群的上同调
2009-6
世界图书出版公司
布朗
306
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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）. The reader who does not have therequired background in topology can simply take these theorems on faith There are a number of exercises, some of which contain results which arereferred to in the text. A few of the exercises are marked with an asterisk towarn the reader that they are more diflicult than the others or that they requiremore background I am very grateful to R. Bieri, J-P. Serre, U. Stammbach, R. Strebel, andC. T. C. Wall for helpful comments on a preliminary version of this book
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IntroductionCHAPTER 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 AmalgamationsCHAPTER 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 TransferCHAPTER 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 GroupCHAPTER 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 CohomologyCHAPTER Ⅶ Equivariant Homology and Spectral Sequences 1. Introduction 2. The Spectral Sequence of a Filtered Complex ……CHAPTER Ⅷ Finiteness ConditionsCHAPTER Ⅸ Euler CharacteristicsCHAPTER Ⅹ Farrell Cohomology TheoryReferencesNotation IndexIndex
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