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Introduction 1

CHAPTER Ⅰ　Some Homological Algebra 4

0．Review of Chain Complexes 4

1．Free Resolutions 10

2．Group Rings 12

3．G-Modules 13

4．Resolutions of Z Over ZG via Topology 14

5．The Standard Resolution 18

6．Periodic Resolutions via Free Actions on Spheres 20

7．Uniqueness of Resolutions 21

8．Projective Modules 26

Appendix. Review of Regular Coverings 31

CHAPTER Ⅱ　The Homology of a Group 33

1．Generalities 33

2．Co-invariants 34

3．The Definition of H*G 35

4．Topological Interpretation 36

5．Hopf's Theorems 41

6．Functoriality 48

7．The Homology of Amalgamated Free Products 49

Appendix. Trees and Amalgamations 52

CHAPTER Ⅲ　Homology and Cohomology with Coefficients 55

0．Preliminaries on ？G and HomG 55

1．Definition of H*(G,M)and H*(G,M) 56

2．Tor and Ext 60

3．Extension and Co-extension of Scalars 62

4．Injective Modules 65

5．Induced and Co-induced Modules 67

6．H* and H* as Functors of the Coefficient Module 71

7．Dimension Shifting 74

8．H* and H* as Functors of Two Variables 78

9．The Transfer Map 80

10．Applications of the Transfer 83

CHAPTER Ⅳ Low Dimensional Cohomology and Group Extensions 86

1．Introduction 86

2．Split Extensions 87

3．The Classification of Extensions with Abelian Kernel 91

4．Application:p-Groups with a Cyclic Subgroup of Index p 97

5．Crossed Modules and H3(Sketch) 102

6．Extensions With Non-Abelian Kernel(Sketch) 104

CHAPTER Ⅴ Products 107

1．The Tensor Product of Resolutions 107

2．Cross-products 108

3．Cup and Cap Products 109

4．Composition Products 114

5．The Pontryagin Product 117

6．Application:Calculation of the Homology of an Abelian Group 121

CHAPTER Ⅵ Cohomology Theory of Finite Groups 128

1．Introduction 128

2．Relative Homological Algebra 129

3．Complete Resolutions 131

4．Definition of ？ 134

5．Properties of ？ 136

6．Composition Products 142

7．A Duality Theorem 144

8．Cohomologically Trivial Modules 148

9．Groups with Periodic Cohomology 153

CHAPTER Ⅶ Equivariant Homology and Spectral Sequences 161

1．Introduction 161

2．The Spectral Sequence of a Filtered Complex 161

3．Double Complexes 164

4．Example:The Homology of a Union 166

5．Homology of a Group with Coefficients in a Chain Complex 168

6．Example:The Hochschild-Serre Spectral Sequence 171

7．Equivariant Homology 172

8．Computation of d1 175

9．Example:Amalgamations 178

10．Equivariant Tate Cohomology 180

CHAPTER Ⅷ Finiteness Conditions 183

1．Introduction 183

2．Cohomological Dimension 184

3．Serre's Theorem 190

4．Resolutions of Finite Type 191

5．Groups of Type FPn 197

6．Groups of Type FP and FL 199

7．Topological Interpretation 205

8．Further Topological Results 210

9．Further Examples 213

10．Duality Groups 219

11．Virtual Notions 225

CHAPTER Ⅸ Euler Characteristics 230

1．Ranks of Projective Modules:Introduction 230

2．The Hattori-Stallings Rank 231

3．Ranks Over Commutative Rings 235

4．Ranks Over Group Rings;Swan's Theorem 239

5．Consequences of Swan's Theorem 242

6．Euler Characteristics of Groups:The Torsion-Free Case 246

7．Extension to Groups with Torsion 249

8．Euler Characteristics and Number Theory 253

9．Integrality Properties of x(Γ) 257

10．Proof of Theorem 9.3;Finite Group Actions 258

11．The Fractional Part of x(Γ) 261

12．Acyclic Covers;Proof of Lemma 11.2 265

13．The p-Fractional Part of x(Γ) 266

14．A Formula for xг(？) 270

CHAPTER Ⅹ Farrell Cohomology Theory 273

1．Introduction 273

2．Complete Resolutions 273

3．Definition and Properties of ？*(Γ) 277

4．Equivariant Farrell Cohomology 281

5．Cohomologically Trivial Modules 287

6．Groups with Periodic Cohomology 288

7．？*(Γ)and the Ordered Set of Finite Subgroups of Γ 291

References 295

Notation Index 301

Index 303