群的上同调

Introduction

CHAPTER Ⅰ Some Homological Algebra

0. Review of Chain Complexes

1. Free Resolutions

2. Group Rings

3. G-Modules

4. Resolutions of Z Over ZG via Topology

5. The Standard Resolution

6. Periodic Resolutions via Free Actions on Spheres

7. Uniqueness of Resolutions

8. Projective Modules

Appendix. Review of Regular Coverings

CHAPTER Ⅱ The Homology of a Group

1. Generalities

2. Co-invariants

3. The Definition of H,G

4. Topological Interpretation

5. Hopf's Theorems

6. Functoriality

7. The Homology of Amalgamated Free Products

Appendix. Trees and Amalgamations

CHAPTER Ⅲ Homology and Cohomology with Coefficients

0. Preliminaries on X G and HomG

1. Definition of H,（G, M） and H*（G, M）

2. Tor and Ext

3. Extension and Co-extension of Scalars

4. Injective Modules

5. Induced and Co-induced Modules

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

7. Dimension Shifting

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

9. The Transfer Map

10. Applications of the Transfer

CHAPTER Ⅳ Low Dimensional Cohomology and Group Extensions

1. Introduction

2. Split Extensions

3. The Classification of Extensions with Abelian Kernel

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

5. Crossed Modules and H3 （Sketch）

6. Extensions With Non-Abelian Kernel （Sketch）

CHAPTER Ⅴ Products

1. The Tensor Product of Resolutions

2. Cross-products

3. Cup and Cap Products

4. Composition Products

5. The Pontryagin Product

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

CHAPTER Ⅵ Cohomology Theory of Finite Groups

1. Introduction

2. Relative Homological Algebra

3. Complete Resolutions

4. Definition of H

5. Properties of H

6. Composition Products

7. A Duality Theorem

8. Cohomologically Trivial Modules

9. Groups with Periodic Cohomology

CHAPTER Ⅶ Equivariant Homology and Spectral Sequences

1. Introduction

2. The Spectral Sequence of a Filtered Complex

3. Double Complexes

4. Example: The Homology of a Union

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

6. Example: The Hochschild-Serre Spectral Sequence

7. Equivariant Homology

8. Computation of

9. Example: Amalgamations

10. Equivariant Tate Cohomology

CHAPTER Ⅷ

Finiteness Conditions

1. Introduction

2. Cohomological Dimension

3. Serre's Theorem

4. Resolutions of Finite Type

5. Groups of Type Fan

6. Groups of Type FP and FL

7. Topological Interpretation

8. Further Topological Results

9. Further Examples

10. Duality Groups

11. Virtual Notions

CHAPTER Ⅸ

Euler Characteristics

1. Ranks of Projective Modules: Introduction

2. The Hattori-Stallings Rank

3. Ranks Over Commutative Rings

4. Ranks Over Group Rings; Swan's Theorem

5. Consequences of Swan's Theorem

6. Euler Characteristics of Groups: The Torsion-Frce Case

7. Extension to Groups with Torsion

8. Euler Characteristics and Number Theory

9. Integrality Properties of

10. Proof of Theorem 9.3; Finite Group Actions

11 The Fractional Part of

12. Acyclic Covers; Proof of Lemma 11.2

13. The p-Fractional Part of

14. A Formula for

CHAPTER Ⅹ

Farrell Cohomology Theory

I. Introduction

2. Complete Resolutions

3. Definition and Properties

4. Equivariant Farrell Cohomology

5. Cohomologically Trivial Modules

6. Groups with Periodic Cohomology

7. the Ordered Set of Finite Subgroups of F

References

Notation Index

Index

This book is based on a course given at Cornell University and intendedprimarily for second-year graduate students. The purpose of the course wasto introduce students who knew a little algebra and topology to a subject inwhich there is a very rich interplay 'between the two. Thus I take neither apurely algebraic nor a purely topological approach, but rather I use bothalgebraic and topological techniques as they seem appropriate The first six chapters contain what I consider to be the basics of the subjectThe remaining four chapters are somewhat more specialized and reflect myown research interests. For the most part, the only pre'requisites for readingthe book are the elements of algebra （groups, rings, and modules, includingtensor products over non-commutative rings） and the elements of algebraictopology （fundamental group, covering spaces, simplicial and CW-complexes, and homology）. There are, however, a few theorems, especially inthe later chapters, whose proofs use slightly more topology （such as theHurewicz theorem or Poincare duality）.