. The stability theorem states that the induced persistence
diagrams should be close to each other, and the figure demonstrates
this is indeed the case.
A few comments on the diagrams:
1. There is a point (0, 3) representing (0, •) indicating the one persistent component.
2. The main dominating feature is the very persistent point terminating at around 1.5. It represents the 1-dimensional hole, i.e.,the homology of S1 . Its precise coordinates tell us more about thegeometry of the sample.
• The birth is between 0 and .5. The precise birth depends onthe edges of the Rips complex going “around the circle”46 . The
maximal gap needed for such circumcision is the birth time.
We can see that such a gap is smallest in the upper-right caseresulting in early birth. On the other hand, the gap is largest in
the upper-left case47 and results in a later birth.
• The terminal scale of this feature is the minimal diameter of an“almost equilateral triangle” reaching “around the circle”.
3. The other points on persistence diagrams are of low persistence andappear48 as an artefact of discretization.

46...and thus generating the 1-cycle.

47

The gap of this sample appears in
the upper-right part48

For example, as the Rips complex
on n points is a discrete collection ofn points at small scales, each suchdiagram will have n many points
indicating persistent 0-dimensional
homology.

persistent homology: stability theorem

151

Figure 10.14: Four approximationsof a circle and the corresponding
persistence diagrams.

Spheres
We next present examples approximating spheres. In Figure 10.15
we present persistence diagrams via Rips complexes of a sample of 100
points on unit spheres: S2 (on left) and S3 (on right); using Euclidean(top) or geodesic distance (bottom).
3

S 3 (1)Figure 10.15: Persistence diagrams
via Rips complexes of samples of one
hundred points sampled from unit
S2 (on left) and S3 (on right) using
Euclidean or intrinsic (geodesic)
distance.

As the only non-trivial homology (except for dimension 0) of S2 is
H2 (S2 ; F ) ⇠= F, we expect a long persistent line in that dimension,
which is indeed the case. In fact, the long 2-dimensional bar clearly
indicates that in both cases the most prominent homology is of rank 1
5
in dimension 2. The same holds for S3 although a denser
sample would
4

T In persistence diagrams in Fig-ure 10.15 there are a lot of shortbars. Some of these are artefactsof discretization, other indicate amore complex structure of persistenthomology reaching beyond the interpretation of the size of homology
representatives of the underlying
space. Interpreting such bars is avery active research topic.

152

introduction to persistent homology

have made the same observation easier in the geodesic case.
To demonstrate the improvement induced by larger density we
present in Figure 10.17 a sequence of diagrams with increasing density.
The underlying space is a cut-o↵ unit sphere, i.e., a 2-dimensional
sphere with a cap above the parallel of circumference approximately
1.5 removed, see Figure 10.16. We take a sample of 100, 200, and 400
points, generate a geodesic distance, and generate persistence diagrams
via Rips filtrations.
Here is what we would expect from resulting persistence diagrams:
1. The cut-o↵ sphere is contractible for small scales and thus initial
clutter of 1-dimensional bars should be decreasing in size as we
increase density.
2. At a certain scale an o↵set of the cut-o↵ sphere will fill in the top
and create a void and hence49 a 2-dimensional homology in the
Cech complex. As Rips complexes are interleaved with Cech complexes50 , we might hope that the same 2-dimensional bar might
appear in our case. That is indeed the case and the mentioned longbar grows with increasing density of the sample.

Figure 10.16: A cut-o↵ sphere as anunderlying space leading to diagramsin Figure 10.17.
49

See Example 10.1.1.

Fig

50

...and hence the persistence diagrams are not too di↵erent.

(3) Note that the lon
3. As is frequently the case, there appear multitudes of short bars wethe 3-dimensiona
choose to ignore at this point.
dimensional bar othe 3-dimensional(4) A pairing of a 3-dcontractible in X
(5) We speculate theodesic circles (i.e
De-noising a function
into them in our
Suppose we are given an approximation of a function f in the formof a discrete set of equally spaced measurements. We can connect
Note that, except for s
the resulting points by edges and obtain a graph G representing ourmeasurements, see the left side of Figure 10.18. Suppose we want
We are able to interpret
to extract the global behaviour of f as in the center of Figure 10.18
by removing the local oscillations we consider to be noise. A way
to achieve it would be to construct the sublevel filtration of the unavoidable as we always
simplicial complex G and choose the threshold # for the noise level.
as the density of our samp
We would then draw the corresponding 0-dimensional persistence
diagram and ignore the points in the shaded #-neighborhood of the
2-dimensional bar appea
diagonal, see the right side of Figure 10.18. As a result we obtainbar. It can be explained
3-dimensional bar born aDuring our experiment
the mentioned procedure.
instances (and aligned wit
Figure 10.17: Four persistence diagrams via Rips complexes of a cut-o↵
sphere, based on samples of 100, 200,
and 400 points.

51

A vertex of G appears at thefunction value it represents. An edgeof G appears as soon as both of its
vertices appear.
52
Each local minimum in our approximation except for • and ⇤generates a point in this neighborhood.
5152

persistent homology: stability theorem

two prominent points •, ⇤ in the persistence diagram. The de-noisedfunction presented in the center of Figure 10.18 can now be obtainedby connecting the critical simplices corresponding to these points:

153

T By adjusting the threshold # wecan adjust the level of details wewant to preserve.

1. Blue birth simplices get connected to the higher endpoint of the redterminal simplex.
2. The only exception is ⇤, which is not a terminal simplex, but gets
added as the highest point in the graph in order to finalize53 ourapproximation by connecting it to ⇤.

The component represented by ⇤can not get terminated as a homology element.
53

1

10.5 Concluding remarks

Figure 10.18: A noisy function, itsreconstruction and the corresponding
persistence diagram. The shaded
region contains a multitude of points
we choose to ignore in our reconstruction.

Recap (highlights) of this chapter• continuous filtrations;• persistence modules;• interleaving;• Stability theorem;

Background and applications
There are three di↵erent proofs of the stability theorem in theliterature. The initial proof was combinatorial54 and did not include
the isometry theorem or interleavings but rather just the continuity ofpersistence diagrams. It motivated a more algebraically oriented proof
using interleavings55 . The third and most direct proof uses explicit
matching56 .
The existence of a decomposition of a persistence module into indecomposable parts is a particular case of a standard approach referred to as the Krull Remak Schmidt principle. The fact that theindecomposable are precisely the interval modules is a special caseof the Gabriel’s theorem. The fact that the indecomposable parts of

54

David Cohen-Steiner, Herbert
Edelsbrunner, and John Harer.
Stability of persistence diagrams.
Discrete & Computational Geometry, 37(1):103–120, 2007. doi:
10.1007/s00454-006-1276-5
55Frédéric Chazal, David CohenSteiner, Marc Glisse, Leonidas J.
Guibas, and Steve Y. Oudot. Proximity of persistence modules andtheir diagrams. In Proceedings of
the Twenty-Fifth Annual Symposium on Computational Geometry, SCG ’09, pages 237–246, New
York, NY, USA, 2009. ACM. doi:
10.1145/1542362.1542407
56
Ulrich Bauer and Michael Lesnick.
Induced matchings of barcodes andthe algebraic stability of persistence.
In Proceedings of the Thirtieth Annual Symposium on Computational
Geometry, SOCG’14, pages 355–364,
New York, NY, USA, 2014. ACM.
doi: 10.1145/2582112.2582168

154

introduction to persistent homology

multi-parameter persistent homology are not as simple as the interval
modules is the major obstacle to exhaustive applications of multiparameter persistence.
For a longer treatment of stability see a book57 . There is also a
recent treatment simplifying some of these ideas58 .
The material presented up to this point represents the core ideas
and properties of persistent homology. From this point on the topics
diverge, with some of the major motivations being:• Theoretical treatment: persistent homology represents a parameterized version of homology and as such there are many ways to
explore the structure further, either by generalizing the framework59 , determining what geometrical properties it encodes60 and
expanding the ideas into other theoretical contexts.
• Practical treatment, mostly associated with data analysis: in thiscontext persistent homology is often viewed as a stable shape descriptor. As a result considerable e↵ort is being invested to incorporate persistent homology into the flow of data analysis, either by
adjusting it to specific data types61 , establishing meaningful probabilistic62 and statistical63 analysis, and to extract relevant features.
• Computational treatment: the aim of this context is to optimize thecomputational resources required to obtain persistence (or at least apart of it) by developing faster algorithms often incorporating various shortcuts (for example, the twist64 .) or additional structure65 .
Currently available software for computing persistent homology
includes (but is not restricted to) Ripser and related Ripserer.jl,
Ripser.py, and Cubical Ripser, Dionysus, PHAT, GUDHI, javaPlex,
Perseus, Eirene, etc.
This list of topics and software is by no means exhaustive.

Appendix: From the interleaving distance to the bottleneck distanceIn this appendix we will provide an explanation that leads to thebottleneck distance by determining the interleaving distance between
pairs of interval modules.
Case 1: distance between an interval module and the zero persistence module. The situation is presented in Figures 10.19 and 10.20.
For 0  p < q let us discuss the interleaving of the interval module F p,q (the bold portion of the figures) and the trivial persistence
module (the grey portion below).

57

Frédéric Chazal, Vin de Silva,
Marc Glisse, and Steve Y. Oudot.
The Structure and Stability of
Persistence Modules. Springer Briefs
in Mathematics. Springer, 2016. doi:
10.1007/978-3-319-42545-0
58
Primož Škraba and Katharine
Turner. Notes on an elementary
proof for the stability of persistence
diagrams, arXiv: 2103.10723, 2021
59

For example, with zig-zag persistence, multi-parameter persistence,
introduction of new constructions ofcomplexes (Witness, selective Rips
complex), etc.
60
For example it encodes, at least
to some degree, shortest homology
bases, intrinsic volumes, geometricshapes, curvatures, dimension, etc.
61
Besides point-clouds, these include
time series, high-dimensional data,
dynamical systems, sensor networks,etc.
62
It turns out there are significant
phase transitions in persistent homology of random processes, etc.
63
This is typically done by mapping
persistence diagrams into Hilbert
space via any of the multitude maps
available, for example persistence
landscapes, persistence silhouette,
persistence silhouette, persistence
images, etc.
64
Chao Chen and Michael Kerber.
Persistent homology computation
with a twist. In Proceedings 27th European Workshop on Computational
Geometry, volume 11, pages 197–200,
2011
65
For example, when computing onedimensional persistence of geodesic
spaces.
pq"

Figure 10.19: interval module F p,q
is not #-interleaved with the trivial
interval if # < (q p)/2.
pq

Figure 10.20: interval module F p,q is#-interleaved with the trivial interval
if # (q p)/2.

persistent homology: stability theorem

155• If the interleaving parameter was # < (q p)/2 as in Figure 10.19,
the composition of the red diagonal maps:– is the trivial map as it factors through the trivial vector space
below;– should have been identity on F by the interleaving condition, as
its domain and target are in [ p, q).
These two observations contradict each other hence the interleaving
parameter is at least (q p)/2.
• For # = ( p q)/2 though, the composition of the diagonal maps
increases the scale parameter by p q, and any such structure mapof F p,q is trivial, hence the #-interleaving consisting of trivial maps
exists, see Figure 10.20.
We conclude that the interleaving distance is # = ( p q)/2.
Case 2: general case. From Case 1 we conclude that for 0 
p0 < q0 the following holds: If F p,q is #-interleaved with F p0 ,q0 for# < ( p q)/2, then [ p0 , q0 )[ p + #, q #) see Figure 10.21. Bysymmetry the opposite also holds: [ p, q)[ p0 + #, q0 #). It is easy
to see these conditions are also sufficient. For minimal # for whichthese two conditions hold we obtain the #-interleaving by mapping
the designated generator of F p,q to the designated generator of F p0 ,q0whenever possible, with other maps being trivial, see Figure 10.23.
It should be apparent from Figure 10.22 that the # in question ismax{| p q|, | p0 q0 |}.
We conclude that F p,q and F p0 ,q0 are max{| p q|, | p0 q0 |} interleaved. However, Case 1 demonstrates that F p,q and F p0 ,q0 are alsomax{( p q)/2, ( p0 q0 )/2}-interleaved by the trivial maps, the interleaving distance between F p,q and F p0 ,q0 ismin{max{| pq |, | p 0q0 |}, max{( pq)/2, ( p0

p

q

"

Figure 10.21: #-interleaving implies
the orange part is non-trivial as theinterleaving maps in the blue region
have to be non-trivial.
p

q

p0
p

q0
q

p0

q0

Figure 10.22: Condition of Figure
10.21 induces two shapes. Theinterleaving distance is the larger #parameter these shapes induce.
p

q

"

p0

q0

Figure 10.23: The interleaving for# = max{| p q|, | p0 q0 |}. The
non-trivial maps are in the shaded
region.

q0 )/2}}.

We now interpret the obtained distanced in the context of persistence diagrams representing interval modules. First note that◆◆✓✓p+q p+q( p q)/2 = d• ( p, q),,22
is the d• distance between ( pq ) and the diagonal D.
Case 1. The interleaving distance between F p,q and the zero persistence module is realized by matching point ( p, q) to the closest pointon the diagonal and computing the resulting d• distance, see Figure
10.24.
Case 2. The distance between F p,q and F p0 ,q0 is the smaller of thefollowing two:

Figure 10.24: Matching a point to D.

Figure 10.25: Matching two points.

156

introduction to persistent homology

1. Either max{| p q|, | p0 q0 |} = d• (( p, q), ( p0 , q0 )), which is the d•distance between the points, see Figure 10.25.
2. Or max{( p q)/2, ( p0 q0 )/2} which can be interpreted as follows:
match each of the points to the closest point on the diagonal D andtake the maximal d• distance, see Figure 10.26.
We have thus interpreted the interleaving distance between interval
modules in the context of persistence modules and obtained the bottleneck distance for diagrams containing at most one point. The crucial
ingredients of the interpretation are the matching and d• . Theorem
10.3.2 essentially states that an optimal interleaving between any pairof persistence modules essentially consists of such matchings: match
some pairs of interval modules from both persistence modules, andthen match the remaining interval modules to D. The interleaving distance (and thus the bottleneck distance) corresponds to the matchingwhose d• -distance of its maximal matching is minimal.

Figure 10.26: Matching each of thetwo points to D.

11
Discrete Morse theory

Homology and persistent homology detect holes in spaces
through the use of algebraic constructions: a simplicial complex generates a chain complex and the resulting homology construction detects
holes. However, functions and vector fields also contain information
about the topology of the domain. In the smooth setting this information is contained in critical points of functions and zeros of vector
fields, a situation which is beautifully described by Morse theory.
In this chapter we will describe discrete Morse theory. As the namesuggests we will delve into the world of discrete functions and discrete
vector fields defined on simplicial complexes. Our main goal will be
to describe how these encode homology, often leading to simplified
representations and faster computations than the standard methods.

11.1 Motivation
We first recall the definition of elementary collapses.
Definition 11.1.1. A simplex in a simplicial complex is a free face
if it is a face of precisely one simplex. This implies that the coface
in question is a maximal simplex
Let K be a simplicial complex, sk 1 ⇢ t k 2 K, and assume s isa free face in K. A removal K ! K \ {s, t } is called an elementary collapse.
Complex K is collapsible to a subcomplex L  K if there isa collapse (i.e., a sequence of elementary collapses) resulting in thesubcomplex L. Complex K is collapsible if it is collapsible to a point.

Remark 11.1.2. We have already proved in Lemma 3.4.6 that an elementary collapse results in a homotopically equivalent space. As aresult, if a simplicial complex K is collapsible to a subcomplex L  K

T If s is a free face in a simplicial

complex K then its only coface is a
maximal simplex in K.

Figure 11.1: An elementary collapse
indicated by an arrow from s into t.

158

introduction to persistent homology

then L ' K. In particular, each collapsible simplicial complex is contractible. The converse does not hold as there exist contractible simplicial complexes without a free face, for example Dunce hat (Figure11.2) and Bing’s house.
Given a simplicial complex K it would be of interest to simplify (i.e.
collapse) it as much as possible. This would, for example, simplify thecomputation of homology groups. One would go about such simplification by repeating the following sequence as long as possible: find afree face and perform the corresponding collapse. An example is given
in Figure 11.3, where the collapses of the first three steps are indicatedby the arrows. One can encode such a collapse by:• drawing all the arrows1 indicating collapses, or• annotate simplices by numbers2 so that the countdown-sequence
encodes3 the collapsing sequence.
Both of these encodings are demonstrated on the right side of Figure
11.3.

a
a
a
a
Figure 11.2: Dunce hat is obtainedby glueing the boundary of a discalong a circle: twice alone one direction and once along the otherdirection. The obtained space can betriangulated but contains no free face
meaning it is not collapsible. However, it turns out to be contractible.

1

An example of a discrete vector
field.
2

A rough example of a discrete
Morse function.
3
10 collapses to 9; 8 collapses to 7,
etc.

6

7
8

5
10

Eventually a collapsing sequence ends when there are no more freefaces. At this point we can resort to another trick that will on one
hand change the structure of a complex, yet still simplify its description in a way. Choose any simplex, declare it to be a critical simplex,
remove it from the complex, and continue with collapsing. In the end
we will form a “complex” consisting of critical simplices. The detailsof the construction will be described throughout this section. At thispoint we only illustrate a geometric interpretation of this idea in termsof “stretching” simplices.
For our motivational purposes let us continue in Figure 11.4 withthe example from Figure 11.3. We are left with a triangle. We choose
one of its edges to be a critical edge and continue with collapsing.
We can imagine that each collapse stretches the critical edge until, atthe end, we are left with two critical simplices: and edge and a point
jointly forming a circle. The resulting space is homotopy equivalent toour original simplicial complex of Figure 11.3, has a simple representation, but is not a simplicial complex. However, its homology can be

9

Figure 11.3: A simplification of asimplicial complex using elementarycollapses and the encoding of theresulting collapse by arrows (discrete vector field) and annotations(discrete Morse function).

a

a

a

Figure 11.4: Declaring the red edge
to be critical, we can collapse theother two edges and obtain a representation of a circle using only two
critical “simplices”.

discrete morse theory

159

computed in the same way as simplicial homology so in e↵ect, we havetransformed the 1-dimensional boundary matrix from 6 columns to 1
column. For another example see Figure 11.5.

x

a

v

w

x

x

b
y

a

x

y
b

z
x

b

z

v

x

w

x

x

a

Figure 11.5: Stretching critical
simplices of a standard triangulationof the torus (on the left) along
the indicated collapses results in astandard representation of a torus asa square with identified sides (on theright).
Critical simplices can be thoughtof as zeros of the resulting discrete
vector field. We have already seen in
the hairy ball theorem that there is aconnection between zeros of smooth
vector fields and topology of thedomain.

11.2 Discrete Morse functions and discrete vector fields
We start by defining functions that encode the collapsing sequencesand deformations.
Definition 11.2.1. Let K be an abstract simplicial complex. A function f : K ! R is a discrete Morse function [DMF] if 8sk 2
K:

T An abstract simplicial complex

1. e1 = |{t k 1 2 K | f (t )is a collection of simplices hence areal function defined on it maps eachsimplex into a real number.

f (s )}|  1 and2. e2 = |{t k+1 2 K | f (t )  f (s )}|  1.
A function g : K ! R respects dimension4 if for each sk 1 ⇢ t k 2 K
we have g(s ) < g(t ). Such a function is a DMF. On the other hand,each DMF almost respects dimension in the sense5 that for each simplex t k at most one exceptional facet and at most one exceptional coface of dimension k + 1 are allowed. The following proposition demonstrates that the two exceptions cannot occur simultaneously.
Proposition 11.2.2. Given the notation of Definition 11.2.1, either
e1 = 0 or e2 = 0.
Proof. Aiming for the contradiction, assume that for s 2 K and forvertices v1 , v2 2 K (0) we havef (s)f (s [ {v1 })But then s [ {v2 } 2 K and we have:f (s [ {v1 , v2 }).

(11.1)As an example think of g(s) =dim(s).
4

5

Putting it di↵erently, for eachsimplex the values of the function
strictly decrease by passing to its
faces with at most one exception,and the values of the function strictlyincreases by passing to its cofaces
with at most one exception.

160

introduction to persistent homology
67
8

1. f (s [ {v2 }) > f (s) as the exceptional coface of s is s [ {v1 }.

5

2. f (s [ {v2 }) < f (s [ {v1 , v2 }) as the exceptional face of s [ {v1 , v2 }is s [ {v1 }.

10

9

3

These two conclusions combine into f (s ) < f (s [ {v1 , v2 }) which
contradicts equation 11.1.
Proposition 11.2.2 implies that simplices with exceptions form disjoint pairs. We will refer to such pairs as regular pairs. A regular pairconsists of a simplex t and its facet s. It encodes an “arrow” s ! t
in the sense of the motivational section and is thus presented as such,see Figure 11.6 for an example. A simplex without any exception6 iscalled7 critical simplex. Given a DMF on a simplicial complex, eachsimplex is either critical or contained in a unique regular pair.

2
Figure 11.6: An example of a DMF
and the resulting discrete vector fieldin blue.
6
I.e., for which e1 = e2 = 0.
7
A critical simplex is not containedin any regular pair.

Proposition 11.2.4.
Let f be a
DMF on a simplicial complex
K. For each i let ni denotethe number of critical simplices of dimension i. Thenc = n0 n1 + n2 . . . .
Proof. Removing a regular pairof simplices does not change c
because simplices are of adjacentdimensions.
8

Gradient vector fields
Definition 11.2.5. Let K be a simplicial complex and p 2 N. Givena discrete vector field on K consisting of pairs {(si , ti )}i2 J , a p-path
is a sequence
p 1

si

p

! ti1

1

p 1

si

2

p

! ti2

p 1

si

3

p

! · · · ! ti

k

p 1

si

such that for each j:
p 1

• (si

j

p 1

• si

j

p

, ti ) is an arrow in the discrete vector field, andjp

is a facet of ti

j 1

.
p 1

p 1

Such a p-path is a cycle if s1 = sk+1 and k 1.
A discrete vector field is acyclic if it admits no cycle.
A few observations concerning Definition 11.2.5:

k +1

1

3

Definition 11.2.3. Let K be an abstract simplicial complex. A discrete
vector field is a disjoint collection of pairs (si , ti ) of simplices fromK such that for each i simplex si is a facet of ti . Each pair of a discrete vector field is referred to as an arrow.
The disjointness condition means that each simplex can be themember of at most one pair of a discrete vector field. The collection ofregular pairs of a DMF forms8 a discrete vector field, see Figure 11.6.
A discrete vector field is called a gradient vector field9 if it is induced
by some DMF in this manner. The arrows constituting a discrete
vector field will be sometimes referred10 to as regular pairs.

2

4

The converse is not true in general
as we will explain the the next
subsection.
9
We will be omitting adjective“discrete” when mentioning gradientvector fields.
10
The reason is twofold: to emphasize that the pair is a part ofthe structure of a discrete vector
field, and to stress the analogy withregular pairs of a DMF.

discrete morse theory

161

1. A critical simplex can only appear as the last simplex of a p-pathin a discrete vector field.
2. Given a DMF f , function values decrease along any p-path in the
induced discrete vector field, i.e.:f (si j )f (ti j ) > f (si j+1 ),8 j.

In particular, f (si1 ) > f (sim ) for all m > 1.
3. Observation 2. implies that each gradient vector field is acyclic.
The following theorem proves the converse.

Theorem 11.2.6. Each acyclic discrete vector field on a simplicial complex K is a gradient vector field, i.e. it is induced by some DMF.
A proof is given in the appendix. As a result we obtain the following theorem.
Theorem 11.2.7. A discrete vector field is gradient vector field i↵ itis acyclic.
We conclude this section by demonstrating how acyclic discrete
vector fields encode collapses.
Proposition 11.2.8. Suppose the critical simplices of an acyclic discrete vector field on K form a subcomplex L  K. Then there exists
a collapse K ! L and thus K ' L.
Proof. We claim there exists a regular pair (s, t ) such that s is a freeface. Assuming for a moment this claim is true, we can remove pair(s, t ) by performing an elementary collapse and proceed by using theclaim on the resulting complex. Thus the inductive argument and theclaim suffice to prove the proposition.
We now turn our attention to proving the claim. Let n denote themaximal dimension of a simplex in K \ L. There exists an n-path in
the discrete vector field. Take a maximal11 such path and let s !
t denote the first regular pair in it. Simplex s is a free face by thefollowing argument:

Figure 11.7: A 2-path in blue ending
in a critical edge, a 2-path in red
ending in a non-critical edge, and a
1-path in orange ending in a criticalvertex.

T Di↵erent DMFs on a simplicial

complex K may induce the same
discrete vector field. For example, if
f is a DMF, then so are e f and 3 f
5, and all of them induce the same
discrete vector field. Our primaryinterest in discrete vector fields
lies in their encodings of collapses
and deformations-simplificationsof a simplicial complex. A DMF
represents a convenient but notunique way of encoding a discrete
vector field.

Corollary 11.2.9. If an acyclic discrete
vector field on K has a single critical
simplex, then that simplex is a vertex
and K is collapsible.
Proof. The statement follows directly from Proposition 11.2.8.

11

Such a path contains a first regularpair because the discrete vector fieldis acyclic.

• s  t as the simplices form a regular pair.
• If s was a facet of another simplex t 0 in K \ L, then the n-simplex
t 0 would be contained in another regular pair12 (s0 , t 0 ), which could
be used to prolong our n-path. This contradicts the maximality of
the chosen n-path.

There are no (n + 1)-simplices in
K \ L.

12

162

introduction to persistent homology• If s was a facet of another simplex t 0 in L, then s 2 L as L is asubcomplex, a contradiction.

11.3 Morse homology
In this section we will explain the proceedure that leads to the computation of homology from a gradient vector field. The geometric idea
behind the theory was presented at the beginning of this chapter: collapsing regular pairs to stretch critical simplices, with the resulting
space having the same homotopy type as the original simplicial complex but fewer “simplices”. In our treatment we will refrain13 fromformally defining the resulting space and instead construct the resulting chain complex directly. However, it might still be helpful to keep
the geometric idea in mind to help navigate the algebraic construction.

Morse chain complex
For the rest of this section we fix a simplicial complex K, a gradient vector field on K, and an (algebraic) field F to provide efficient
algebraic constructions. For each i let ni denote the number of critical
i-simplices.
Definition 11.3.1. Let p 2 {0, 1, . . .}. A Morse p-chain is a fornppp
mal sum Âi=1 li si with li 2 F and si being an oriented critical simplex of dimension p in K for each i.
The p-dimensional Morse chain group C p is the vector space consisting of all Morse p-chains.
Observe that C p ⇠= F n p . In order to obtain a chain complex we alsoneed to define boundary maps. These are based on oriented14 paths indiscrete vector fields.
Definition 11.3.2. Let p 2 {0, 1, . . .}. An oriented p-path from an
p 1
p 1
oriented simplex s1
to an oriented simplex sk+1 is a p-path
p 1

s1

p

! t1

p 1

s2

p

! t2

p 1

s3

p

! · · · ! tk

2. does not match sj+1 .

A formal definition of resulting
spaces would require a significant
amount of additional material fromalgebraic topology. This would
include a formal treatment of CW
complexes, i.e., spaces obtainedby inductively glueing discs. We
have actually mentioned several
presentations of such constructions
when presenting torus, Klein bottle
and projective plane by drawing asquare with identifications along theedges, when defining the dunce hat,and in one of the previous appendicesin the context of relative homology.

T As with the usual homology,

multiplying an oriented simplex by
1 changes its orientation.

14

Paths in a discrete vector field are
directed by definition. The adjective“orientable” refers to the fact thatthe simplices forming the path areoriented in a certain way.

p 1

sk+1

consisting of oriented simplices, such that for each j the orientationinduced by tj on its facets:
1. matches sj , and13

T One could say that the simplices

ti j in an oriented p-path are oriented
consistently along the path.

discrete morse theory

Given an oriented critical p-simplex t, let d(t ) denote the collectionof all of its facets with the induced orientation arising from t. For
each15 oriented critical ( p 1)-simplex s defineat,s =

Âs0 2d(t )|{ oriented paths from s0 to s}|163

Given an oriented critical ( p 1)simplex s observe that at,s counts
di↵erent paths than at, s .
15

as the number of oriented p-paths from elements of d(t ) to s.
Definition 11.3.3. The boundary map d of the Morse chain complex is defined as follows: for each oriented critical p-simplex t define
np 1

d p t = Â (at,si
i =1

where s1 , . . . , sn p 1 are critical ( ptation.

T The oriented paths constituting

at, si )si ,1)-simplices with a fixed orien-

Examples will be provided below when demonstrating the computation of Morse homology, see also Figures 11.8 and 11.9. It turns outthat d2 = 0.
Definition 11.3.4. The Morse chain complex is the chain complex
defined as
d

d

d

∂

d

d

· · · ! Cn ! Cn 1 ! · · · ! C1 ! C0 ! 0.

Morse homology
We may now define Morse homology as the homology arising from
the Morse chain complex.
Definition 11.3.5. Let p 2 {0, 1, . . .}. The Morse homology of a
gradient vector field on K is defined as
H p (K; F ) = ker d p / Im d p+1 .

Theorem 11.3.6. Morse homology is isomorphic to the standard (simplicial) homology:
H p (K; F ) ⇠= H p (K; F ).
Corollary 11.3.7 (Weak Morse inequalities). For each p the numberof critical p simplices is greater or equal to the corresponding Betti
number: n p b p .

the boundary map model how arrows
stretch the boundary of a p-critical
simplex towards critical ( p 1)simplices.

164

introduction to persistent homology

Example 11.3.8. Given the situation of Figure 11.8 there is one criticaledge hb, ei and one critical vertex h ai. Thus C1 ⇠= C0 ⇠= F with theother Morse chain groups being trivial.
Let us determine dhb, ei:
1. d(hb, ei) = {hei,

hbi}.

d

2. There is one oriented 1-path from hei to h ai:

hei ! he, aich a i.

3. In a similar fashion there is one oriented 1-path fromh a i.

hbi tob

4. Observations 2. and 3. imply ahb,ei,h ai = 1 and ahb,ei, hai = 1.
5. dhb, ei = (ahb,ei,h ai

ahb,ei, hai ) · h ai = 0aeFigure 11.8: A gradient vector field
on a simplicial complex. Criticaledges are colored in red. There is one
oriented 1-path from hei to h ai andone oriented 1-path from hbi to h ai.

The resulting Morse chain complex is of the form
0

· · · ! 0 ! F ! F ! 0.
The resulting homology is trivial in dimensions two and above, andnontrivial below: H0 (K ) ⇠= H1 ( K ) ⇠= F.
Example 11.3.9. Let us compute the Morse homology of a torus.
x

a

v

w

b
y

x

x

b
y

b
y

a

v

⌧

a

v

x

x

y

b
y

a

v

⌧

z
x

w

w

z

z

x

x

v

w

• critical edges a (red) and b (blue);• critical triangle t.

x

y

b
y

a

v

w

z

z

x

x

v

w

x
y

⌧

Given the triangulation and the gradient vector field on a torus presented on the leftmost part of Figure 11.9 we determine the following
critical simplices:• critical vertex x in purple;

x

⌧

w

z

z

x

x

z

v

w

x

Figure 11.9: A triangulation of atorus, a gradient vector field, andpaths generating the Morse boundaryfrom Example 11.3.8.

discrete morse theory

165

We orient the critical simplices according to visualizations in the otherparts in Figure 11.9. The resulting Morse chain complex is of the form

· · · ! 0 ! F ! F2 ! F ! 0.
We next determine the Morse boundary of t. Only two simplices ofd(t ) are the starting simplices of oriented 2-chains ending in a criticaledge:• From the diagonal edge of d(t ) there are two oriented 2-paths16 tocritical edges17 h ai and hbi.
• From the top edge of d(t ) there are two oriented 2-paths18 to critical edges19 h ai and hbi.
• There are oriented 2-paths starting in the vertical edge of d(t ) but
none of them ends in a critical edge.
Combining these three cases we concluded( t ) =h ai

16

Drawn in black on the center-leftpart of Figure 11.9.
17
The center-right part of Figure11.9 contains opaque green arrows
indicating the orientations of theedges contained in the oriented 2paths. The terminal edges of theoriented 2-paths are h ai and hbi.
18
Drawn in black on the rightmost
part of Figure 11.9.
19
The two oriented 2-paths di↵er
only in the last simplex.

hbi + h ai + hbi = 0.

In a similar way we conclude that da = db = 0. The resulting Morsechain complex is of the form
00

· · · ! 0 ! F ! F2 ! F ! 0.
The resulting homology is trivial in dimensions three and above, andnontrivial below: H0 (K ) ⇠= H2 ( K ) ⇠= F and H1 (K ) ⇠= F2 .

Generating DMFs and gradient vector fields
Using discrete Morse theory depends on the ability to generate
DMFs and gradient vector fields with as few critical simplices as possible. The weak Morse inequalities show that the lower bounds for thenumbers of critical simplices are Betti numbers. A DMF on a simplicial complex is perfect if the number of critical p-simplices coincides
with the pth Betti number. In terms of the numbers of critical simplices, perfect DMFs are optimal DMFs. Not every simplicial complex
admits a perfect DMF: an example is the Dunce hat.
There is a simple algorithm to generate a perfect DMF on a graph.
For each component generate a gradient vector field as follows:• Find a spanning tree.
• Choose a critical vertex.
• Define a gradient vector field pointing towards the critical vertex
along the spanning tree, see Figure 11.11.

Figure 11.10: In this figure we provide a geometric justification for
the way the orientation carries forward through oriented 2-paths. The
bottom-right part is a snapshot fromthe center-right part Figure 11.9
indicating how the orientation of the
diagonal edge carries on through thearrow to the other two edges of thetriangle. The first three parts of thisfigure indicate how such an orientation on the two edges is obtained bydeforming the oriented diagonal edge
along the arrow of a discrete vector
field.

166

introduction to persistent homology

The mentioned construction can be generalized to higher dimensional simplicial complexes: keep adding arrows while making surethat the acyclicity condition is preserved. However, better results are
typically obtained through more elaborate designs.

11.4 Concluding remarks
Recap (highlights) of this chapter• Discrete Morse functions

Figure 11.11: A graph (left) and ablue spanning tree (right) with agradient vector field pointing to thechosen critical vertex (red). The
edges not contained in the tree (red)are the critical edges.

• Gradient vector fields• Morse homology

Background and applications
Smooth Morse theory was developed20 by Marston Morse in thefirst part of the twentieth century. Amongst its results it relates critical points and gradient flows of a generic function on a manifold tothe homology of a manifold. Its discrete version21 has been introduced
at the turn of the millennium. The past two decades saw a considerable development of discrete Morse theory from various directions,
including computational aspects, developing analogies between discreteand smooth results, etc. For a textbook presenting smooth and discrete point of view see a book22 , for recent applications see a book23 .
An algorithm for generating a discrete morse function is given in apaper24 .
An echo of the smooth Morse theory is the hairy ball theorem: thetopology of a domain is connected to zeros of vector fields and thus toextrema of functions. In a similar way, an echo of the discrete Morsetheory is our proof of the Euler-Poincaré formula, where we essentiallyonly counted the maxima of the x-coordinate function. Theorem 11.2.6
is a discrete variant of the assigning of a potential function to a vector
field.
Generalized discrete Morse theories can be used to prove25 thatČech complexes collapse onto alpha complexes in Euclidean spaces.
Several computer programs use discrete Morse theory to a di↵erent
degree to assist26 with computations of homology. The theory can also
be used as a preprocessing tool or a framework within which to analize
discrete functions.

A proof of Theorem 11.2.6
We first introduce some preliminary notions. Given a simplicial
complex K the Hasse diagram of K is a directed graph defined as

20

Marston Morse. The Calculus of
Variations in the Large. American
Mathematical Society Colloquium
Publication. Vol. 18. New York, 1934
21
Robin Forman. A user’s guide todiscrete morse theory. Sém. Lothar.
Combin., 48, 12 2001

22

Kevin P. Knudson. Morse Theory: Smooth and Discrete. World
scientific, 2015. doi: 10.1142/9360
23
Tamal K. Dey and Yusu Wang.
Computational Topology for Data
Analysis. Cambridge: Cambridge
University Press, 2022. doi:
10.1017/9781009099950
24

Henry King, Kevin Knudson, andNeža Mramor. Generating Discrete
Morse Functions from Point Data.
Experiment. Math. 14 (4) 435 – 444,
2005
25

Ulrich Bauer and Herbert Edelsbrunner. The morse theory of Čech
and Delaunay complexes. Transactions of the American Mathematical Society, 369:1, 06 2016. doi:
10.1090/tran/6991
26
For example, using simplification
using emergent pairs in Ripser. On
the other hand Perseus is actuallybased on a discrete Morse theory.

discrete morse theory167dfollows:c

1. The nodes are the simplices of K;
2. Directed edges correspond to pairs (simplex, a facet). In particular,
each n-simplex is the source of n + 1 directed edges.
An example is provided in Figure 11.12. The directed edges in thegraph represent the containment of a facet.
Given an empty discrete vector field on a simplicial complex K,
the directed edges also represent the direction of descent of the corresponding DMF. In this trivial case, all the simplices are critical, there
are no exceptions and the DMF respects dimension. An obvious choiceof a DMF in this case is the dimension function of a simplex. For illustrative purposes let us discuss how we could obtain such a function
from the Hasse diagram in an inductive manner:• Assign the smallest value, say 0, to all minimal nodes of a directed
graph;• Assign the second smallest value, say 1, to all nodes whose lower
set27 has already been enumerated;• Proceed by induction: In step number n assign the nth smallest
value, say n 1, to all nodes whose lower set has already been
enumerated;

b

a

e

hb, d, eihb, dihb, eihbi

hd, eihdi

hc, dihei

ha, eihci

ha, bihai

Figure 11.12: A simplicial complex
and its Hasse diagram. Hasse diagrams are typically drawn in levels
corresponding to the dimensions ofsimplices.

27

The lower set of a node s is thecollection of all nodes which appear
as the target of a directed edge
starting at s.
dcThis inductive construction of function works for any acyclic directed graph and will be used in our eventual proof.
Given a non-trivial discrete vector field on a simplicial complex K
we define a modified Hasse diagram of K by reverting the directionof the directed edges corresponding to the regular pairs, see Figure11.12 for an example. A modified Hasse diagram encodes the sufficientconditions on a DMF to generate the initial discrete vector field. The
above inductive procedure on such a diagram will produce a required
DMF i↵ the diagram itself is acyclic as a directed graph.
Lemma 11.4.1. The modified Hasse diagram of an acyclic discrete
vector field is acyclic.
Proof. Since each simplex can be a member of at most one regularpair in a discrete vector field, a cycle in the modified Hasse diagram
H cannot contain consecutive directed edges corresponding to regularpairs. As directed edges of H either end in a simplex of dimension
1 higher (in the case of regular pairs) or lower (in the case of edges
encoding the facet relation) than the dimension of the initial simplex,
any cycle in H has to be an alternating concatenation of these two
types. As a result, a cycle in H corresponds to a p-cycle in the initial

b

a

e

hb, d, eihb, dihb, eihbi

hd, eihdi

hc, dihei

ha, eihci

ha, bihai

Figure 11.13: The modified Hasse
diagram, the reverted edges are red.

168

introduction to persistent homology

d 4

discrete vector field, which is non-existent by the main assumption.

5

6
c
3

5

A proof of Theorem 11.2.6. Given an acyclic discrete vector field, thecorresponding modified Hasse diagram is acyclic by Lemma 11.4.1.
Thus the inductive procedure above results in a suitable DMF, see
Figure 11.14.

2
b

4

3
e
2

1
a
0

1
4
hb, d, ei

3hb, di3hb, eihbi
2

5hd, eihdi
45hc, dihei
2

1ha, eihci
6

1
ha, bihai
0Figure 11.14: The modified Hasse
diagram and the DMF (in red) constructed by the inductive proceedure.

Index

Abelian group, 81
abstract simplex, 37
abstract simplicial complex, 37
affine combination, 33
affine hull, 33
affine independence, 34
Alexander duality, 102alpha complex, 65
annotation, 138ball, 12
barcode, 121, 125
barycentric coordinates, 21, 34
barycentric subdivision, 20
basis, 79
Betti number, 89
birth simplex, 129body, 36bottleneck distance, 146
boundary, 89
boundary map, 87
boundary of a manifold, 48
boundary point, 48
Brouwer fixed point, 111
Brouwer fixed point theorem,
111
chain, 86
chain complex, 88
chain group, 86
classification algorithm forsurfaces, 55
close manifold, 48

closed surface, 48
coface, 35
collapse, 157
column echelon form, 93
combinatorial manifold, 49
component, 17
connected space, 17
connected sum, 52
consistent orientation, 51
continuous filtration, 138
continuous map, 13
contractible space, 16
convex combination, 21
convex hull, 21
convex set, 21
critical scale, 139, 141
critical simplex, 160
cubical complex, 106
cycle, 89Čech complex, 61Čech filtration, 62
Dante, 57
Delaunay triangulation, 25
diameter, 59
dimension of a simplex, 35
dimension of a simplicial
complex, 36
direct sum, 142disc, 14
discrete Morse function, 159
discrete vector field, 160
distance, 11170

introduction to persistent homology

Dowker duality, 73elder rule, 122elementary collapse, 44, 96, 157
elementary divisors, 104
Euler characteristic, 22, 41, 97
Euler-Poincaré formula, 22
exact sequence, 114
face, 35
facet, 35
field, 75
fields of remainders, 76
filtration, 123
filtration function, 138
free face, 157
full simplex, 98
functoriality, 111
fundamental class, 99
geodesic metric, 12geometric realization, 38
geometric simplex, 35
geometric simplicial complex,
35
gradient vector field, 160
Gromov-Hausdor↵ distance,
144
group, 81
hairy ball theorem, 112
Hausdor↵ distance, 143
homeomorphism, 14
homology group, 89
homology representative, 94
homomorphism, 82
homotopic maps, 15
homotopy, 15
homotopy equivalence, 16
homotopy equivalent spaces, 16
image, 79
incremental expansion, 96
induced maps, 110

induced orientation, 51
Inscribed angle theorem, 28
interior of a manifold, 48
interior point, 48
interleaving, 68
interleaving distance, 140, 143interval module, 142
isometry, 13
isomorphism, 79, 82, 142
Jaccard distance, 12
Jung’s theorem, 62
kernel, 79
Klein bottle, 48, 100
line sweep, 23
link, 37
locally Delaunay, 26
manifold, 48
mapper, 66
matching distance, 146
MaxMin edge, 27
Mayer-Vietoris exact sequence,
115
metric, 11
metric space, 11
MiniBall algorithm, 70
Moebius band, 16Morse homology, 163
multiplicity, 126
nerve, 62
nerve theorem, 64o↵sets, 139orientable surface, 51
oriented simplex, 50
oriented triangulation, 51
p-path, 160
partial matching, 146
path, 13

INDEX

path component, 17
path connected space, 17
perfect DMF, 165
persistence diagram, 126
persistence module, 141
persistent Betti numbers, 123
persistent homology, 123
pivot, 128
projective plane, 48
quotient, 76
rank, 83
regular pairs, 160
relative homology, 119representing cycles, 94
retraction, 111
Rips complex, 59
Rips filtration, 60
row echelon form, 92

skeleton, 36
Smith normal form, 94, 104
sphere, 14
star, 37
subcomplex, 36
subdivision, 36
sublevel filtration, 138
surface, 48
tangent vector field, 112
terminal simplex, 129
torus, 39
triangulation, 19, 36
vector space, 78
Vietoris complex, 73
Voronoi diagram, 25
Voronoi region, 24
Wasserstein distances, 147

simplicial approximation, 43
simplicial map, 42

zig-zag lemma, 117

171

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