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AuthorBrown, Kenneth S. author
TitleCohomology of Groups [electronic resource] / by Kenneth S. Brown
ImprintNew York, NY : Springer New York, 1982
Connect tohttp://dx.doi.org/10.1007/978-1-4684-9327-6
Descript X, 306 p. online resource

SUMMARY

As a second year graduate textbook, Cohomology of Groups introduces students to cohomology theory (involving a rich interplay between algebra and topology) with a minimum of prerequisites. No homological algebra is assumed beyond what is normally learned in a first course in algebraic topology. The basics of the subject are given (along with exercises) before the author discusses more specialized topics


CONTENT

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 -- II The Homology of a Group -- 1. Generalities -- 2. Co-invariants -- 3. The Definition of H*G -- 4. Topological Interpretation -- 5. Hopfs Theorems -- 6. Functoriality -- 7. The Homology of Amalgamated Free Products -- Appendix. Trees and Amalgamations -- III Homology and Cohomology with Coefficients -- 0. Preliminaries on ?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 -- 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) -- 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 -- VI Cohomology Theory of Finite Groups -- 1. Introduction -- 2. Relative Homological Algebra -- 3. Complete Resolutions -- 4. Definition of ?* -- 5. Properties of ?* -- 6. Composition Products -- 7. A Duality Theorem -- 8. Cohomologically Trivial Modules -- 9. Groups with Periodic Cohomology -- 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 d1 -- 9. Example: Amalgamations -- 10. Equivariant Tate CohoMology -- VIII Finiteness Conditions -- 1. Introduction -- 2. CohoMological Dimension -- 3. Serreโ{128}{153}s Theorem -- 4. Resolutions of Finite Type -- 5. Groups of Type FPn -- 6. Groups of Type FF and FL -- 7. Topological Interpretation -- 8. Further Topological Results -- 9. Further Examples -- 10. Duality Groups -- 11. Virtual Notions -- IX Euler Characteristics -- 1. Ranks of Projective Modules: Introduction -- 2. The Hattori-Stallings Rank -- 3. Ranks Over Commutative Rings -- 4. Ranks Over Group Rings; Swanโ{128}{153}s Theorem -- 5. Consequences of Swanโ{128}{153}s Theorem -- 6. Euler Characteristics of Groups: The Torsion-Free 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 ??(A) -- X Farrell Cohomology Theory -- 1. Introduction -- 2. Complete Resolutions -- 3. Definition and Properties of ?*(?)277 -- 4. Equivariant Farrell Cohomology -- 5. Cohomologically Trivial Modules -- 6. Groups with Periodic Cohomology -- 7. ?*(?) and the Ordered Set of Finite Subgroups of ? -- References -- Notation Index


Mathematics Group theory Mathematics Group Theory and Generalizations



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