Discrete Vector Fields

and the Cohomology of Certain

Arithmetic and Crystallographic Groups

Bui Anh Tuan

Supervisor: Prof. Graham Ellis

National University of Ireland, Galway

March 26, 2015

Bui Anh Tuan  Page 1

Introduction

Introduction

Let G be a group. Some popular questions in homological algebra are:

1 How to find a classifying space for G? Ans: Discrete Vector Field. Eg. G is finite group (See [1]).

2 How to compute the cohomology ring for G? Ans: Discrete Vector Field.

Introduction

Let G be a group. Some popular questions in homological algebra are:

1 How to find a classifying space for G? Ans: Discrete Vector Field. Eg. G is finite group (See [1]).

2 How to compute the cohomology ring for G? Ans: Discrete Vector Field.

Introduction

Let G be a group. Some popular questions in homological algebra are:

1 How to find a classifying space for G? Ans: Discrete Vector Field. Eg. G is finite group (See [1]).

2 How to compute the cohomology ring for G? Ans: Discrete Vector Field.

Introduction

Let G be a group. Some popular questions in homological algebra are:

1 How to find a classifying space for G? Ans: Discrete Vector Field. Eg. G is finite group (See [1]).

2 How to compute the cohomology ring for G? Ans: Discrete Vector Field.

Introduction

Let G be a group. Some popular questions in homological algebra are:

1 How to find a classifying space for G? Ans: Discrete Vector Field. Eg. G is finite group (See [1]).

2 How to compute the cohomology ring for G? Ans: Discrete Vector Field.

Discrete Vector Fields

Definition 1.1

A discrete vector field V on a regular CW-complex X is a collection of arrows s → t where

s, t are cells and any cell is involved in at most one arrow, dim(t) = dim(s) + 1, s lies in the boundary of t.

source cell target cell critical cells

Discrete Vector Fields

Definition 1.1

A discrete vector field V on a regular CW-complex X is a collection of arrows s → t where

s, t are cells and any cell is involved in at most one arrow, dim(t) = dim(s) + 1, s lies in the boundary of t.

source cell target cell critical cells

Definition 1.2

A discrete vector field is admissible if for every source cell s, the length of any path starting from s is bounded by a fixed integer λ(s).

Theorem 1.3 (Whitehead/Forman)

If X is a regular CW-space with admissible discrete vector field then there is a homotopy equivalence

X ≃ Y

where Y is a CW-space whose cells correspond to those of X not involved in any arrow.

Definition 1.2

A discrete vector field is admissible if for every source cell s, the length of any path starting from s is bounded by a fixed integer λ(s).

Theorem 1.3 (Whitehead/Forman)

If X is a regular CW-space with admissible discrete vector field then there is a homotopy equivalence

X ≃ Y

where Y is a CW-space whose cells correspond to those of X not involved in any arrow.

Discrete Vector Fields

Theorem 1.4 ([11])

Let C(X) = (C_{∗}(X), d_{∗}, V) be the cellular chain complex of a contractible CW-space X together with a contracting discrete vector field V. Then the contracting homotopy on C_{∗}(X) can be defined as the sum of all maximal V-paths starting from any n-cell σ^{n},

h_{n}(σ^{n}) = \{ \begin{matrix}0 & if σ^{n} is not a source \\ ∑ σ_{i}^{n+1} & ∂_{n+1}(∑ σ_{i}^{n+1}) contains just the one \\ & source σ^{n}of dimension n \end{matrix}

One can implement this contraction h_{∗} by using the equivalent recursive formula as follows:

h_{n}(σ^{n}) = ε(σ^{n}, V (σ^{n})) (V (σ^{n}) − ∑_{σ′∈β_{n}^{s}−σ^{n}} ε(σ′, V (σ^{n}))h_{n}(σ′))

Discrete Vector Fields

Theorem 1.4 ([11])

Let C(X) = (C_{∗}(X), d_{∗}, V) be the cellular chain complex of a contractible CW-space X together with a contracting discrete vector field V. Then the contracting homotopy on C_{∗}(X) can be defined as the sum of all maximal V-paths starting from any n-cell σ^{n},

h_{n}(σ^{n}) = \{ \begin{matrix}0 & if σ^{n} is not a source \\ ∑ σ_{i}^{n+1} & ∂_{n+1}(∑ σ_{i}^{n+1}) contains just the one \\ & source σ^{n}of dimension n \end{matrix}

One can implement this contraction h_{∗} by using the equivalent recursive formula as follows:

h_{n}(σ^{n}) = ε(σ^{n}, V (σ^{n})) (V (σ^{n}) − ∑_{σ′∈β_{n}^{s}−σ^{n}} ε(σ′, V (σ^{n}))h_{n}(σ′))

Introduction

Discrete Vector Field and Contracting Homotopy

Example 1.5 (SL_{2}(ℤ) and modular tree)

Example 1.6 (SL_{2}(ℤ) and modular tree)

Content

1 Arithmetic Groups

(with Graham Ellis, published in Journal of Algebra, 2013)

2 Crystallographic Groups

(submitted to Journal of Symbolic Computation, 2015)

3 Mod2-Cohomology ring of SL_{2}(O_{−2}[1/2]]) (jointwork with Alexander de Rahm, in preparation.)

Content

1 Arithmetic Groups

(with Graham Ellis, published in Journal of Algebra, 2013)

2 Crystallographic Groups

(submitted to Journal of Symbolic Computation, 2015)

3 Mod2-Cohomology ring of SL_{2}(O_{−2}[1/2]]) (jointwork with Alexander de Rahm, in preparation.)

Content

1 Arithmetic Groups

(with Graham Ellis, published in Journal of Algebra, 2013)

2 Crystallographic Groups

(submitted to Journal of Symbolic Computation, 2015)

3 Mod2-Cohomology ring of SL_{2}(O_{−2}[1/2]]) (jointwork with Alexander de Rahm, in preparation.)

Arithmetic Groups

Cohomology

of Arithmetic Groups

Arithmetic Groups

Introduction

Definition 2.1

Arithmetic group SL_{2}(ℤ[1/m]) is defined to be group of

\{A = ( \begin{matrix}a & b \\ c & d \end{matrix} ) : det(A) = 1 and a, b, c, d ∈ ℤ[1/m]\}.

where m is a square free positive integer.

\underline{Problems:} 1 Compute homology groups of SL_{2}(ℤ[1/m]). 2 Investigate the cohomology ring of SL_{2}(ℤ[1/m]).

Definition 2.1

Arithmetic group SL_{2}(ℤ[1/m]) is defined to be group of

\{A = ( \begin{matrix}a & b \\ c & d \end{matrix} ) : det(A) = 1 and a, b, c, d ∈ ℤ[1/m]\}.

where m is a square free positive integer.

\underline{Problems:} 1 Compute homology groups of SL_{2}(ℤ[1/m]). 2 Investigate the cohomology ring of SL_{2}(ℤ[1/m]).

This work is motivated by

1 A question of Keven Hutchinson: structure of H_{3}(SL_{2}(ℚ), ℤ). ℤ[1/2] ⊂ ℤ[1/2.3] ⊂ ℤ[1/2.3.5] ⊂ ⋯ ⊂ ℤ[1/2.3.5...] = ℚ.

2 The paper “On the cohomology of SL(2, ℤ[1/p])” by A. Adem and N. Naffah [12] for primes p only.

This work is motivated by

1 A question of Keven Hutchinson: structure of H_{3}(SL_{2}(ℚ), ℤ). ℤ[1/2] ⊂ ℤ[1/2.3] ⊂ ℤ[1/2.3.5] ⊂ ⋯ ⊂ ℤ[1/2.3.5...] = ℚ.

2 The paper “On the cohomology of SL(2, ℤ[1/p])” by A. Adem and N. Naffah [12] for primes p only.

Arithmetic Groups

Methods

Theorem 2.2 (J.P. Serre)

SL_{2}(ℤ[1/(mp)]) ≅ SL_{2}(ℤ[1/m]) ∗_{Γ_{0}^{m}(p)} SL_{2}(ℤ[1/m]),

where p is a prime and Γ_{0}^{m}(p) is the congruence subgroup of SL_{2}(ℤ[1/m]) of level p.

\underline{Ideas:}

1 Factorize m = p_{1}p_{2}..p_{r}. 2 Use the result of SL(2, ℤ[1/p]), where p is a prime as in Adem’s

paper. 3 or SL(2, ℤ) = C_{6} ∗_{C_{2}} C_{4}. 4 or SL(2, ℤ) = ∗_{C_{2}}C_{3}. (produce smaller resolution than in (3))

Theorem 2.2 (J.P. Serre)

SL_{2}(ℤ[1/(mp)]) ≅ SL_{2}(ℤ[1/m]) ∗_{Γ_{0}^{m}(p)} SL_{2}(ℤ[1/m]),

where p is a prime and Γ_{0}^{m}(p) is the congruence subgroup of SL_{2}(ℤ[1/m]) of level p.

\underline{Ideas:}

1 Factorize m = p_{1}p_{2}..p_{r}. 2 Use the result of SL(2, ℤ[1/p]), where p is a prime as in Adem’s

paper. 3 or SL(2, ℤ) = C_{6} ∗_{C_{2}} C_{4}. 4 or SL(2, ℤ) = ∗_{C_{2}}C_{3}. (produce smaller resolution than in (3))

Theorem 2.2 (J.P. Serre)

SL_{2}(ℤ[1/(mp)]) ≅ SL_{2}(ℤ[1/m]) ∗_{Γ_{0}^{m}(p)} SL_{2}(ℤ[1/m]),

where p is a prime and Γ_{0}^{m}(p) is the congruence subgroup of SL_{2}(ℤ[1/m]) of level p.

\underline{Ideas:}

1 Factorize m = p_{1}p_{2}..p_{r}. 2 Use the result of SL(2, ℤ[1/p]), where p is a prime as in Adem’s

paper. 3 or SL(2, ℤ) = C_{6} ∗_{C_{2}} C_{4}. 4 or SL(2, ℤ) = ∗_{C_{2}}C_{3}. (produce smaller resolution than in (3))

Theorem 2.2 (J.P. Serre)

SL_{2}(ℤ[1/(mp)]) ≅ SL_{2}(ℤ[1/m]) ∗_{Γ_{0}^{m}(p)} SL_{2}(ℤ[1/m]),

where p is a prime and Γ_{0}^{m}(p) is the congruence subgroup of SL_{2}(ℤ[1/m]) of level p.

\underline{Ideas:}

1 Factorize m = p_{1}p_{2}..p_{r}. 2 Use the result of SL(2, ℤ[1/p]), where p is a prime as in Adem’s

paper. 3 or SL(2, ℤ) = C_{6} ∗_{C_{2}} C_{4}. 4 or SL(2, ℤ) = ∗_{C_{2}}C_{3}. (produce smaller resolution than in (3))

Theorem 2.2 (J.P. Serre)

SL_{2}(ℤ[1/(mp)]) ≅ SL_{2}(ℤ[1/m]) ∗_{Γ_{0}^{m}(p)} SL_{2}(ℤ[1/m]),

where p is a prime and Γ_{0}^{m}(p) is the congruence subgroup of SL_{2}(ℤ[1/m]) of level p.

\underline{Ideas:}

1 Factorize m = p_{1}p_{2}..p_{r}. 2 Use the result of SL(2, ℤ[1/p]), where p is a prime as in Adem’s

paper. 3 or SL(2, ℤ) = C_{6} ∗_{C_{2}} C_{4}. 4 or SL(2, ℤ) = ∗_{C_{2}}C_{3}. (produce smaller resolution than in (3))

Theorem 2.2 (J.P. Serre)

SL_{2}(ℤ[1/(mp)]) ≅ SL_{2}(ℤ[1/m]) ∗_{Γ_{0}^{m}(p)} SL_{2}(ℤ[1/m]),

where p is a prime and Γ_{0}^{m}(p) is the congruence subgroup of SL_{2}(ℤ[1/m]) of level p.

\underline{Ideas:}

1 Factorize m = p_{1}p_{2}..p_{r}. 2 Use the result of SL(2, ℤ[1/p]), where p is a prime as in Adem’s

paper. 3 or SL(2, ℤ) = C_{6} ∗_{C_{2}} C_{4}. 4 or SL(2, ℤ) = ∗_{C_{2}}C_{3}. (produce smaller resolution than in (3))

\underline{Problem:}

How to investigate

G_{1} ∗_{A} G_{2}

if we have known G_{1}, G_{2} and A?.

\underline{Solution:}

C.T.C Wall [9]: if we have free resolutions with contracting homotopy for G_{1}, G_{2} and A, we can construct a free resolution for G.

\underline{Problem:}

How to investigate

G_{1} ∗_{A} G_{2}

if we have known G_{1}, G_{2} and A?.

\underline{Solution:}

C.T.C Wall [9]: if we have free resolutions with contracting homotopy for G_{1}, G_{2} and A, we can construct a free resolution for G.

Contracting homotopy = Discrete Vector Field (DVF)

\underline{New problem:}

Find a admissible DVF on a space that G_{1} ∗_{A} G_{2} acts on.

Bui Anh Tuan  Page 14

Contracting homotopy = Discrete Vector Field (DVF)

\underline{New problem:}

Find a admissible DVF on a space that G_{1} ∗_{A} G_{2} acts on.

Bui Anh Tuan  Page 14

Home?_{^{1} Maps}_{^{2} Compass}_{^{3} Taxi}

DVF?

1 Store the tree 2 Our Algorithm 3 Our HAP package

Home?_{^{1} Maps}_{^{2} Compass}_{^{3} Taxi}

DVF?

1 Store the tree 2 Our Algorithm 3 Our HAP package

Home?_{^{1} Maps}_{^{2} Compass}_{^{3} Taxi}

DVF?

1 Store the tree 2 Our Algorithm 3 Our HAP package

Home?_{^{1} Maps}_{^{2} Compass}_{^{3} Taxi}

DVF?

1 Store the tree 2 Our Algorithm 3 Our HAP package

Home?_{^{1} Maps}_{^{2} Compass}_{^{3} Taxi}

DVF?

1 Store the tree 2 Our Algorithm 3 Our HAP package

Home?_{^{1} Maps}_{^{2} Compass}_{^{3} Taxi}

DVF?

1 Store the tree 2 Our Algorithm 3 Our HAP package

Home?_{^{1} Maps}_{^{2} Compass}_{^{3} Taxi}

DVF?

1 Store the tree 2 Our Algorithm 3 Our HAP package

Home?_{^{1} Maps}_{^{2} Compass}_{^{3} Taxi}

DVF?

1 Store the tree 2 Our Algorithm 3 Our HAP package

Euclidean Algorithm

Let g be an element in G = H ∗_{A} K.

1 Decompose g into product of elements in G_{1} and G_{2}.

g = h_{1}k_{1}h_{2}k_{2}...h_{n}k_{n}

where h_{i} ∉ K and k_{i} ∉ H (by a slightly variance of Euclidean Algorithm).

2 Find the way ”home”. Remember that,

1 The tree can be made by sending a ”closed” edge under the group action.

Euclidean Algorithm

Let g be an element in G = H ∗_{A} K.

1 Decompose g into product of elements in G_{1} and G_{2}.

g = h_{1}k_{1}h_{2}k_{2}...h_{n}k_{n}

where h_{i} ∉ K and k_{i} ∉ H (by a slightly variance of Euclidean Algorithm).

2 Find the way ”home”. Remember that,

1 The tree can be made by sending a ”closed” edge under the group action.

Arithmetic Groups

Results m = n =

1  2  3  4  5
6  0  ℤ_{2} ⊕ ℤ  ℤ_{4} ⊕ ℤ_{8} ⊕ (ℤ_{3})^{2}  ℤ_{2}  ℤ_{4} ⊕ ℤ_{8} ⊕ (ℤ_{3})^{2}
10  ℤ_{3}  ℤ_{4} ⊕ ℤ  ℤ_{2} ⊕ ℤ_{8} ⊕ (ℤ_{3})^{4}  ℤ_{4}  ℤ_{2} ⊕ ℤ_{8} ⊕ (ℤ_{3})^{4}
14  ℤ_{3} ℤ_{2} ⊕ ℤ_{3} ⊕ ℤ ℤ_{4} ⊕ ℤ_{8} ⊕ (ℤ_{3})^{2} ⊕ ℤ^{2}  (ℤ_{2})^{3} ⊕ (ℤ_{3})^{2}  ℤ_{4} ⊕ ℤ_{8} ⊕ (ℤ_{3})^{2}
15  ℤ_{4}  ℤ_{4} ⊕ ℤ ℤ_{2} ⊕ (ℤ_{4})^{2} ⊕ (ℤ_{3})^{2} ⊕ ℤ^{2}  ℤ_{2} ⊕ (ℤ_{4})^{2}  ℤ_{2} ⊕ (ℤ_{4})^{2} ⊕ (ℤ_{3})^{2}
21  ℤ_{4} ℤ_{2} ⊕ ℤ_{3} ⊕ ℤ ℤ_{2} ⊕ (ℤ_{4})^{3} ⊕ ℤ_{3} ⊕ ℤ^{2}  (ℤ_{2})^{3} ⊕ ℤ_{3}  ℤ_{2} ⊕ (ℤ_{4})^{3} ⊕ ℤ_{3}
22  ℤ_{3} ℤ_{2} ⊕ ℤ_{5} ⊕ ℤ  ℤ_{4} ⊕ ℤ_{8} ⊕ (ℤ_{3})^{4}  ℤ_{2}  ℤ_{4} ⊕ ℤ_{8} ⊕ (ℤ_{3})^{4}
26  ℤ_{3} ℤ_{4} ⊕ ℤ_{3} ⊕ ℤ ℤ_{2} ⊕ ℤ_{8} ⊕ (Z_{3})^{2} ⊕ ℤ^{4} (ℤ_{2})^{4} ⊕ ℤ_{4} ⊕ (ℤ_{3})^{2} ℤ_{2} ⊕ ℤ_{8} ⊕ (ℤ_{3})^{2}
30  0  ℤ_{2} ⊕ ℤ_{4} ⊕ ℤ  −−  −−  −−
33  ℤ_{4} ℤ_{2} ⊕ ℤ_{5} ⊕ ℤ ℤ_{2} ⊕ (ℤ_{4})^{3} ⊕ (ℤ_{3})^{2} ⊕ ℤ^{2}  (ℤ_{2})^{3}  ℤ_{2} ⊕ (ℤ_{4})^{3} ⊕ (ℤ_{3})^{2}
34  ℤ_{3}  ℤ_{16} ⊕ ℤ  ℤ_{2} ⊕ ℤ_{8} ⊕ (ℤ_{3})^{4} ⊕ ℤ^{2}  (ℤ_{2})^{2} ⊕ ℤ_{4}  ℤ_{2} ⊕ ℤ_{8} ⊕ (ℤ_{3})^{4}
35 ℤ_{4} ⊕ ℤ_{3} ℤ_{4} ⊕ ℤ_{3} ⊕ ℤ ℤ_{2} ⊕ (ℤ_{4})^{2} ⊕ (ℤ_{3})^{2} ⊕ ℤ^{6} (ℤ_{2})^{5} ⊕ (ℤ_{4})^{2} ⊕ (ℤ_{3})^{2} ℤ_{2} ⊕ (ℤ_{4})^{2} ⊕ (ℤ_{3})^{2}
38 ℤ_{3} ℤ_{2} ⊕ ℤ_{9} ⊕ ℤ ℤ_{4} ⊕ ℤ_{8} ⊕ (ℤ_{3})^{2} ⊕ ℤ^{4} (ℤ_{2})^{5} ⊕ (ℤ_{3})^{2} ℤ_{4} ⊕ ℤ_{8} ⊕ (ℤ_{3})^{2}
39  ℤ_{4} ℤ_{4} ⊕ ℤ_{3} ⊕ ℤ ℤ_{2} ⊕ (ℤ_{4})^{2} ⊕ ℤ_{3} ⊕ ℤ^{6} (ℤ_{2})^{5} ⊕ (ℤ_{4})^{2} ⊕ ℤ_{3} ℤ_{2} ⊕ (ℤ_{4})^{2} ⊕ ℤ_{3}
42  0 (ℤ_{2})^{2} ⊕ ℤ_{3} ⊕ ℤ  −−  −−  −−
46  ℤ_{3} ℤ_{2} ⊕ ℤ_{11} ⊕ ℤ ℤ_{4} ⊕ ℤ_{8} ⊕ (ℤ_{3})^{4} ⊕ ℤ^{2}  (ℤ_{2})^{3}  ℤ_{4} ⊕ ℤ_{8} ⊕ (ℤ_{3})^{4}
Table : Homology groups H_{n}(SL_{2}(ℤ[1/m]), ℤ)

Bui Anh Tuan  Page 17

Bottlenecks

HAP [3] package for the GAP [8] computational algebra system and has been used to compute H_{n}(SL_{2}(ℤ[1/m]), ℤ) for all m ≤ 50, m ≠ 30, 42 and all n ≥ 0.

The two main bottlenecks:

1 the ℤG-rank of R_{n}^{G} in higher degrees n; G = SL_{2}(ℤ[1/(2 · 3 · 5 · 7 · 11 · 13)]. Produced resolution R_{n}^{G} has ℤG-rank 1075200 for all n ≥ 7.

2 the length of the boundary ∂(e^{n}) of certain generators of R_{n}^{G} in higher degrees. G = SL_{2}(ℤ[1/30]), one generator f of R_{4}^{G} has boundary ∂_{4}(f) involving 1359551 ℤ-free generators of R_{3}^{G}.

⇒ Apply Tietze type reductions to R_{∗}^{G}

Bottlenecks

HAP [3] package for the GAP [8] computational algebra system and has been used to compute H_{n}(SL_{2}(ℤ[1/m]), ℤ) for all m ≤ 50, m ≠ 30, 42 and all n ≥ 0.

The two main bottlenecks:

1 the ℤG-rank of R_{n}^{G} in higher degrees n; G = SL_{2}(ℤ[1/(2 · 3 · 5 · 7 · 11 · 13)]. Produced resolution R_{n}^{G} has ℤG-rank 1075200 for all n ≥ 7.

2 the length of the boundary ∂(e^{n}) of certain generators of R_{n}^{G} in higher degrees. G = SL_{2}(ℤ[1/30]), one generator f of R_{4}^{G} has boundary ∂_{4}(f) involving 1359551 ℤ-free generators of R_{3}^{G}.

⇒ Apply Tietze type reductions to R_{∗}^{G}

Bottlenecks

HAP [3] package for the GAP [8] computational algebra system and has been used to compute H_{n}(SL_{2}(ℤ[1/m]), ℤ) for all m ≤ 50, m ≠ 30, 42 and all n ≥ 0.

The two main bottlenecks:

1 the ℤG-rank of R_{n}^{G} in higher degrees n; G = SL_{2}(ℤ[1/(2 · 3 · 5 · 7 · 11 · 13)]. Produced resolution R_{n}^{G} has ℤG-rank 1075200 for all n ≥ 7.

2 the length of the boundary ∂(e^{n}) of certain generators of R_{n}^{G} in higher degrees. G = SL_{2}(ℤ[1/30]), one generator f of R_{4}^{G} has boundary ∂_{4}(f) involving 1359551 ℤ-free generators of R_{3}^{G}.

⇒ Apply Tietze type reductions to R_{∗}^{G}

Bottlenecks

HAP [3] package for the GAP [8] computational algebra system and has been used to compute H_{n}(SL_{2}(ℤ[1/m]), ℤ) for all m ≤ 50, m ≠ 30, 42 and all n ≥ 0.

The two main bottlenecks:

1 the ℤG-rank of R_{n}^{G} in higher degrees n; G = SL_{2}(ℤ[1/(2 · 3 · 5 · 7 · 11 · 13)]. Produced resolution R_{n}^{G} has ℤG-rank 1075200 for all n ≥ 7.

2 the length of the boundary ∂(e^{n}) of certain generators of R_{n}^{G} in higher degrees. G = SL_{2}(ℤ[1/30]), one generator f of R_{4}^{G} has boundary ∂_{4}(f) involving 1359551 ℤ-free generators of R_{3}^{G}.

⇒ Apply Tietze type reductions to R_{∗}^{G}

Arithmetic Groups

Further work

1 Find a way to reduce the ℤG-rank of R_{n}^{G}. 2 Find a way to reduce the length of the boundary ∂(e^{n}). 3 Check Kevin Hutchinson’s conjecture:

H_{2}(SL_{2}(ℤ[1/(2 · 3 · 5 ⋯ p_{r})]) = ℤ ⊕ ℤ_{2} ⊕ ℤ_{4} ⋯ ⊕ ℤ_{p_{r}−1}

Crystallographic Groups

Crystallographic Groups

Bui Anh Tuan  Page 20

Crystallographic Groups

Introduction

Tilling Euclidean plane by regular polygons:

Figure : Triangles  Figure : Squares  Figure : Hexagons

Tilling Euclidean plane by regular polygons:

Figure : Triangles  Figure : Squares  Figure : Hexagons

Tilling Euclidean plane by regular polygons:

Figure : Triangles  Figure : Squares  Figure : Hexagons

Tilling Euclidean space by polytopes:

Tilling Euclidean space by polytopes:

Tilling Euclidean space by polytopes:

Translations Translations Translations Rotations Translations Rotations Translations Rotations Reflections Translations Rotations Reflections Glide Reflections

⟹ Isometries of the plane ⟹ Crystallographic groups Translations Rotations Reflections Glide Reflections

⟹ Isometries of the plane ⟹ Crystallographic groups

Crystallographic Groups

Definition

Definition 3.1

A crystallographic group Γ is a discrete group of isometries of an n-dimensional Euclidean space ℝ^{n} having a bounded fundamental domain. If G acts freely on ℝ^{n} then we call G a Bieberbach group.

Theorem 3.2 (Bieberbach)

1 Any n-dimensional crystallographic group Γ contains n linearly independent translations; the group of linear parts of the transformations from Γ is finite, we call it the point group of Γ.

2 Two crystallographic groups are equivalent if and only if they are

isomorphic as abstract groups. 3 For any n, there are up to equivalence only finitely many

n-dimensional crystallographic groups.

Definition

Definition 3.1

A crystallographic group Γ is a discrete group of isometries of an n-dimensional Euclidean space ℝ^{n} having a bounded fundamental domain. If G acts freely on ℝ^{n} then we call G a Bieberbach group.

Theorem 3.2 (Bieberbach)

1 Any n-dimensional crystallographic group Γ contains n linearly independent translations; the group of linear parts of the transformations from Γ is finite, we call it the point group of Γ.

2 Two crystallographic groups are equivalent if and only if they are

isomorphic as abstract groups. 3 For any n, there are up to equivalence only finitely many

n-dimensional crystallographic groups.

Crystallographic Groups

Classification of Crystallographic groups

dim Cryst.groups Bieberbach group
2  17  2
3  219  10
4  4783  74
5  222018  1060
6  28927915  38746

Table : Classification of low dimensional cryst.groups Bui Anh Tuan  Page 34

Existing methods

\underline{Problems:}

1 Cohomology of crystallographic groups. 2 Cohomology ring structure.

Existing methods:

1 Marc Röder [7]

1 Bieberbach groups 2 use convex hull computation 3 no cohomology ring structure implemented.

2 Wall’s extension method [9]

1 works for all crystallographic groups; 2 produces big ℤG-resolutions; 3 memory problem.

Existing methods

\underline{Problems:}

1 Cohomology of crystallographic groups. 2 Cohomology ring structure.

Existing methods:

1 Marc Röder [7]

1 Bieberbach groups 2 use convex hull computation 3 no cohomology ring structure implemented.

2 Wall’s extension method [9]

1 works for all crystallographic groups; 2 produces big ℤG-resolutions; 3 memory problem.

Existing methods

\underline{Problems:}

1 Cohomology of crystallographic groups. 2 Cohomology ring structure.

Existing methods:

1 Marc Röder [7]

1 Bieberbach groups 2 use convex hull computation 3 no cohomology ring structure implemented.

2 Wall’s extension method [9]

1 works for all crystallographic groups; 2 produces big ℤG-resolutions; 3 memory problem.

Existing methods

\underline{Problems:}

1 Cohomology of crystallographic groups. 2 Cohomology ring structure.

Existing methods:

1 Marc Röder [7]

1 Bieberbach groups 2 use convex hull computation 3 no cohomology ring structure implemented.

2 Wall’s extension method [9]

1 works for all crystallographic groups; 2 produces big ℤG-resolutions; 3 memory problem.

Existing methods

\underline{Problems:}

1 Cohomology of crystallographic groups. 2 Cohomology ring structure.

Existing methods:

1 Marc Röder [7]

1 Bieberbach groups 2 use convex hull computation 3 no cohomology ring structure implemented.

2 Wall’s extension method [9]

1 works for all crystallographic groups; 2 produces big ℤG-resolutions; 3 memory problem.

Existing methods

\underline{Problems:}

1 Cohomology of crystallographic groups. 2 Cohomology ring structure.

Existing methods:

1 Marc Röder [7]

1 Bieberbach groups 2 use convex hull computation 3 no cohomology ring structure implemented.

2 Wall’s extension method [9]

1 works for all crystallographic groups; 2 produces big ℤG-resolutions; 3 memory problem.

Existing methods

\underline{Problems:}

1 Cohomology of crystallographic groups. 2 Cohomology ring structure.

Existing methods:

1 Marc Röder [7]

1 Bieberbach groups 2 use convex hull computation 3 no cohomology ring structure implemented.

2 Wall’s extension method [9]

1 works for all crystallographic groups; 2 produces big ℤG-resolutions; 3 memory problem.

Existing methods

\underline{Problems:}

1 Cohomology of crystallographic groups. 2 Cohomology ring structure.

Existing methods:

1 Marc Röder [7]

1 Bieberbach groups 2 use convex hull computation 3 no cohomology ring structure implemented.

2 Wall’s extension method [9]

1 works for all crystallographic groups; 2 produces big ℤG-resolutions; 3 memory problem.

Existing methods

\underline{Problems:}

1 Cohomology of crystallographic groups. 2 Cohomology ring structure.

Existing methods:

1 Marc Röder [7]

1 Bieberbach groups 2 use convex hull computation 3 no cohomology ring structure implemented.

2 Wall’s extension method [9]

1 works for all crystallographic groups; 2 produces big ℤG-resolutions; 3 memory problem.

Crystallographic Groups

Existing methods

\underline{New method:}

1 faster than Wall’s extension method; 2 more general than Röder’s method.

Crystallographic Groups

Existing methods

\underline{New method:}

1 faster than Wall’s extension method; 2 more general than Röder’s method.

New method

We focus on cubical tessellation

1 high proportion of low-dimensional crystallographic groups admit such a cubical fundamental region (eg. 50% of 4-dim cryst.groups);

2 easily give explicit formulas for a contracting homotopy.

New method

We focus on cubical tessellation

1 high proportion of low-dimensional crystallographic groups admit such a cubical fundamental region (eg. 50% of 4-dim cryst.groups);

2 easily give explicit formulas for a contracting homotopy.

New method

We focus on cubical tessellation

1 high proportion of low-dimensional crystallographic groups admit such a cubical fundamental region (eg. 50% of 4-dim cryst.groups);

2 easily give explicit formulas for a contracting homotopy.

New method

1 Introduce G-full basis and define that a cryst.group is cubical if there

exists a G-full basis. 2 Procedure for finding G-full basis

Algorithm 3.1 G-full basis Input: a set of crystallographic matrices that generate a crystallographic group G.

Output: “fail” or a G-full basis.

Procedure:

1: Choose a basis B_{T} for a T -lattice. 2: Slice up B_{T} to d parts where d is a divisor of the order of point group

to obtain B_{δ}. 3: Check if B_{δ} is a G-full basis.

Results

Example 3.3

Let G be the crystallographic group generated by

(\begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 1 \end{matrix}), (\begin{matrix}−1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1/2 & 0 & 1 \end{matrix}), (\begin{matrix}1 & 0 & 0 & 0 \\ 0 & −1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 1/2 & 1 \end{matrix}).

H^{0}(G, ℤ) = ℤ H^{1}(G, ℤ) = ℤ H^{2}(G, ℤ) = ℤ_{2}^{2} H^{3}(G, ℤ) = ℤ_{2} H^{n}(G, ℤ) = 0, for all n ≥ 4. Cohomology ring: H^{∗}(G, ℤ) = ℤ[x, y, z]/ < x^{2}, 2xy, xz, y^{2}, z^{2}, yz, 2y, 2z >, where deg(x) = 1, deg(y) = deg(z) = 2.

Results

Example 3.3

Let G be the crystallographic group generated by

(\begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 1 \end{matrix}), (\begin{matrix}−1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1/2 & 0 & 1 \end{matrix}), (\begin{matrix}1 & 0 & 0 & 0 \\ 0 & −1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 1/2 & 1 \end{matrix}).

H^{0}(G, ℤ) = ℤ H^{1}(G, ℤ) = ℤ H^{2}(G, ℤ) = ℤ_{2}^{2} H^{3}(G, ℤ) = ℤ_{2} H^{n}(G, ℤ) = 0, for all n ≥ 4. Cohomology ring: H^{∗}(G, ℤ) = ℤ[x, y, z]/ < x^{2}, 2xy, xz, y^{2}, z^{2}, yz, 2y, 2z >, where deg(x) = 1, deg(y) = deg(z) = 2.

Results

Example 3.3

Let G be the crystallographic group generated by

(\begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 1 \end{matrix}), (\begin{matrix}−1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1/2 & 0 & 1 \end{matrix}), (\begin{matrix}1 & 0 & 0 & 0 \\ 0 & −1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 1/2 & 1 \end{matrix}).

H^{0}(G, ℤ) = ℤ H^{1}(G, ℤ) = ℤ H^{2}(G, ℤ) = ℤ_{2}^{2} H^{3}(G, ℤ) = ℤ_{2} H^{n}(G, ℤ) = 0, for all n ≥ 4. Cohomology ring: H^{∗}(G, ℤ) = ℤ[x, y, z]/ < x^{2}, 2xy, xz, y^{2}, z^{2}, yz, 2y, 2z >, where deg(x) = 1, deg(y) = deg(z) = 2.

Results

Example 3.3

Let G be the crystallographic group generated by

(\begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 1 \end{matrix}), (\begin{matrix}−1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1/2 & 0 & 1 \end{matrix}), (\begin{matrix}1 & 0 & 0 & 0 \\ 0 & −1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 1/2 & 1 \end{matrix}).

H^{0}(G, ℤ) = ℤ H^{1}(G, ℤ) = ℤ H^{2}(G, ℤ) = ℤ_{2}^{2} H^{3}(G, ℤ) = ℤ_{2} H^{n}(G, ℤ) = 0, for all n ≥ 4. Cohomology ring: H^{∗}(G, ℤ) = ℤ[x, y, z]/ < x^{2}, 2xy, xz, y^{2}, z^{2}, yz, 2y, 2z >, where deg(x) = 1, deg(y) = deg(z) = 2.

Crystallographic Groups

Results

\begin{matrix}Dimension & T otal & F ound & F ail & Gram(P) ≠ I_{n} \\ 2 & 17 & 12 & 5 & 0 \\ 3 & 219 & 114 & 19 & 86 \\ 4 & 4783 & 1996 & 716 & 2071 \end{matrix}

Table : Statistical result for dimension 2,3 and 4.

Bui Anh Tuan  Page 40

Crystallographic Groups

Comparisons

Example 3.4

Let G be the 3550th group of the 4-dim cryst.groups in GAP’s library.

1 Wall’s extension method takes 58183 ms to construct a free resolution

with the rank 1, 6, 18, 38, 66, 102 respect to six first degrees. 2 Our method give a free resolution in 3104 ms and the corresponding

rankss are 1, 3, 6, 7, 7, 7.

Example 3.4

Let G be the 3550th group of the 4-dim cryst.groups in GAP’s library.

1 Wall’s extension method takes 58183 ms to construct a free resolution

with the rank 1, 6, 18, 38, 66, 102 respect to six first degrees. 2 Our method give a free resolution in 3104 ms and the corresponding

rankss are 1, 3, 6, 7, 7, 7.

Example 3.4

Let G be the 3550th group of the 4-dim cryst.groups in GAP’s library.

1 Wall’s extension method takes 58183 ms to construct a free resolution

with the rank 1, 6, 18, 38, 66, 102 respect to six first degrees. 2 Our method give a free resolution in 3104 ms and the corresponding

rankss are 1, 3, 6, 7, 7, 7.

Example 3.5

Let G be the 2nd group of the 3-dim cryst.groups in GAP’s library.

1 Röder’s method may produce a fundamental region in a complicated shape.

2 Our method found one cubical fundamental region

gap> G:=SpaceGroup(3,2);;
gap> B:=CrystGFullBasis(G);
[[[1,0,0],[0,1,0],[0,0,1/2]],[1/2,1/2,1/4]]

Example 3.5

Let G be the 2nd group of the 3-dim cryst.groups in GAP’s library.

1 Röder’s method may produce a fundamental region in a complicated shape.

2 Our method found one cubical fundamental region

gap> G:=SpaceGroup(3,2);;
gap> B:=CrystGFullBasis(G);
[[[1,0,0],[0,1,0],[0,0,1/2]],[1/2,1/2,1/4]]

Further works

1 Conjecture: A crystallographic group admits a cubical fundamental region if and only if its point group is a subgroup of the group of all automorphism of an n-cube.

2 Which cryst.group admits a hexagoncylinder fundamental region.

3 Which cryst.group admits a truncatedoctahedron fundamental region.

Further works

1 Conjecture: A crystallographic group admits a cubical fundamental region if and only if its point group is a subgroup of the group of all automorphism of an n-cube.

2 Which cryst.group admits a hexagoncylinder fundamental region.

3 Which cryst.group admits a truncatedoctahedron fundamental region.

Further works

1 Conjecture: A crystallographic group admits a cubical fundamental region if and only if its point group is a subgroup of the group of all automorphism of an n-cube.

2 Which cryst.group admits a hexagoncylinder fundamental region.

3 Which cryst.group admits a truncatedoctahedron fundamental region.

Mod2-Cohomology ring of SL_{2}(O_{−2}[1/2]])

Mod2-Cohomology ring of

SL_{2}(O_{−2}[1/2])

Bui Anh Tuan  Page 44

Mod2-Cohomology ring of SL_{2}(O_{−2}[1/2]])

Introduction

\underline{Problem:} Calculate Mod2 cohomology ring H^{∗}(SL_{4}(ℤ[1/2]), 𝔽_{2})

Jointwork with Alexander Rahm & Gael Collinet: SL_{4}(ℤ[1/2]) acts on a cell complex with stabilizers: ▶ G_{1} = SL_{2}(ℤ[i][1/2]) (H^{∗}(G_{1}, 𝔽_{2}) is computed by Nicolas Weiss), ▶ G_{2} = SL_{2}(ℤ[\sqrt{2}][1/2]), ▶ G_{3} = SL_{2}(ℤ[\sqrt{−2}][1/2])

Mod2-Cohomology ring of SL_{2}(O_{−2}[1/2]])

Introduction

\underline{Problem:} Calculate Mod2 cohomology ring H^{∗}(SL_{4}(ℤ[1/2]), 𝔽_{2})

Jointwork with Alexander Rahm & Gael Collinet: SL_{4}(ℤ[1/2]) acts on a cell complex with stabilizers: ▶ G_{1} = SL_{2}(ℤ[i][1/2]) (H^{∗}(G_{1}, 𝔽_{2}) is computed by Nicolas Weiss), ▶ G_{2} = SL_{2}(ℤ[\sqrt{2}][1/2]), ▶ G_{3} = SL_{2}(ℤ[\sqrt{−2}][1/2])

Mod2-Cohomology ring of SL_{2}(O_{−2}[1/2]])

Quillen’s conjecture

Conjecture (Quillen)

Let l be a prime number. Let K be a number field with ζ_{l} ∈ K, and S a finite set of places containing the infinite places and the places over l. Then the composition res_{K,S} ∘ δ makes H^{∗}(GL_{n}(O_{K,S}), 𝔽_{l}) a free module over the cohomology ring H_{cts}^{∗}(GL_{n}(ℂ), 𝔽_{l}) ≅ 𝔽_{l}[c_{1}, ...c_{n}].

Positive cases in which the conjecture has been established are

1 n = l = 2, O_{K,S} = ℤ[1/2] by Mitchell, 2 n = 3, l = 2, O_{K,S} = ℤ[1/2] by Henn, 3 n = 2, l = 3, O_{K,S} = ℤ[ζ_{3}][1/3] by Anton.

Counterexamples:

1 n ≥ 32 and l = 2 by Dwyer,

2 n ≥ 14 and l = 2 by Henn and Lannes, 3 n ≥ 27 and l = 3 by Anton.

Bui Anh Tuan  Page 47

Quillen’s conjecture

Conjecture (Quillen)

Let l be a prime number. Let K be a number field with ζ_{l} ∈ K, and S a finite set of places containing the infinite places and the places over l. Then the composition res_{K,S} ∘ δ makes H^{∗}(GL_{n}(O_{K,S}), 𝔽_{l}) a free module over the cohomology ring H_{cts}^{∗}(GL_{n}(ℂ), 𝔽_{l}) ≅ 𝔽_{l}[c_{1}, ...c_{n}].

Positive cases in which the conjecture has been established are

1 n = l = 2, O_{K,S} = ℤ[1/2] by Mitchell, 2 n = 3, l = 2, O_{K,S} = ℤ[1/2] by Henn, 3 n = 2, l = 3, O_{K,S} = ℤ[ζ_{3}][1/3] by Anton.

Counterexamples:

1 n ≥ 32 and l = 2 by Dwyer,

2 n ≥ 14 and l = 2 by Henn and Lannes, 3 n ≥ 27 and l = 3 by Anton.

Bui Anh Tuan  Page 47

Quillen’s conjecture

Conjecture (Quillen)

Let l be a prime number. Let K be a number field with ζ_{l} ∈ K, and S a finite set of places containing the infinite places and the places over l. Then the composition res_{K,S} ∘ δ makes H^{∗}(GL_{n}(O_{K,S}), 𝔽_{l}) a free module over the cohomology ring H_{cts}^{∗}(GL_{n}(ℂ), 𝔽_{l}) ≅ 𝔽_{l}[c_{1}, ...c_{n}].

Positive cases in which the conjecture has been established are

1 n = l = 2, O_{K,S} = ℤ[1/2] by Mitchell, 2 n = 3, l = 2, O_{K,S} = ℤ[1/2] by Henn, 3 n = 2, l = 3, O_{K,S} = ℤ[ζ_{3}][1/3] by Anton.

Counterexamples:

1 n ≥ 32 and l = 2 by Dwyer,

2 n ≥ 14 and l = 2 by Henn and Lannes, 3 n ≥ 27 and l = 3 by Anton.

Bui Anh Tuan  Page 47

Quillen’s conjecture

Conjecture (Quillen)

Let l be a prime number. Let K be a number field with ζ_{l} ∈ K, and S a finite set of places containing the infinite places and the places over l. Then the composition res_{K,S} ∘ δ makes H^{∗}(GL_{n}(O_{K,S}), 𝔽_{l}) a free module over the cohomology ring H_{cts}^{∗}(GL_{n}(ℂ), 𝔽_{l}) ≅ 𝔽_{l}[c_{1}, ...c_{n}].

▶ G_{3} = SL_{2}(ℤ[\sqrt{−2}][1/2]): n = l = 2 and O_{K,S} = ℤ[\sqrt{−2}] is still open.

Results

dim_{𝔽_{2}}H^{q}(SL_{2}(ℤ[\sqrt{−2}][1/2]), 𝔽_{2}) = \begin{cases}2, & q = 4k + 6 \\ 1, & q = 4k + 5 \\ 3, & q = 4k + 4 \\ 4, & q = 4k + 3 \\ 1, & q = 2 \\ 0, & q = 1. \end{cases} For the ring

structure, this shows that H^{∗}(SL_{2}(ℤ[\sqrt{−2}][1/2]), 𝔽_{2}) is isomorphic to 𝔽_{2}[e_{4}](x_{2}, x_{3}, y_{3}, z_{3}, s_{3}, x_{4}, s_{4}, s_{5}, s_{6}), where the polynomial class e_{4} is in degree 4.

This satisfies the Quillen’s conjecture.

Mod2-Cohomology ring of SL_{2}(O_{−2}[1/2]])

Further works

1 Calculate H^{∗}(SL_{2}(ℤ[\sqrt{2}][1/2]), 𝔽_{2}). 2 Verify Quillen’s conjecture for n = l = 2 and O_{K,S} = ℤ[\sqrt{2}]. 3 Calculate H^{∗}(SL_{4}(ℤ[1/2]), 𝔽_{2}).

References

Ellis, G., 2004. Computing resolutions of groups. Joural of Symbolic Computation.

Adem, A., K. Dekimpe, Petrosyan, N., and Putrycz, B., 2012. On generators of crystallographic groups and actions on flat orbifolds. Journal of Group Theory 15(4), 553-561.

Ellis, G., 2008. hap – Homological Algebra programming, Version 1.8, an official package for the gap computational algebra system.

(http://www.gap-system.org/Packages/hap.html)

Ellis, G., An introduction to Computational Homotopy. To be published.

Forman, R., 1998. Morse theory for cell complexes. Advances in Mathematics 134, 90–145.

Oppenorth, J., Plesken, W., and Schulz, T., 2008. CARAT – [C]rystallographic [A]lgo[R]ithms [A]nd [T]ables.

(http://wwwb.math.rwth-aachen.de/carat/index.html).

References

Röder, M., 2013. hapcryst – A HAP extension for crytallographic groups, Version 0.1.11, an official package for the gap computational algebra system.

(http://www.gap-system.org/Packages/hapcryst.html)

The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.7.7.

(http://www.gap-system.org)

Wall, C.T.C., 1961. Resolutions of extensions of groups. Proc. Cambridge Philos. Soc. 57, 251–255.

Dekimpe, K., and Petrosyan, N., 2009. Homology of Hantzsche-Wendt groups. AMS Discrete Groups and Geometric Structures 501, 87–102.

Ellis, Graham and Fintan Hegarty,. 2014. Computational homotopy of finite regular CW-spaces. Journal of Homotopy and Related Structures 25-54.

References

Adem, A., Naffah, N., On the cohomology of SL2(Z[1/p]), in: Geometry and Cohomology in Group Theory, Durham, 1994, in: London Math. Soc. Lecture Note Ser., vol. 252, 1998, pp. 19.