Cohomology Of Groups (Graduate Texts In Mathematics)

Aimed at second year graduate students, this text introduces them 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, and the basics of the subject, as well as exercises, are given prior to discussion of more specialized topics. Contents: 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. Serre'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; Swan's Theorem.- 5. Consequences of Swan'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.