Discrete vector fields and the cohomology of certain arithmetic and crystallographic groups

We devise and implement an algorithm for computing a finiteZG-equivariantCW-space with nice cell stabilizer groups and a contracting discrete vectorfield, where G = SL2Z[1/m] for any pos

Discrete vector fields

Bui Anh Tuan

Supervisor: Professor Graham Ellis

School of Mathematics, Statistics and Applied Mathematics

National University of Ireland, Galway

January 2015

1.1 Outline of the thesis 2

1.2 Main goals of the thesis 2

1.3 Review of necessary background material 4

1.3.1 CW-spaces 4

1.3.2 Cellular chain complexes 7

1.3.3 (Co)homology of groups and ring structure 8

2 Vector fields and perturbations 11 2.1 Wall’s technique 12

2.1.1 Proof for theorem 2.0.7 14

2.2 Discrete vector fields 16

2.3 Proof of theorem 2.0.6 and related algorithm 18

3 Homology of SL2(Z[1/m]) 24 3.1 Resolutions for groups acting on trees 26

i

Contents ii

3.1.1 Proof of Theorem 3.1.1 and related algorithms 28

3.2 Resolution for SL(2,Z[1/m]) 31

3.2.1 Theoretical methods 31

3.2.2 Practical results 35

4 Crystallographic groups with cubical fundamental domain 38 4.1 Introduction 39

4.1.1 Wall’s extension method 41

4.1.2 R¨oder’s method 42

4.1.3 Our contribution 43

4.2 Some definitions 43

4.3 Cubical fundamental domain for crystallographic groups 46

4.4 Non-free resolutions for crystallographic groups 52

4.4.1 Construction of non-free resolutions 52

4.4.2 Contracting homotopy for cubical case 55

4.5 Free resolutions for crystallographic groups 56

4.6 Cup product, cohomology ring structure and a proof for Proposition 4.1.1 57

4.7 Three dimensional Bieberbach groups 58

4.7.1 First group 59

4.7.2 Second group 60

4.7.3 Third group 61

4.7.4 Fourth group 62

Contents iii

4.7.5 Fifth group 63

4.7.6 Fourth group 65

4.7.7 Sixth group 66

4.7.8 Seventh group 67

4.7.9 Eighth group 68

4.7.10 Ninth group 70

4.7.11 Tenth group 70

4.8 Experimental results 70

4.8.1 Comparison to Wall’s extension method 71

4.8.2 Comparison to R¨oder’s method 71

5 Bianchi groups 74 5.1 Reviews of SL(2,O−2) 75

5.2 H1(SL(2,O−2)[1/2],Z) 78

5.2.1 ”non-standard” conjugation with the uniformizerπ = √ −2 79

5.2.2 Natural injection 80

5.3 Algorithmetic method for finding a contracting homotopy 81

avail-2 We devise and implement an algorithm for computing a finiteZG-equivariantCW-space with nice cell stabilizer groups and a contracting discrete vectorfield, where G = SL2(Z[1/m]) for any positive integer m (See Algorithm

3.2.1.)

3 We implement a function which inputs a non-free ZG-resolution and outputsfinitely many terms of a freeZG-resolution RG

∗ ofZ, where G = SL2(Z[1/m])for any positive integer m (See Algorithm2.1.1.)

4 We devise and implement an algorithm for computing finitely many terms of afreeZH-resolution RH

∗ ofZ for H a finite index subgroup of G = SL2(Z[1/m]).(See Algorithm3.2.2.)

5 We devise and implement an algorithm that attempts to find a cubical mental cell for a cubical crystallographic group G (See Algorithm 4.3.1)

funda-6 We devise and implement an algorithm that inputs a crystallographic groupGtogether with a cubical fundamental cell and outputs a finite ZG-equivariantCW-space with nice cell stabilizer groups and a contracting discrete vectorfield (See Algorithm 4.4.1)

iv

8 We devise and implement an algorithm for computing a freeZG-resolution RG

10 We implement a function for calculating a free ZΓ-resolution of Z for Γ thecongruence subgroup of level √

−2 of G = SL2(O−2)

Chapter 1

Introduction

1

1.1 Outline of the thesis 2

This thesis has seven chapters

Chapter 1 includes two sections In the first section we present the main goal ofthe thesis, namely methods for calculating the homology of SL(2,Z[1/m]) and thecohomology ring structure of certain crystallographic groups The second sectionrecalls standard material that will be used in the thesis

Chapter 2 introduces discrete vector fields in the context of group cohomology andexplains how they can be used to find a contracting homotopy on a classifying space

We also describe a homological perturbation lemma that can be used to construct

a free resolution from a non-free resolution

Chapter 3is devoted to constructing a free resolution for SL(2,Z[1/m]) where m isany positive integer, and calculating the homology of such groups We describe analgorithm for computing finitely many terms of a free ZG-resolution of Z for G afinite index subgroup of SL(2,Z[1/m] An implementation of the algorithm is used

to determine the integral homology groups Hn(SL(2,Z[1/m]), Z) for all integers

m≤ 50 (m 6= 36, 42) , and n ≥ 0

Chapter 4 provides a method for calculating the cohomology ring structure for clidean crystallographic groups with cubical fundamental domain We describe al-gorithms for attemping to decide if a given crystallogrpahic group admits a cubicalfundamental domain; and calculating the resulting cellular chain complex as a ZG-resolution We also give a method for computing the contracting homotopy on thechain complex Finally we provide a computer method for calculating the cohomol-ogy cup product

This thesis has two main topics which are Arithmetic groups and Crystallographicgroups

1.2 Main goals of the thesis 3

The work on arithmetic groupsSL(2,Z[1/m]) was motivated by a question of KevinHutchinson In his paper [18], Hutchinson stated that he is interested in the precisestructure ofH3(SL2(Q), Z) and the answer is still unknown This chapter is writtennot to solve Hutchinson’s question but to use it as the motivation since we all knowthe class of integral rings Z[1/m] is very close to the field of rational numbers Q asfollows,

Z[1/2] ⊂ Z[1/2.3] ⊂ Z[1/2.3.5] ⊂ · · · ⊂ Z[1/2.3.5 ] = Q

The work is also a generalization of the problem stated in the paper ”On the mology of SL(2,Z[1/p])” by A Adem and N Naffah [1] in which the authors onlysolve the problem for primes p

coho-The main goal of this part is to present a new method for calculating the grouphomology of the arithmetic groups SL(2,Z[1/m])

Crystallographic groups have been studied for many years and are still an activetopic of research Some recently published works in the area are [2, 5, 7] InGAP[14], we know two methods for calculating the group homology of crystallographicgroups

One of those two methods is developed and implemented by Graham Ellis, presented

by a function in HAP [9] His method based on a technique of C.T.C Wall [28]about group extensions T → G → P The technique can be applied to a crystallo-graphic group given some methods for computing the free resolutions RT

∗ and RP

∗.The free ZG-resolution can be constructed by combining those resolutions Thismethod works for all crystallographic groups and yields cohomology rings for suchgroups However, for some cases, it produces big ZG-resolutions or doesn’t stopafter one hour of running and some give an error of exceeding permitted memory.Another method provided by Marc R¨oder [23] His algorithm uses convex hull com-putations to construct a fundamental domain for a Bieberbach group and produces

a finite regular CW-space for such a group The disadvantages are that R¨oder siders only Biebebach groups and uses computational geometry software Moreover,for some groups, the fundamental domain produced by this method may have acomplicated shape which we hardly see a accomplishable method to find a contract-ing homotopy on the cellular chain complexes involving in order to compute the

con-1.3 Review of necessary background material 4

cohomology rings for such groups

By seeing the disadvantages of those methods above, we was interested in finding anew method which could be faster than Ellis and Wall’s and applied to many groupsrather than R¨oder’s At the time we are writing this thesis, we limit our goal tothe groups that admit a cubical fundamental domain The reasons are that: weobserve that a high proportion of low-dimensional crystallographic groups admit acubical fundamental domain; and we can easily give explicit formula for a contract-ing homotopy on the cellular chain complex involving Furthermore, by subdividingthe cubical fundamental domain, we can obtain the Bredon homology which by thelimit of this thesis, we will not mention here

The main goals of this part are to introduce: (i) new algorithms which attempt tocalculate the group (co)homology of certain crystallographic groups; (ii) a methodfor computing the cohomology ring structures for those groups who was successfullyapplied in (i)

This section recalls the basic definitions and concepts needed in the thesis Thematerial is standard and taken from [17, 22, 16] with little or no modification

1+x2

2+ + x2

n

1.3 Review of necessary background material 5

Definition 1.3.1 [16] An n-cell is a space homeomorphic to the open n-diskint(Dn) A cell is a space which is an n-cell for some n ≥ 0 We say the dimension

of such a cell is n

Definition 1.3.2 [16] A cell-decomposition (or cell-structure) of a space X is afamily E = {eα|α ∈ I} of subspaces of X such that each eα is a cell and

X =aα∈I

cell-• Axiom 1: (Characteristic Maps) For each n-cell e ∈ E there is a map ϕe :

Dn → X restricting to a homeomorphism ϕe|int(D n ):int(Dn)→ e and taking

Sn−1 into Xn−1

• Axiom 2: (Closure Finiteness) For any cell e ∈ E the closure ¯e intersects only

a finite number of other cells in E

• Axiom 3: (Weak Topology) A subset A ⊆ X is closed iff A ∩ ¯e is closed in Xfor each e∈ E

The restrictions ϕe|∂D n are called the attaching maps

Notice that we can recover X (up to homeomorphism) from a knowledge of X0 andthe attaching maps The recovery is described in [17] as follows:

(1) Start with a discrete setX0, whose points are regarded as 0-cells

(2) Inductively, form the n-skeleton Xn from Xn−1 by attaching n-cells en

α viamaps ϕα : Sn−1 → Xn−1 This means that Xn is the quotient space of thedisjoint union Xn−1`

αDn

α of Xn−1 with a collection of n-disks Dn

α under the

1.3 Review of necessary background material 6

identificationsx∼ ϕα(x) for x ∈ ∂Dn

α Thus as a set Xn=Xn−1`

αen

α whereeachen

α is an open n-cell

(3) One can either stop this inductive process at a finite stage, setting X = Xnfor some n < ∞, or one can continue indefinitely, setting X =SnXn In thelatter case X is given the week topology: A set A ⊂ X is open (or closed) iff

A∩ Xn is open (or closed) in Xn for each n

Definition 1.3.5 [15] A map f : X → Y between CW-spaces is cellular if itsatisfies f (Xn)⊆ Yn for all n

Definition 1.3.6 [16] A CW-subspace of a CW-space X is a union A of cells in

X such that the closure of each cell is also contained in A; therefore, it is also aCW-space

In other words, given a CW-space X = S

α∈Ieα and subset I0

⊂ I, the union

Y =S

α∈I 0eα is a CW-subspace of X if ¯eα ⊂ Y for all α ∈ I0

Thus Xn is a CW-subspace of X, for every n

Let X be a CW-space and Y be a CW-subspace of X We say that (X,Y) is aCW-pair For an arbitrary CW-space X, we have CW-pairs (X, Xn) for all n andsimilarly (Xn, Xm) forn ≥ m

Proposition 1.3.1 [15] Let (X, Y ) be a CW-pair Then the quotient space X/Ycan be turned in a CW-space such that the quotient map X → X/Y is cellular.Definition 1.3.7 [17] A G-space is a CW-space together with an action of G on

X which permutes cells That means, for each g ∈ G a homeomorphism x → gx of

X such that the image gσ of any cell σ is again a cell

1.3 Review of necessary background material 7

Let R be a ring We mainly interested in the integers R = Z and the group ring

R =ZG, where G is a group

Definition 1.3.8 [29] A chain complex C∗ = (Cn, dn)n∈N of R-modules is a quence of homomorphisms of R-modules

se-· se-· se-· dn+2 //Cn+1 d n+1 //Cn d n //Cn−1d n−1 //· · · d2 //C1 d 1 //C0 d 0 //0

such that dndn+1 = 0 for all n The chain complex C∗ is called an exact sequence if

Imdn+1 = Kerdn for all n

We say that the chain complex is of length n if Ck = 0 for k > n and Cn 6= 0 Werefer to the homomorphisms dn as a boundary homomorphisms The module Ck issaid to be of degree k

Definition 1.3.9 [29] The k-dimensional homology of a chain complexC∗ overR

is the R−module

Hk(C∗) = Kerdk/Im dn+1.And the k-dimensional cohomology of C∗ is

Hk(C∗) = Hk(HomR(C∗, R))

Suppose that X is a CW-space

Definition 1.3.10 [17] The cellular chain complex C∗(X) is the chain complexthat Cn(X) is the free abelian group generated by all n-cells in X

Cn(X) =

b nMi=1

Zen i

where bn is the number of n-cells; and the boundary map can be defined by usingsimplicial homology The definition of the boundary is slightly non-trivial and wewill not described it here See [17] for the standard definition

For the purpose of this thesis, we are mainly interested in regular CW-spaces In

1.3 Review of necessary background material 8

this case, we use a more practical construction for the cellular chain complex based

on the following proposition

Proposition 1.3.2 [21] Let X be a regular CW-space and letCk be the free abeliangroup formally generated by the k-cells of X Let d0

k : Ck → Ck1 be any sequence ofhomomorphisms satisfying

Then the chain complex (C∗, d0

∗) and the chain complex C∗(X) of Definition 1.3.10

have isomorphic homology Hn(C∗) ∼=Hn(C0

∗), for n ≥ 0

Definition 1.3.11 [20] The homology groups of a CW-space X are defined to bethe homology groups of the cellular chain complex C∗(X)

This section recalls details on homology and cohomology of group via topology Themain reference of this section is K S Brown’s book [4]

Definition 1.3.12 [4] Let R be a ring and M a (left) R-module A resolution of

M is an exact sequence of R-modules

//F2 ∂ 2 //F1 ∂ 1 //F0  //M //0

If each Fi is projective (free), then this is called a projective (free) resolution

1.3 Review of necessary background material 9

Definition 1.3.13 [4] Let G be a group and ε : F → Z a projective resolution of

Z over ZG, where ZG is the integral group ring of G and Z can be considered as aZG-module with trivial G-action We define the n-th integral homology groups of Gby

Hn(G,Z) = Hn(F ⊗ZGZ)and the n-th integral cohomology groups of G by

Hn(G,Z) = Hn(HomZG(F,Z))

If X is a G-space then the action of G on X induces an action of G on the cellularchain complex C(X) which thereby becomes a chain complex of G-modules More-over, the canonical augmentation  : C0(X) → Z (defined by (v) = 1 for every0-cells v of X) is a map of G-modules In this case, we say that X is a free G-space

if the action of G freely permutes the cells of X And so that, each chain module

Cn(X) is a free ZG-module with one basis element for every G-orbit of n-cells.Proposition 1.3.3 [4] Let X be a contractible free G-space Then the augmentedcellular chain complex of X is a free resolution of Z over ZG

Proposition 1.3.4 [4] Let X be a free G-space then

Propo-The cup product in cohomology is an operation which turns the cohomology groups

of a group G into a graded ring: H∗(G, R) = L

k≥0Hk(G, R) Details on cup

1.3 Review of necessary background material 10

products can be found in [4] In this section, we introduce a construction of the cupproduct which is implemented in HAP [9]

for all n ≥ 0 with h−1 = 0 and δ0 = 0

For any groupG there is a bilinear mapping:

Hp(G,Z) ⊕ Hq(G,Z) → Hp+q(G,Z), (u, v) 7→ uv

called the cup product The product is associative, and uv = (−1)pqvu Thecup product gives a multiplication on the direct sum of the cohomology groups

H∗(G,Z) = ⊕k∈ NHk(G,Z) This multiplication turns H∗(G;Z) into a ring and now

is called integral cohomology ring of G The construction of the cup product is asfollows: The cohomology classu∈ Hp(G,Z) is represented by a cocycle u : RG

p → Zwhich induces a chain mapping un:RG

n → RG

n−p (for n > p− 1) by recursion usingthe contracting homotopy as the diagram below:

RG n−p //RG

n−p−1

h n−p−1

RG n−p−2

h qq n−p−2 //

RG 1

h 1

RG 0

h 0

Zh

un−1

RG n−2

un−2

RG p+1 δ1 //

Chapter 2

Vector fields and perturbations

11

2.1 Wall’s technique 12

In this chapter, we prove the following useful computational result

Theorem 2.0.6 Let G = SL2(Z) There is a free ZG-resolution RG

∗ of Z withprecisely two free generators in each degree greater than 1 and with boundary homo-morphism and contracting homotopy explicitly described below

To prove this theorem we need

(i) a perturbation technique of Wall [28] and

(ii) a discrete vector field on a contractible CW-space X with (non-free) action of

In this chapter we also give a proof for the following theorem in section

Theorem 2.0.7 Let G be a discrete group and X be an n-dimensional regularcontractibleG-space such that for each cell σ of X, the stabilizer Gσ fixesσ pointwise.Suppose that all stabilizer subgroups are contained in a fix periodic group Then there

is a free periodic ZG-resolution in degree greater than n

LetG be a discrete group acting on a contractible CW-space X in such a way that theaction permutes cells Each cell e in X has stabilizer group Ge={g ∈ G : g.e = e}whose elements need not stabilize e point-wise Let [e] denote the equivalence class

of cells in the orbit of e under the action of G, and let Orb(n) denote the set ofequivalence classes of n-dimensional cells The cellular chain complex C∗(X) of X

is an exact sequence of ZG-modules with H0(C∗(X)) =Z and with

Cp(X) = M

[e]∈Orb(p)

ZG ⊗ZG eZe

2.1 Wall’s technique 13

where each Ze is a copy of the integers endowed with an “orientation” action of

Ge The chain complex C∗(X) is a ZG-resolution of Z but generally not free Themethod below gives a construction that inputs the non-free ZG-resolution C∗(X)together with the freeZGe-resolutionsRG e

∗ , and outputs a freeZG-resolution RG

∗ ofZ

Suppose that for each class [e] we are given a free ZGe-resolution RG e

∗ of Z Bydefining

d0(x) for all x ∈ Ap,q+1 Then we can construct the module homomorphisms dk :

2.1 Wall’s technique 14

Ap,q → Ap−k,q+k−1 and the differential d of Proposition 2.1.1 by first lifting δ to

d1 :Ap,0 → Ap−1,0 and then recursively defining dk=−h0(Pk

i=1didk−i) on free erators of the module Ap,q Furthermore, if H0(C∗) ∼=Z and each Cp is free abelian,

gen-we can construct abelian group homomorphismsh :⊕p+q=nAp,q → ⊕p+q=n−1Ap,q isfying dhd(x) = d(x) by setting h(ap,q =h0(a + p, q)− hd+h0(ap,q) +ε(ap,q) for freegenerators ap,q of the abelian group Ap,q Here d+ = Pp

sat-i=1 and, for q ≥ 1, ε = 0.For q = 0 we define ε = h1− h0d+h1+hd+h1+hd+h0d+h1 where h1 :Ap,0 → Ap+1,0

is an abelian group homomorphism induced by a contracting homotopy on C∗

The above construction can be implemented in the form of the following algorithm

Algorithm 2.1.1 Converting a non-free ZG-resolution to a free ZG-resolutionInput:

• a non-free ZG-resolution C∗ of Z together with free ZGe-resolution of Z forall stabilizersGe with contracting homotopy, and

• an integer n ≥ 0

Output: The first n + 1 terms of a free ZG-resolution RG

∗ of Z Moreover, if theinput C∗ is together with a contracting homotopy then the output also togetherwith a contracting homotopy

Procedure:

1: The nth-term RG

n is computed by the formula2.1;

2: The boundary is constructed by using Proposition 2.1.1;

3: If the input together with a contracting homotopy then the contracting topy of RG

homo-∗ is constructed by using Proposition 2.1.2

Algorithm2.1.1has already been implemented inHAP package without a ing homotopy The contracting homotopy implementation is new and has been done

contract-by the author

LetG be a discrete group and X be an n-dimensional regular contractible G-space.Let [ei] denote the equivalence class of i-cells in the orbit of ei under the action of

∗ can be regarded as a free ZGe i-resolution of

Z Following the construction using Wall’s technique, we have a free ZG-resolution

k

So that, for all k≥ n,

RG k+m =RG

We now prove the periodicity of the differential maps d∗ :RG

∗ → RG

∗−1Lemma 2.1.3

d1p,q =d1p,q+mwhere d1

p,q:Ap,q→ Ap−1,q, for all 1≤ p ≤ n and q ≥ 0

The periodicity of dk : Ap,q → Ap−k,q+k−1 can be obtained by induction starting

2.2 Discrete vector fields 16

from the periodicity of d1 in Lemma 2.1.3

Assume that the periodicity ofdkare proved for allk ≤ N, means that dk

p,q=dk p,q+m

We will prove that dN+1

p,q =dN+1p,q+m By Proposition 2.1.2,

dN +1− p, q = −h0(

NX+1 i=1

In this section we recall the concept of discrete vector field and introduce theoreticaland algorithmic methods to construct a contracting homotopy on a CW-space as anapplication of discrete vector field Material is mainly taken from [13, 12, 24, 19]

Let X be a CW-space, whenever a cell τ ∈ X is attached to a cell σ, we call σ is

a face of τ ; a face of codimension 1 is called a facet If the attaching map ϕτ of

τ restricts to a homeomorphism on the preimage ϕ−1

τ (σ) and the closure of ϕ−1

2.2 Discrete vector fields 17

• 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

An arrow s → t involves two cells s and t, where s is called the source and t thetarget of such arrow We also write t = V (s) and s = V−1(t) A cell which is notinvolved in any arrow is called a critical cell A cell basis βn is canonically divided

by the vector fieldV into three components βn =βt

n) is made of the target (respectively source, critical)n-cells

Definition 2.2.3 [13, 24] Let V be a discrete vector field on CW-space X Then

a V -path γ of length r from a k-dimensional cell σk to a k-dimensional cell τk is asequence of arrows ((s0 → t0), (s1 → t1), , (sr−1 → tr−1))⊂ V satisfying:

• σk=s0 and τk in the boundary of tr−1,

• si+1 lies in the boundary of ti for all 0≤ i < r − 1

A V-path γ is said to be a cycle if s0 lies in the boundary of tr−1

Definition 2.2.4 [13] A discrete vector field is admissible if there is no cycle andfor every source cell s, the length of any path starting from s is bounded by a fixedinteger λ(s), i.e there is no path of infinite length

Theorem 2.2.1 [13,30] If X is a regular CW-space with admissible discrete vectorfield then there is a homotopy equivalence

X ' Y

where Y is a CW-space whose cells are in one-one correspondence with the criticalcells of X

Proof of Theorem 2.2.1 can be found in [30]

We now consider a very special case when the CW-space X is contractible, meansthat it can be retracted to a single-point-space If there is a discrete vector fielddescribing such a retraction, we shall call it a contracting vector field

2.3 Proof of theorem 2.0.6 and related algorithm 18

Figure 2.1: DVF on cubical CW-space

Figure 2.1 is an example of a contracting discrete vector field on a 2-dimensionalCW-space with the cubical cell structure

Theorem 2.2.2 [13, 11, 19] Let C(X) = (C∗(X), d∗, β∗) be the cellular chaincomplex of a contractible CW-space X together with a contracting discrete vectorfield V Then the contracting homotopy on C(X) can be defined as the sum of allmaximal V-paths starting from any n-cell σn,





Consider the matrices

2.3 Proof of theorem 2.0.6 and related algorithm 19

Figure 2.2: Portion of the cubic tree indicated in red

Note that S4 = (ST )6 = 1 and that the group SL2(Z) is generated by S and T

We let M denote the group with presentation hs, u | s2 = u3 = 1i Following [6]

we construct the cubic tree T by taking the left cosets of U = hui in M as vertices,and joining cosets xU and yU by an edge if, and only if, x−1y ∈ UsU Thus thevertex U is joined to sU , usU and u2sU The vertices of this tree are in one-to-onecorrespondence with all reduced words in s, u and u2 that, apart from the identity,end in s We say that a sequence of vertices x0U , x1U , , xnU is a rooted path if

x0 = 1 and there is an edge between xiU and xi+1U for 0 ≤ i ≤ n − 1 A rootedpath is described by a reduced word in s, u and u2 ending in s

The group M can be realized as the modular group of transformations z 7→ (mz +n)/(pz + q), m, n, p, q integers with mq− np = 1, of the upper half complex plane

H = {x + iy ∈ C : y > 0} which is generated by the two transformations a(z) =

−1/z and u(z) = z + 1 The cubic tree can thus be embedded in H Figure 2.2

illustrates this embedding by showing a portion of the cubic tree in bold againstthe tessellation of H by triangular fundamental domains for the modular group.The surjection SL2(Z) → M, S 7→ s, T 7→ s−1u and left multiplication in M yield

an action of SL2(Z) on the cubic tree T Under this action there is one orbit ofvertices and one orbit of edges The stabilizer group of any vertex is conjugate to

C6 =hST i The stabilizer group of any edge is conjugate to C4 =hSi The cellularchain complex C∗(T ) thus has the form

C∗(T ) : Z[SL2(Z)] ⊗ZC 4 Ze δ

−→ Z[SL2(Z)] ⊗ZC 6 Z (2.4)

whereC6acts trivially onZ and where Zedenotes the integers with non-trivial action

2.3 Proof of theorem 2.0.6 and related algorithm 20

ofC4 The boundary homomorphismδ is induced by the equivariant homomorphism

on the chain complex2.5 The existence of this latter contracting homotopy impliesthat 2.5 is exact and thus a free ZG-resolution of Z

In order to implement the contracting homotopy on 2.5 we need to provide a tracting homotopy conC∗(T ) We do this by specifying a discrete vector field on thecubic tree with just one critical vertex and no critial edge Above, we have identifiedthe vertices of T with the left cosets of U ≤ M But since we are constructing a

con-2.3 Proof of theorem 2.0.6 and related algorithm 21ZG-resolution for G = SL(2, Z) it is more practical to identify the vertices of Twith the left cosets of the group hST i in G Each left coset consists of six matrices.

We use the following two rules to construct a discrete vector field on T Supposethat a vertex AhST i of T is represented by a matrix

1 If A∈ hST i then the vertex AhST i is critical

2 Otherwise, the vector field has an arrow from vertexAhST i to the edge taining this vertex and the vertexBhST i with matrix B defined by

T−1A if Sign(a) = Sign(c) and|a| = |c|,

T A if Sign(a) =−Sign(c) and |a| ≥ |c|

Rules 1 and 2 provide a recursive algorithm for expressing an arbitrary matrixA∈ G

as a product of S, T, T−1 and −I where I is the identity matrix For instance, thematrix

2.3 Proof of theorem 2.0.6 and related algorithm 22Algorithm 2.3.1 Decompose elements of SL2(Z)

Input: An element g ∈ SL2(Z)

Output: A product of S, T, T−1 and −I

Procedure:

1: Follow the method described in rules 1 and 2; and the above example

Algorithm 2.3.2 Cellular chain complex C∗(T )

Input: Void

Output: The cellular chain complex C∗(T ), where T is the cubic tree that G acts

on The chain complex is together with a contracting homotopy

Procedure:

1: Find the chain conplex C∗(T ) by using 2.4

2: The contracting homotopy on C∗(T ) is calculated using Algorithm 2.3.1

The computation of the cohomology ring H∗(G,Z) for a group G is described inSection 1.3.3

The following GAP session constructs a freeZG-resolution RG

∗ ofZ for G = SL2(Z),

up to degreed = 3, and uses it to compute a minimal set X that generates H∗(G,Z)

up to degree 2 The setX consists of two homogeneous cohomology classes of degrees

2.3 Proof of theorem 2.0.6 and related algorithm 23gap> IntegralCohomologyGenerators(R,0);

H∗(SL2(Z), Z) = Z12[x2]where x2 is the generator of H2(G)

Chapter 3

24

in thehap [9] package for the gap [14] computational algebra system and has beenused to compute Hn(SL2(Z[1/m]), Z) for all m ≤ 50, m 6= 30, 42 and all n ≥ 0.When m = 30 or m = 42 the implementation is practical only for n = 1, 2 Table

?? summarizes these computations

The homology of Hn(SL2(Z[1/m]), Z) is known for all primes m = p by work ofAdem and Naffah [1] and these prime cases are thus omitted from Table ?? Thetable also omits cases where m is divisible by the square of a prime as these coincidewith square free cases It is known by work of Williams and Wisner [31] that thehomology is a finite group with only 2-torsion and 3-torsion for n > k + 1 when

m = p1p2 pk is a product of k primes In fact, from our algorithm we can provethat the homology is periodic of period 2 in degrees > k + 1 (See lemma 3.2.1 for

a proof.)

Each completed row of the table thus describes the integral homology of a group forall degrees n

3.1 Resolutions for groups acting on trees 26

Table 3.1: Homology groups Hn(SL2(Z[1/m]), Z)

The algorithm works inductively by expressing a square free integer m in the form

m = pm0 with p a prime, and using the decomposition

SL2(Z[1/m]) ∼=SL2(Z[1/m0

])∗Γ 0 (p)SL2(Z[1/m0

])

as an amalgamated free product of two distinct copies ofSL2(Z[1/m0]) inSL2(Z[1/m])

over the congruence subgroup Γ0(p)⊂ SL2(Z[1/m]) A special case of a

homolog-ical perturbation technique given in [10] (which, in turn, is based on a result of

Wall [28]) is used to construct a resolution for SL2(Z[1/m]) from resolutions for

SL2(Z[1/m0]) and Γ0(p) Although our primary interest is for integral coefficients,

our implementation of the algorithm in HAP[9] can be used for arbitrary finitely

generated coefficient modules

In this section, we will give a contruction of a free resolution RG

∗ with contractinghomotopy for any amalgamated free product G = G1 ∗AG2 And by that give a

3.1 Resolutions for groups acting on trees 27

proof for the following theorem

Theorem 3.1.1 A free resolution RG

∗ with contracting homotopy can be built gorithmically) for an amalgamated free product G = G1∗AG2 if we are given

(al-1 free resolutions RG 1

∗ , RG 2

∗ , RA

∗, all together with a contracting homotopy, and

2 a contracting homotopy on the cellular chain complex of the tree associated tothe amalgamated product

The tools for the construction are a discrete vector field and a practical version ofWall’s lemma which was mentioned in Chapter2 Before going into the construction,

we recall some needed basic concepts

A graph consists of:

(i) a non-empty set X (called vertices),

(ii) a set Y (called oriented edges),

(iii) a mapY → X × X which sends any e ∈ Y to the pair (o(e); t(e)) of its originvertex o(e) and the terminal vertex t(e) and

(iv) a map from Y to itself which sends each edge e to its inverse edge ¯e which isdifferent from e and has its origin and terminus switched and so that ¯¯e = e

A path of length n in a graph is a concatenation e1 en of edges where ei starts

at the vertex where ei−1 ends for i = 2; ; n We say that the path is from o(e1)

to t(en) A circuit is a path as above where en ends at the initial vertex of e1 Acircuit of length 1 is called a loop A graph is said to be connected if each pair ofvertices is contained in some path

A tree is a connected non-empty graph without circuits A group G is said to act

on a graph Γ = (X; Y ) if G acts on the set X in such a way that G takes edges toedges In particular, G preserves an orientation of Γ if, and only if, it acts withoutinversion i.e., ge6= ¯e for any edge e and any g ∈ G A group G acts freely on Γ if itacts without inversion and a vertex can be fixed only by the identity element

3.1 Resolutions for groups acting on trees 28

In this thesis, we will use the equivalent definition below

Definition 3.1.1 A tree is a contractible 1-dimensional CW-space Then a treeXwhich group G acts on is a G-space and we call X a G-tree

If G acts without inversion on T , one can define the quotient graph of Γ by G in anatural manner It is defined to be the graph whose vertex set is the set X/G oforbits of vertices of Γ under the G-action and the edges are G-orbits of edges of Γ

Definition 3.1.2 [25] LetG be a group acting on a graph Γ A fundamental domainfor the action of G is a subgraph ∆ ⊂ Γ such that ∆ ' X/G, the isomorphism beinginduced from the quotient map from Γ→ Γ/X

Proposition 3.1.2 [25] Let G be a group acting on a tree T A fundamentaldomain for the action of G exists if and only if T /G is a tree

Theorem 3.1.3 [25] Let G be a group acting on a graph Γ Let T a segment in

Γ be a fundamental domain for the action of G Let P ; Q be the vertices of T and

e = (y; ¯y) be the geometric edge of T Let GP; GQ and Gy =Gy ¯ be the stabilizers of

P ; Q and y respectively Then the following are equivalent:

(i) Γ is a tree

(ii) The canonical homomorphismGP∗G yGQ → G induced by the inclusion GP →

G and GQ → G is an isomorphism

Theorem 3.1.4 [25] Let G = G1∗AG2 be an amalgam of two groups Then there

is a tree Γ (and only one, up to isomorphism) on which G acts, with fundamentaldomain a segment such that if the vertices of this segment are {P ; Q} and the edgesare {y; ¯y} then G1 ' GP, G2 ' GQ and A' Gy

The construction of a free resolution for G = G1∗AG2 below is also the proof of theTheorem 3.1.1

3.1 Resolutions for groups acting on trees 29

Consider the group G = G1 ∗AG2 acts on the tree T with details as in Theorem

3.1.4 Then T is a G-space with the 0-cells are vertices of T and 1-cells are the pairs

of edges of T One can see that T has two orbits of 0-cells whose representativesare P, Q respectively and one orbit of 1-cells whose representative is y Let [e]denote the equivalence class of cells in the orbit of e under the action of G, andletOrb(n) denote the set of equivalence classes of n-dimensional cells The cellularchain complex of length 1 ofT ,

∗ , and outputs a freeZG-resolution RG

∗ of Z In this sections we shall need this construction only for1-dimensional spaces T So from this point on we recall the construction in thisparticularly easy case The boundary homomorphism C1(X) → C0(X) induces aZG-equivariant homomorphism of ZG-chain complexes

n−1⊗ZG eZG

∂ n−1

which we view as a bicomplex The free ZG-resolution RG

∗ is the total complex ofthis bicomplex That is RG

n =A1,n−1⊕ A0,n where

A1,n−1 = M

[e]∈Orb(1)

(RG e n−1⊗ZG e Ze) ⊗G e ZG, A0,n= M

[v]∈Orb(0)

RG v

n ⊗G eZG (3.1)

3.1 Resolutions for groups acting on trees 30

The boundary homomorphism is

dn:A1,n−1⊕A0,n → A1,n−2⊕A0,n−1, x⊕y 7→ ∂n−1(x)⊕(−1)nδn−1(x)+∂n(y) (3.2)

Recall that a contracting homotopy on theZG-resolution RG

∗ is a family ofZ-linearhomomorphisms hn:RG

• a non-free ZG-resolution (of length 1) C∗ of Z together with free

ZGe-resolution of Z with contracting homotopy for all stabilizers Ge

• an integer n ≥ 0

Output: The first n terms of a freeZG-resolution RG

∗ of Z Moreover, if the input

C∗ is together with a contracting homotopy then the output also together with acontracting homotopy

Procedure:

1: The nth-term RG

n is computed by the formula3.1;

2: The boundary is constructed by using formula 3.2;

3: If the input together with a contracting homotopy then the contracting topy of RG

homo-∗ is constructed by using formula 3.3.This algorithm is a special case of Algorithm 2.1.1 when it applies to a non-freeZG-resolution of length 1 Proposition 2.1.1 in this case, the differential map d is

3.2 Resolution for SL(2,Z[1/m]) 31

easier to implement since dk = 0 for all k > 1 And we just need to implement

the contracting homotopy for the case q = 0 So that this case can be implemented

more efficiently Moreover, this special case can be used to prove precisely Lemma

3.2.1 about the periodicity of the homology of G

The contracting homotopy on the cellular chain complex of the tree associated to

the amalgamated product can be defined by using the contracting vector field An

example for constructing such a vector field is described in Section3.2

We could use the above construction, together with the standard resolution RC m

∗for finite cyclic groups Cm, to produce a free ZG-resolution RG

∗ for G = SL2(Z) ∼=

C4∗C 2 C6 However, for computational efficiency we prefer to use a slightly smaller

resolution that is described in the section 2.3

])-3.2 Resolution for SL(2,Z[1/m]) 32resolution RSL2 (Z[1/m 0 ])

∗ using the construction described in Section 3.1.1

In principle one can apply this technique recursively to obtain a freeZSL2(resolution for anym For the recursion to work we need an algorithm for a contract-ing homotopy on the tree associated to the amalgamated sum Such an algorithmboils down to one for expressing an arbitrary matrixA ∈ SL2(Z[1/m]) as a product

Z[1/m])-of generators Z[1/m])-of each Z[1/m])-of the two copies Z[1/m])-of SL2(Z[1/m0]) The required algorithm is

a slight variant of the Euclidean type algorithm 2.3.1 described in section2.3.All the above information has been implemented in HAP as follows:

Algorithm 3.2.1 A ZG-equivariant CW-space for G = SL2(Z[1/m]) ∗Γ m

0 (p)

SL2(Z[1/m])

Input: A pair of integers (m, p)

Output: A non free ZG-resolution with stabilizers and contracting homotopy.Procedure:

1: Find the chain conplex C∗(T ) by using 2.4

2: The contracting homotopy on C∗(T ) is calculated using a light variant of thealgorithm 2.3.1

3.2 Resolution for SL(2,Z[1/m]) 33Algorithm 3.2.2 Free resolution for G = SL(2,Z[1/m])

Input: A pair of integers (m, n)

Output: The first n terms of a freeZG-resolution of Z with contracting homotopy.Procedure:

5: RSp:= Conjugated resolution of RS by the matrix P:=[[1,0],[0,p]]

6: C:= The chain complex constructed by algorithm 3.2.1 with stabilizers’ tions are RS, RSp and RH

resolu-7: R:= the free resolution constructed by using algorithm 2.1.1 with the inputs Cand n

return R

EndFunction

In practical, we can use the lemma below to reduce the run-time

Lemma 3.2.1 Let m = p1p2 pn is a product of k primes, then the homology of

SL2(Z[1/m]) is periodic of period 2 in degree > k + 1

Proof LetG = SL2(Z[1/m]) The lemma is proved if we can show that the freeZG-resolution RG

∗ is periodic of period 2 in degree > k + 1 as follows:

for all n > k + 1, the ZG-modules

∗ is periodic ofperiod 2 in degree> k

3.2 Resolution for SL(2,Z[1/m]) 34Following the algorithm in section 3.1.1, the free ZG-resolution RG

∗ can be structed by combining the free ZH-resolution RH

con-∗ , its conjugated resolution and theresolution of its congruence subgroup In this case, G acts on tree T with the fun-damental domain for this action a segment in which the edgey joining two vertices

P, Q Then T can be seen as a G−space with a single orbit of 1-cells represented

by e1 and two orbits of 0-cells represented by e1

0, e2

0 is respectively Γm0

0 (p), H and the conjugate HP where

P = diag(1, p) We know that Γm0

0 (p) is a subgroup of both H and HP Fix theresolutionRH

∗ forH and we choose the conjugated resolution RH P

∗ forHP and ZΓm0

0 resolution RH

The periodicity of the boundary map in equation3.5can be proved by using formulae

3.2 By induction, ∂ is periodic, we now just only need to prove the periodicity ofδ