simplicial complex

simplicial complex

An abstract

simplicial complex

K

is a

collection

of nonempty

finite sets

with the

property

that for any

element

σ

∈

K

, if

τ

⊂

σ

is a nonempty subset, then

τ

∈

K

. An
element of

K

of

cardinality

n

+

1

is called an

n

-

simplex

. An element of an element of

K

is called a

vertex

. In what follows, we may occasionally identify a vertex

v

with its corresponding

singleton set

{

v

}

∈

K

; the reader
will be alerted when this is the case.

The

standard

n

-complex

, denoted by

Δ

n

, is the
simplicial complex consisting of all nonempty subsets of

{

0

,

1

,

…

,

n

}

.

1

Geometry of a simplicial complex

Let

K

be a simplicial complex, and let

V

be the set of vertices of

K

. Although there is an established notion of

infinite

simplicial
complexes, the geometrical treatment of simplicial complexes is much simpler in the finite case and so for
now we will assume that

V

is a finite set of cardinality

k

.

We introduce the

vector space

ℝ

V

of formal

ℝ

–

linear combinations

of elements of

V

; i.e.,

ℝ

V

:=

{

a

1

⁢

V

1

+

a

2

⁢

V

2

+

⋯

+

a

k

⁢

V

k

∣

a

i

∈

ℝ

,

V

i

∈

V

}

,

and the vector space

operations

are defined by formal

addition

and

scalar multiplication

. Note that we may regard each vertex in

V

as a
one-term formal sum, and thus as a point in

ℝ

V

.

The

geometric realization

of

K

, denoted

|

K

|

, is the subset
of

ℝ

V

consisting of the union, over all

σ

∈

K

, of
the convex hull of

σ

⊂

ℝ

V

. If we fix a

bijection

ϕ

:

V

→

{

1

,

…

,

k

}

, then the vector space

ℝ

V

is

isomorphic

to the

Euclidean vector space

ℝ

k

via

ϕ

, and the set

|

K

|

inherits a metric from

ℝ

k

making it into a

metric space

and

topological space

. The

isometry

class of

K

is

independent

of the choice of the bijection

ϕ

.

Examples:

1.

Δ

2

=

{

{

0

}

,

{

1

}

,

{

2

}

,

{

0

,

1

}

,

{

0

,

2

}

,

{

1

,

2

}

,

{

0

,

1

,

2

}

}

has

V

=

3

, so its realization

|

Δ

2

|

is a subset
of

ℝ

3

, consisting of all points on the

hyperplane

x

+

y

+

z

=

1

that are inside or on the

boundary

of the

first octant

. These
points form a

triangle

in

ℝ

3

with one face, three edges,
and three vertices (for example, the convex hull of

{

0

,

1

}

∈

Δ

2

is the edge of this triangle that lies in the

x

⁢

y

–plane).

2.

Similarly, the realization of the standard

n

–simplex

Δ

n

is an

n

–dimensional

tetrahedron

contained

inside

ℝ

n

+

1

.

3.

A triangle without

interior

(a “wire frame” triangle) can be
geometrically realized by starting from the simplicial complex

{

{

0

}

,

{

1

}

,

{

2

}

,

{

0

,

1

}

,

{

0

,

2

}

,

{

1

,

2

}

}

.

Notice that, under this procedure, an element of

K

of cardinality 1
is geometrically a vertex; an element of cardinality 2 is an edge;
cardinality 3, a face; and, in general, an element of cardinality

n

is realized as an

n

-face inside

ℝ

V

.

In general, a

triangulation

of a topological space

X

is a
simplicial complex

K

together with a

homeomorphism

from

|

K

|

to

X

.

2

Homology of a simplicial complex

In this

section

we define the

homology

and

cohomology groups

associated to a simplicial complex

K

. We do so not because the
homology of a simplicial complex is so intrinsically interesting in
and of itself, but because the resulting homology theory is identical
to the

singular homology

of the associated topological space

|

K

|

,
and therefore provides an

accessible

way to calculate the latter

homology groups

(and, by

extension

, the homology of any space

X

admitting a triangulation by

K

).

As before, let

K

be a simplicial complex, and let

V

be the set of
vertices in

K

. Let the chain group

C

n

⁢

(

K

)

be the

subgroup

of the

exterior algebra

Λ

⁢

(

ℝ

V

)

generated by all elements of the form

V

0

∧

V

1

∧

⋯

∧

V

n

such that

V

i

∈

V

and

{

V

0

,

V

1

,

…

,

V

n

}

∈

K

. Note that we are ignoring here the

ℝ

–vector space

structure

of

ℝ

V

; the group

C

n

⁢

(

K

)

under this

definition

is merely a

free abelian group

, generated by the

alternating

products

of the above form and with the

relations

that are
implied by the properties of the

wedge product

.

Define the

boundary map

∂

n

:

C

n

⁢

(

K

)

⟶

C

n

-

1

⁢

(

K

)

by the

formula

∂

n

⁡

(

V

0

∧

V

1

∧

⋯

∧

V

n

)

:=

∑

j

=

0

n

(

-

1

)

j

⁢

(

V

0

∧

⋯

∧

V

j

^

∧

⋯

∧

V

n

)

,

where the hat notation means the term under the hat is left out of the
product, and extending linearly to all of

C

n

⁢

(

K

)

. Then one checks
easily that

∂

n

-

1

∘

∂

n

=

0

, so the collection
of chain groups

C

n

⁢

(

K

)

and boundary maps

∂

n

forms a

chain
complex

𝒞

⁢

(

K

)

. The

simplicial homology

and cohomology
groups of

K

are defined to be that of

𝒞

⁢

(

K

)

.

Theorem

:

The simplicial homology and cohomology groups of

K

,
as defined above, are canonically isomorphic to the singular homology
and cohomology groups of the geometric realization

|

K

|

of

K

.

The proof of this theorem is considerably more difficult than what we
have done to this point, requiring the techniques of

barycentric
subdivision

and

simplicial approximation

, and we refer the interested
reader to

[

1

]

.

References

1

Munkres, James.

Elements of

Algebraic
Topology

, Addison–Wesley, New York, 1984.

Title

simplicial complex

Canonical

name

SimplicialComplex

Date of creation

2013-03-22 12:34:46

Last modified on

2013-03-22 12:34:46

Owner

djao (24)

Last modified by

djao (24)

Numerical id

11

Author

djao (24)

Entry type

Definition

Classification

msc 55U10

Classification

msc 54E99

Related topic

HomologyTopologicalSpace

Related topic

CWComplex

Defines

simplicial homology

Defines

simplicial cohomology

Defines

triangulation

Defines

abstract simplicial complex

Defines

abstract

n

-simplex

Generated on Sat Feb 10 11:46:09 2018 by

LaTeXML