Topological invariants of aperiodic tilings
Franz Gähler
Mathematics, University of Bielefeld

Mathematics of Quasi-Periodic Order
Kyoto, 21 to 23 June 2010

Franz Gähler

Topological invariants of aperiodic tilings

Examples of Topological Invariants
You all know Euler’s formula relating the number of faces, edges andvertices of a polyhedron:nf − ne + nv = 2
The number 2 is actually a topological invariant of the 2-sphere S 2 .
It is called the Euler characteristic χ(S 2 ).
A polyhedron represents a decomposition of S 2 into cells. A spacecomposed of such cells is called a cell complex. χ does not depend on thedecomposition that is chosen.
Franz Gähler

Topological invariants of aperiodic tilings

Homology of Finite Cell Complexes
Given a cell complex, we can consider formal linear combinations of k-cells,
forming so-called chain groups Ck under addition. In the polyhedron case,
we have C2 = Znf , C1 = Zne , C0 = Znv .
There are natural boundary maps ∂k : Ck → Ck−1 . The boundary of ak-cell is the sum of the cells in its boundary. This gives a sequence ofgroups and maps
∂∂

∂

∂

32
100 −→
C2 −→C1 −→C0 −→0The quotients Hk = ker(∂k )/im(∂k+1 ) are called homology groups, and are
topological invariants of the cell complex. Their ranks bk = rk(Hk ) areP
called Betty numbers, and χ = k (−1)k bk .
Franz Gähler

Topological invariants of aperiodic tilings

Cohomology of Finite Cell Complexes
For finite cell complexes, cohomology is almost the same as homology.
We consider formal linear combinations of k-cells, forming this time
so-called co-chain groups C k under addition. In the polyhedron case, weagain have C 2 = Znf , C 1 = Zne , C 0 = Znv .
The natural maps between co-chain groups are the co-boundary maps
δk : C k−1 → C k . δk is simply the transpose of ∂k .
We now have a sequence of groups and maps
δδ

δ

δ

32
100 ←−
C2 ←−
C1 ←−
C0 ←−
0The quotients H k = ker(δk+1 )/im(δk ) are called co-homology groups, andare topological invariants of the cell complex.
Franz Gähler

Topological invariants of aperiodic tilingsKlein’s Bottle and Torsionv

e1v

For this cell complex, we have ∂2 c = 2e1 ,and ∂1 ≡ 0. Thus, we gete2cv

e1e2

H1 = ker(∂1 )/im(∂2 ) = Z2 /2Z= Z ⊕ (Z/2Z) = Z ⊕ Z2v

H1 contains torsion elements - elements of finite order.
In cohomology, the torsion appears in a different dimension:
H 2 = Z2 ,Franz Gähler

H1 = H0 = Z

Topological invariants of aperiodic tilingsProperties of Tilings
I finite number of local patterns(finite local complexity)
I repetitivity
I well-defined patch frequencies
I translation module
I local isomorphism(LI classes)
I mutual local derivability

Franz Gähler

Topological invariants of aperiodic tilings

The Hull of a Tiling
Let T be a tiling of Rd , of finite local complexity.
We introduce a metric on the set of translates of T :Two tilings have distance < , if they agree in a ball of radius 1/ aroundthe origin, up to a translation < .
The hull ΩT is then the closure of {T − x|x ∈ Rd }.
ΩT is a compact metric space, on which Rd acts by translation.
If T is repetitive, every orbit is dense in ΩT .
ΩT then consists of the LI class of T .

Franz Gähler

Topological invariants of aperiodic tilings

Approximating the Hull by Cell Complexes
We define a sequence of cellular (CW-)spaces Ωn approximating Ω.
The d-cells of Ω0 are the interiors of the tiles; two tile boundaries areidentified if they are shared somewhere in the tiling.

Franz Gähler

Topological invariants of aperiodic tilings

The Cells of the Octagonal Tiling

Cells of first orderapproximant of theoctagonal tiling.
Franz Gähler

Topological invariants of aperiodic tilings

Approximating the Hull by Cell Complexes
We define a sequence of cellular (CW-)spaces Ωn approximating Ω.
The d-cells of Ω0 are the interiors of the tiles; two tile boundaries areidentified if they are shared somewhere in the tiling.
For Ωn we proceed as for Ω0 , except that we first label the tiles according
to their nth corona (collared tiles).
There are natural, continuous cellular mappings h : Ωn → Ωn−1 , andinduced homomorphisms h∗ : H ∗ (Ωn−1 ) → H ∗ (Ωn ).
∞Ω then is the inverse limit lim←− Ωn , consisting of all sequences {xk }k=0 ,with xk ∈ Ωk and h(xk ) = xk−1 .
∗The cohomology of Ω is the direct limit H ∗ (Ω) ∼= lim−→H (Ωn )Franz Gähler

Topological invariants of aperiodic tilings

Cohomology of Substitution Tilings
The appromimants Ωn of the hull were introduced by Anderson andPutnam (AP), Ergod. Th. & Dynam. Sys. 18, 509 (1998).
They used a single CW-space Ω0 and the mapping Ω0 → Ω0 induced bysubstitution, and take the inverse limit of the iterated mapping. This isequivalent to iterated refinements according to the nth corona, for some n.
This inverse limit using a single Ωn is easier to control, but is limited tosubstitution tilings.
Using a sequence of Ωn is more general, but the limit is hard to control.
However, the approach may be of conceptual interest.
Franz Gähler

Topological invariants of aperiodic tilings

Quasiperiodic Projection Tilings
Irrational sections through aperiodic klotz tiling.
We assume polyhedral
acceptance domains withrationally oriented faces.
Such tilings are called canonical projection tilings.
Forrest-Hunton-Kellendonk computed their cohomology for low
co-dimensions in terms of acceptance domains.
Here, we shall use a different approach.
Franz Gähler

Topological invariants of aperiodic tilings

Kalugin’s Approach
Irrational sections through aperiodic klotz tiling.
Disregarding singular cut
positions, points in unit cellparametrize tilings.
For proper parametrisation, torus has to be cut up.
This is done is steps −→ inverse limit construction!
Cohomology of n-torus cut up along set Ar satisfies−→ H k (Ωr ) −→ Hn−k−1 (Ar ) −→ Hn−k−1 (Tn ) −→ H k+1 (Ωr ) −→P. Kalugin, J. Phys. A: Math. Gen. 38, 3115 (2005).

Franz Gähler

Topological invariants of aperiodic tilingsSimplifying the Set of Cuts
H∗ (Ar ) and thus H ∗ (Ωr ) depends only on homotopy type of Ar .
We assume polyhedral acceptance domains with rationally oriented faces−→ with increasing r , pieces of Ar grow together.
For r sufficiently large, Ar is a union of thickened affine tori.
Homotopy type of Ar stabilizes at finite r0 !
Often, we can replace Ar by equivalent arrangement Ã of thin tori.
For computing H∗ (Ã): replace Ã by its simplicial resolution, A.
For icosahedral tilings, Ã consists of 4-tori, intersecting in 2-tori and 0-tori.
For codimension-2 tilings, there are only 2-tori and 0-tori.
Franz Gähler

Topological invariants of aperiodic tilings

Kalugins Exact Sequences – 2D Case

Kalugin’s long exact sequence can be split; for tilings of dimension 2 andco-dimension 2, it reads:
αk+10→− Sk →− H k (Ω) →− H4−k−1 (A) −−−→ H4−k−1 (T6 ) →− Sk+1 →− 00 →−

H4 (T4 ) →
−0 →
−40 →
−0 →
−

H3 (T ) →−S20→−→
−H 0 (Ω) →−1H (Ω) →−2H (Ω) →−0→−

H3 (T4 ) →−S1→−0H2 (A) →−4H2 (T ) →
−

S2→
−

0

H1 (A) →−4H1 (T ) →−0H0 (A) →−400→−H0 (T ) →−We need to determine H∗ (T4 ), H∗ (A), Sk = coker αk , and derive H ∗ (Ω) from that.

Franz Gähler

Topological invariants of aperiodic tilings

Mayer-Vietoris Spectral Sequence
1 of Mayer-Vietoris double complex for H (A):First page Ek,`∗⊕θ∈I1 Λ2 Γθ⊕θ∈I1 Λ1 Γθ
L L
ZL1Z 0θ⊕θ∈I1 ZL0

As A is connected, the only differential left has rank L1 + L0 − 1, so that we get:
H0 (A) = Z
H1 (A) = ⊕θ∈I1 Λ1 Γθ ⊕ Zf
H2 (A) = ⊕θ∈I1 Λ2 Γθ
where f =θθ∈I1 L0 − L1 − L0 + 1.

PFranz Gähler

Topological invariants of aperiodic tilings

Cohomology of the HullKalugins exact sequences can now be solved:
H 0 (Ω) = Z
H 1 (Ω) = Λ3 Γ ⊕ ker α2H 2 (Ω) = Λ2 Γ/hΛ2 Γθ iθ∈I1 ⊕ ker α3The ker αk are free groups, whose ranks are computable.
Torsion can only occur in coker α2 = Λ2 Γ/hΛ2 Γθ iθ∈I1 .
Geometrically, ker αk consists of closed k-chains which are non-trivial in
Hk (A), but are exact in the full torus. Thus, they are boundaries of(k + 1)-chains of T4 .
Franz Gähler

Topological invariants of aperiodic tilings

Examples
Cohomology of some 2D tilings from the literature:
H2H1H0χlines

nameZ8Z5Z

4alongPenroseZ24 ⊕Z25

Z5Z

20

betweenTübingen Triangle

Z9Z5Z

5

along

Ammann-Beenker

Z14 ⊕Z2Z5Z

10betweencolored Ammann-Beenker

Z28

Z7

Z

22

along/between

Franz Gähler

Shield, Socolar

Topological invariants of aperiodic tilings

The 3D Case
Similar to the 2D case, except that Kalugin’s exact sequences are much
more difficult to solve.
In particular, this is so for the torsion part. Only some examples could be
solved; for the general case, some extra ideas are required.
αk+10→− Sk →− H k (Ω) →− H6−k−1 (A) −−−→ H6−k−1 (T6 ) →− Sk+1 →− 0
In all icosahedral examples, we have torsion in H2 (A), and may have torsion
in S3 . This leads to group extension problems.
F. Gähler, J. Hunton, J. Kellendonk, Z. Kristallogr. 223, 801-804 (2009).
Franz Gähler

Topological invariants of aperiodic tilings

Icosahedral Examples
Cohomology of some icosahedral tilings from the literature:
H3H2H1H0χ

planesΓ

Z20 ⊕Z2Z16Z7Z

105-foldFDanzer

Z181 ⊕Z

Z72 ⊕Z2Z12Z120

mirrorPAmmann-Kramer

Z331 ⊕Z202 ⊕Z4Z102 ⊕Z42 ⊕Z4Z12Z

240

mirrorFdual can. D6Z205 ⊕Z22

Z72Z7Z145

3,5-foldF

canonical D6

2

Even the simplest of all icosahedral tilings have torsion!
Formulae have to be evaluated by computer (GAP programs). Combinatorics ofintersection tori are determined with (descendants of) programs from the GAP
package Cryst (B. Eick, F. Gähler, W. Nickel, Acta Cryst. A53, 467-474 (1997)).
Franz Gähler

Topological invariants of aperiodic tilingsMutual Local Derivability
One tiling must be locallyconstructible from the other, andvice versa.
Tilings must have sametranslation module.
Acceptance domains of one tilingmust be constructible by finite
unions and intersections ofacceptance domains of the other.

MLD induces a bijection between LI classes.

Franz Gähler

Topological invariants of aperiodic tilings

MLD Classification
Both cohomology and MLD class are determined by the arrangement of singularspaces A, and how the lattice Γ acts on it.
To fix an MLD class, we fix a space group and orbit representatives of the singularspaces.
To make MLD classification finite, we consider
I singular spaces in special orientations
I restricted number of orbits
I some non-genericity condition, likeI

closeness conditionI

existence of non-generic intersections

I

singular spaces pass through special points
Franz Gähler

Topological invariants of aperiodic tilings

MLD Relationships
We fix a space group, and compare different singular sets A, generated from“interesting” orbit representatives. Different singular sets may define the same
MLD class!
Singular sets may be related by translation, or by inflation. These are local
transformations, and so they define same MLD class.
There are also non-local transformations normalizing the space group, like the*-map. This leads to an MD relationship, but not to MLD!
The full translation symmetry Γ̃ of the singular set may be larger than thetranslation symmtery Γ of the tiling.
MLD relationship may be symmetry-preserving (S-MLD) or not. MLD bytranslation is symmetry-preserving only of translation normalizes the space group.
Franz Gähler

Topological invariants of aperiodic tilings

Cohomology of Octagonal MLD Classes
H2H1H0χlines|Γ̃/Γ|multcc

Z9Z5Z

54A

12 (tr)x

Z12Z5Z

84A

12 (tr,inf)

x

Z28

Z9Z

20

4A+4A

42 (tr)

x

Z33

Z9Z

25

4A+4A

14 (tr,inf)xgenremarks
Ammann-Beenkerx1)

Z40

Z9Z

328A

1∞Z14 ⊕Z2Z5Z

104B

22 (tr)x

Z20 ⊕Z2Z5Z

164B

22 (tr,inf)x

Z48 ⊕Z2Z9Z

40

4B+4B

82 (tr)x

Z58 ⊕Z2Z9Z

50

4B+4B

24 (tr,inf)

xZ72 ⊕Z2Z9Z

648B

2∞Z23Z8Z

164A+4B

12 (tr)xZ24Z8Z

174A+4B

12 (tr)xZ29Z8Z

224A+4B

12 (tr,inf)

x2)

Z29

Z8Z

224A+4B

12 (tr,inf)

x2)

Z35

Z8Z

28

4A+4B

14 (tr,inf)

xZ36

Z8Z

29

4A+4B

14 (tr,inf)x1) MLD class splits in two S-MLD classes

Franz Gählerx1)
x1)
xdecorated Ammann-Beenker

2) inequivalent, different combinatorics

Topological invariants of aperiodic tilings

Cohomology of Decagonal MLD Classes
H2H1H0χlines|Γ̃/Γ|multcc

Z8Z5Z

45A

11x

Z14Z5Z

105A

13 (inf)

x

Z33

Z10Z

24

5A+5A

13 (inf)

x

Z34

Z10

Z

25

5A+5A

13 (inf)

x

Z37

Z10

Z

28

10A

12 (inf)

Z49

Z10

Z

40

10A

1∞

24Z5Z

205B

51x

Z5

Z

50

5B

53 (inf)

x

Z10

Z

120

5B+5B

53 (inf)

x

Z10

Z

125

5B+5B

53 (inf)

x

Z10

Z

140

10B

52 (inf)Z ⊕Z25
Z54 ⊕Z25
Z129 ⊕Z25
Z134 ⊕Z25
Z149 ⊕Z25
Z209 ⊕Z25
49genremarks
Penrosexgen. Penrose (γ=1/2)
xTübingen Triangle
xZ10Z

200

10B

5∞ZZ10Z

40

5A+5B

11x

Z69

Z10Z

60

5A+5B

13 (inf)

xZ79

Z10Z

70

5A+5B

13 (inf)

xZ93

Z10Z

84

5A+5B

13 (inf)

xZ94

Z10

Z

85

5A+5B

13 (inf)

x1)

Z94

Z10Z

85

5A+5B

13 (inf)x1)x1) swapped by *-map, which exchanges physical and internal space (non-local equivalence)Franz Gähler

Topological invariants of aperiodic tilings

Cohomology of Dodecagonal MLD Classes
H2H1H0χlines|Γ̃/Γ|multcc

Z28

Z7Z

226A

11x

Z33

Z7Z

276A

11xgenremarks
Socolar tiling

Z42

Z7Z

366A

12 (inf)

x

Z100

Z13

Z

88

6A+6A

41x

Z112

Z13

Z

100

6A+6A

12 (inf)x1)

Z120

Z13

Z

108

6A+6A

41x2)

Z129

Z13

Z

117

6A+6A

12 (inf)

x

Z112

Z13

Z

100

12A

12 (inf)

1)

Z120

Z13

Z

108

12A

12 (inf)

2)

Z144

Z13Z

132

12A

16 (inf)

Z156

Z13Z

144

12A

1∞Z59

Z12Z

48

6A+6B11xZ68

Z12Z

57

6A+6B

11xZ69

Z12Z

58

6A+6B

12 (inf)

xZ87

Z12Z

76

6A+6B

14 (inf)

xZ92

Z12Z

81

6A+6B

14 (inf)

xZ95

Z12Z

84

6A+6B

14 (inf)x1) not equivalentFranz Gählerxx
decorated Socolar tiling2) not equivalent
Topological invariants of aperiodic tilings