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November 1, 2010 – 11:57

Preliminary version Please send corrections and suggestions to clara.loeh@uni-muenster.de

Clara Löh

Group Cohomology & Bounded Cohomology

An introduction for topologists

Clara Löh clara.loeh@uni-muenster.de http://wwwmath.uni-muenster.de/u/clara.loeh/

Mathematisches Institut Georg-August-Universität Göttingen Bunsenstr. 3–5 37073 Göttingen Germany

Current address: Mathematisches Institut WWU Münster Einsteinstr. 62 48149 Münster Germany

Contents

1 Group cohomology 1 1.1 Introduction 2 1.2 The domain categories for group (co)homology 5 1.3 Group cohomology, topologically 9

1.3.1 Classifying spaces 9

1.3.2 (Co)Homology with twisted coefficients 12

1.3.3 Group cohomology, topologically 15

1.3.4 Group (co)homology, first examples 17

1.3.5 Products and free products 20

1.4 Group cohomology, combinatorially 25 1.4.1 The bar resolution 25

1.4.2 Group cohomology, combinatorially 27

1.4.3 Application: Cyclic Galois extensions (Hilbert 90) 28

1.4.4 Application: Group extensions with Abelian kernel 30

1.5 Group cohomology via derived functors 39 1.5.1 Right/left exact functors 40

1.5.2 Derived functors, schematically 42

1.5.3 Projective and injective resolutions 44

1.5.4 The fundamental lemma of homological algebra 49

1.5.5 Derived functors, construction 53

1.5.6 Group cohomology, via derived functors 59

1.5.7 Group cohomology, axiomatically 60

1.5.8 Computing group (co)homology – summary 68

iv Contents

1.6 Group cohomology and group actions 69 1.6.1 Application: Groups acting on spheres I 69 1.6.2 (Co)Homology of finite cyclic groups 72 1.6.3 Application: Groups acting on spheres II 75 1.6.4 Application: Classifying p-groups with a unique

subgroup of order p 76 1.6.5 Application: Groups acting on spheres III 79 1.6.6 Group actions on highly connected spaces 81 1.6.7 Application: The Hurewicz homomorphism in degree 2 82 1.6.8 Preview: Group actions and spectral sequences 83

1.7 Cohomology of subgroups 85 1.7.1 Induction, coinduction, and restriction 85 1.7.2 Shapiro’s lemma 88 1.7.3 The transfer 91 1.7.4 Action on (co)homology of subgroups 98 1.7.5 Decomposing group (co)homology into primary parts 104 1.7.6 Application: Generalising the group-theoretic transfer 107

1.8 Product structures 109 1.8.1 A multiplicative structure for group cohomology 110 1.8.2 Algebraic preliminaries: tensor products of resolutions 112 1.8.3 The cross-product 115 1.8.4 The cup-product 117 1.8.5 The cap-product 127

1.9 Tate cohomology and periodic cohomology 129 1.9.1 Tate cohomology – definition 130 1.9.2 Relative homological algebra 134 1.9.3 Tate cohomology and ordinary group (co)homology 136 1.9.4 The product structure on Tate cohomology 138 1.9.5 Periodic cohomology 142 1.9.6 Characterising groups with periodic cohomology 146

1.10 The Hochschild-Serre spectral sequence 151 1.10.1 Spectral sequences in a nutshell 152 1.10.2 Some classic spectral sequences 156 1.10.3 The Hochschild-Serre spectral sequence 157 1.10.4 Sample computations for group extensions 160

1.11 Exercises 167

Contents v

2 Bounded cohomology 179 2.1 Introduction 180

2.2 The domain category for bounded cohomology 183

2.3 Homology of normed chain complexes 189

2.3.1 Normed chain complexes 189

2.3.2 Semi-norms in homology 192

2.3.3 The equivariant setting 196

2.4 Bounded cohomology, topologically 199

2.4.1 Bounded cohomology of spaces 200

2.4.2 Elementary properties of bounded cohomology 202

2.4.3 Bounded cohomology of groups 206

2.5 Bounded cohomology, combinatorially 209

2.5.1 The Banach bar resolution 209

2.5.2 Bounded cohomology, combinatorially 210

2.5.3 The second bounded cohomology of free groups 212

2.5.4 Application: Quasi-morphisms 215

2.5.5 Comparing the topological and the combinatorial definition of bounded cohomology 218

2.6 Amenable groups 223

2.6.1 Amenable groups via means 223

2.6.2 Examples of amenable groups 224

2.6.3 Inheritance properties of amenable groups 226

2.6.4 Geometric characterisations of amenable groups 227

2.6.5 Application: Characterising amenable groups via bounded cohomology 228

2.7 Bounded cohomology, algebraically 233

2.7.1 Relative homological algebra, Banach version 233

2.7.2 Bounded cohomology, algebraically 239

2.8 The mapping theorem 243

2.8.1 The mapping theorem, algebraically 243

2.8.2 The mapping theorem, topologically 245

2.8.3 Duality and mapping theorems in `1-homology 250

2.9 Application: Simplicial volume 251

2.9.1 Functorial semi-norms and degree theorems 252

2.9.2 Simplicial volume and negative curvature 253

2.9.3 Simplicial volume and Riemannian volume 257

vi Contents

2.9.4 Simplicial volume and the mapping theorem in bounded cohomology 258

2.9.5 Simplicial volume, inheritance properties 259 2.9.6 Open problems 261

2.10 Exercises 263

1 Group cohomology

1.1

Introduction

What is group cohomology? Group cohomology is a contravariant functor turning groups and modules over groups into graded Abelian groups. I.e., on objects group cohomology looks like

Hn(G;A),

where – the number n ∈ N is the degree in the grading of the graded Abelian

group H∗(G;A), – the first parameter is a (discrete) group G, – and the second (“Abelian”) parameter is a ZG-module A, the so-

called coefficients. Similarly, group homology is a covariant functor turning groups and mod- ules over groups into graded Abelian groups.

How can we construct group cohomology? There are three main (equiv- alent) descriptions of group (co)homology:

– Topologically (via classifying spaces) – Combinatorially (via the bar resolution) – Algebraically (via derived functors).

Why is group cohomology interesting? First, group cohomology is an interesting theory in its own right providing a beautiful link between alge- bra and topology. Second, group (co)homology helps to solve the following problems:

– Given two groups, what extension groups with the given “kernel” and the given “quotient” group do there exist? (Section 1.4.4)

– How do cyclic Galois field extensions look like? (Hilbert 90) (Sec- tion 1.4.3)

– How surjective is the Hurewicz homomorphism in degree 2? (Sec- tion 1.6.7)

– Which (finite) groups can act freely on spheres? (Section 1.6 and 1.9)

2

1.1 Introduction 3

Overview

In the first part of the semester we will study the following topics: – Understand and compare the three basic descriptions of group (co)ho-

mology – First applications of group (co)homology – Transfer – Product structures – Cohomology of finite groups and periodic cohomology – The Hochschild-Serre spectral sequence

In the second part of the semester we will look at a functional analytic variant of group cohomology, called bounded cohomology, and its applica- tion to the simplicial volume.

4 1.1 Introduction

1.2 The domain categories

for group (co)homology

The basic algebraic objects in the world of group (co)homology are group rings and modules over group rings.

Definition 1.2.1 (Group ring). Let G be a group. The (integral) group ring of G is the ring ZG (sometimes also denoted Z[G] to avoid misunder- standings)

– whose underlying Abelian group is the free Z-module ⊕

g∈G Z · g, – and whose multiplication is the Z-linear extension of composition

in G, i.e.:

· : ZG× ZG −→ ZG(∑ g∈G

ag · g, ∑ g∈G

bg · g ) 7−→

∑ g∈G

∑ h∈G

ag · bg−1h · g

(where all sums are “finite”).

Example 1.2.2 (Group rings). – We have Z[1] ∼= Z. – The group ring Z[Z] ∼= Z[t, t−1] of the integers is nothing but the ring

of Laurent polynomials. – For all n ∈ N>0 we have Z[Z/n] ∼= Z[t]/(tn − 1).

Group rings and modules over group rings occur, for example, naturally in topology:

Example 1.2.3. Let X be a topological space and let G be a discrete group that acts continuously on X.

– Let n ∈ N. Then the G-action on X induces a ZG-module structure

ZG× Cn(X; Z) −→ Cn(X; Z)

5

6 1.2 The domain categories for group (co)homology

on the chain group Cn(X; Z), given by

G×map(∆n, X) −→ map(∆n, X) (g, σ) 7−→

( t 7→ g · σ(t)

) .

– It is not difficult to see that the differential of the singular chain complex C∗(X; Z) is a ZG-morphism; hence, C∗(X; Z) is naturally a ZG-chain complex.

Notice: if X is a CW-complex and if G acts cellularly on X, then also the cellular chain complex Ccell∗ (X; Z) naturally is a ZG-chain complex.

A fundamental special case is the following: If X is a connected CW- complex (with a chosen base point in the universal covering of X), then the singular/cellular chain complex of the universal covering of X naturally is a free Zπ1(X)-chain complex.

Definition 1.2.4 (Invariants and coinvariants). Let G be a discrete group and let A be a (left) ZG-module. We call the submodule

AG := {a ∈ A | ∀g∈G g · a = a}

the invariants of A. W