Cohomology of Groups (Graduate Texts in Mathematics, No. 87)

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......Page all_19785_to_00314.cpc0005.djvu Introduction......Page all_19785_to_00314.cpc0009.djvu 0. Review of Chain Complexes......Page all_19785_to_00314.cpc0012.djvu 1. Free Resolutions......Page all_19785_to_00314.cpc0018.djvu 2. Group Rings......Page all_19785_to_00314.cpc0020.djvu 3. G-Modules......Page all_19785_to_00314.cpc0021.djvu 4. Resolutions of Z Over ZG via Topology......Page all_19785_to_00314.cpc0022.djvu 5. The Standard Resolution......Page all_19785_to_00314.cpc0026.djvu 6. Periodic Resolutions via Free Actions on Spheres......Page all_19785_to_00314.cpc0028.djvu 7. Uniqueness of Resolutions......Page all_19785_to_00314.cpc0029.djvu 8. Projective Modules......Page all_19785_to_00314.cpc0034.djvu Appendix. Review of Regular Coverings......Page all_19785_to_00314.cpc0039.djvu 1. Generalities......Page all_19785_to_00314.cpc0041.djvu 2. Co-invariants......Page all_19785_to_00314.cpc0042.djvu 3. The Definition of H_* G......Page all_19785_to_00314.cpc0043.djvu 4. Topological Interpretation......Page all_19785_to_00314.cpc0044.djvu 5. Hopf's Theorems......Page all_19785_to_00314.cpc0049.djvu 6. Functoriality......Page all_19785_to_00314.cpc0056.djvu 7. The Homology of Amalgamated Free Products......Page all_19785_to_00314.cpc0057.djvu Appendix. Trees and Amalgamations......Page all_19785_to_00314.cpc0060.djvu 0. Preliminaries on \tensor_G and Hom_G......Page all_19785_to_00314.cpc0063.djvu 1. Definition of H_*(G,M) and H^*(G,M)......Page all_19785_to_00314.cpc0064.djvu 2. Tor and Ext......Page all_19785_to_00314.cpc0068.djvu 3. Extension and Co-extension of Scalars......Page all_19785_to_00314.cpc0070.djvu 4. Injective Modules......Page all_19785_to_00314.cpc0073.djvu 5. Induced and Co-induced Modules......Page all_19785_to_00314.cpc0075.djvu 6. Hi_* and H^* as Functors of the Coefficient Module......Page all_19785_to_00314.cpc0079.djvu 7. Dimension Shifting......Page all_19785_to_00314.cpc0082.djvu 8. H_* and H^* as Functors of Two Variables......Page all_19785_to_00314.cpc0086.djvu 9. The Transfer Map......Page all_19785_to_00314.cpc0088.djvu 10. Applications of the Transfer......Page all_19785_to_00314.cpc0091.djvu 1. Introduction......Page all_19785_to_00314.cpc0094.djvu 2. Split Extensions......Page all_19785_to_00314.cpc0095.djvu 3. The Classification of Extensions with Abelian Kernel......Page all_19785_to_00314.cpc0099.djvu 4. Application: p-Groups with a Cyclic Subgroup of Index p......Page all_19785_to_00314.cpc0105.djvu 5. Crossed Modules and H^3 (Sketch)......Page all_19785_to_00314.cpc0110.djvu 6. Extensions With Non-Abelian Kernel (Sketch)......Page all_19785_to_00314.cpc0112.djvu 1. The Tensor Product of Resolutions......Page all_19785_to_00314.cpc0115.djvu 2. Cross-products......Page all_19785_to_00314.cpc0116.djvu 3. Cup and Cap Products......Page all_19785_to_00314.cpc0117.djvu 4. Composition Products......Page all_19785_to_00314.cpc0122.djvu 5. The Pontryagin Product......Page all_19785_to_00314.cpc0125.djvu 6. Application: Calculation of the Homology of an Abelian Group......Page all_19785_to_00314.cpc0129.djvu 1. Introduction......Page all_19785_to_00314.cpc0136.djvu 2. Relative Homological Algebra......Page all_19785_to_00314.cpc0137.djvu 3. Complete Resolutions......Page all_19785_to_00314.cpc0139.djvu 4. Definition of \hat{H}^*......Page all_19785_to_00314.cpc0142.djvu 5. Properties of \hat{H}^*......Page all_19785_to_00314.cpc0144.djvu 6. Composition Products......Page all_19785_to_00314.cpc0150.djvu 7. A Duality Theorem......Page all_19785_to_00314.cpc0152.djvu 8. Cohomologically Trivial Modules......Page all_19785_to_00314.cpc0156.djvu 9. Groups with Periodic Cohomology......Page all_19785_to_00314.cpc0161.djvu 2. The Spectral Sequence of a Filtered Complex......Page all_19785_to_00314.cpc0169.djvu 3. Double Complexes......Page all_19785_to_00314.cpc0172.djvu 4. Example: The Homology of a Union......Page all_19785_to_00314.cpc0174.djvu 5. Homology of a Group with Coefficients in a Chain Complex......Page all_19785_to_00314.cpc0176.djvu 6. Example: The Hochschild-Serre Spectral Sequence......Page all_19785_to_00314.cpc0179.djvu 7. Equivariant Homology......Page all_19785_to_00314.cpc0180.djvu 8. Computation of d^1......Page all_19785_to_00314.cpc0183.djvu 9. Example: Amalgamations......Page all_19785_to_00314.cpc0186.djvu 10. Equivariant Tate Cohomology......Page all_19785_to_00314.cpc0188.djvu 1. Introduction......Page all_19785_to_00314.cpc0191.djvu 2. Cohomological Dimension......Page all_19785_to_00314.cpc0192.djvu 3. Serre's Theorem......Page all_19785_to_00314.cpc0198.djvu 4. Resolutions of Finite Type......Page all_19785_to_00314.cpc0199.djvu 5. Groups of Type FP_n......Page all_19785_to_00314.cpc0205.djvu 6. Groups of Type FP and FL......Page all_19785_to_00314.cpc0207.djvu 7. Topological Interpretation......Page all_19785_to_00314.cpc0213.djvu 8. Further Topological Results......Page all_19785_to_00314.cpc0218.djvu 9. Further Examples......Page all_19785_to_00314.cpc0221.djvu 10. Duality Groups......Page all_19785_to_00314.cpc0227.djvu 11. Virtual Notions......Page all_19785_to_00314.cpc0233.djvu 1. Ranks of Projective Modules: Introduction......Page all_19785_to_00314.cpc0238.djvu 2. The Hattori-Stallings Rank......Page all_19785_to_00314.cpc0239.djvu 3. Ranks Over Commutative Rings......Page all_19785_to_00314.cpc0243.djvu 4. Ranks Over Group Rings; Swan's Theorem......Page all_19785_to_00314.cpc0247.djvu 5. Consequences of Swan's Theorem......Page all_19785_to_00314.cpc0250.djvu 6. Euler Characteristics of Groups: The Torsion-Free Case......Page all_19785_to_00314.cpc0254.djvu 7. Extension to Groups with Torsion......Page all_19785_to_00314.cpc0257.djvu 8. Euler Characteristics and Number Theory......Page all_19785_to_00314.cpc0261.djvu 9. Integrality Properties of \chi(\Gamma)......Page all_19785_to_00314.cpc0265.djvu 10. Proof of Theorem 9.3; Finite Group Actions......Page all_19785_to_00314.cpc0266.djvu 11. The Fractional Part of \chi(\Gamma)......Page all_19785_to_00314.cpc0269.djvu 12. Acyclic Covers; Proof of Lemma 11.2......Page all_19785_to_00314.cpc0273.djvu 13. The p-Fractional Part of chi(\Gamma)......Page all_19785_to_00314.cpc0274.djvu 14. A Formula for \chi_Gamma(\mathcal{A})......Page all_19785_to_00314.cpc0278.djvu 2. Complete Resolutions......Page all_19785_to_00314.cpc0281.djvu 3. Definition and Properties of \hat{H}^*(\Gamma)......Page all_19785_to_00314.cpc0285.djvu 4. Equivariant Farrell Cohomology......Page all_19785_to_00314.cpc0289.djvu 5. Cohomologically Trivial Modules......Page all_19785_to_00314.cpc0295.djvu 6. Groups with Periodic Cohomology......Page all_19785_to_00314.cpc0296.djvu 7. \hat{H}^*(\Gamma) and the Ordered Set of Finite Subgroups of \Gamma......Page all_19785_to_00314.cpc0299.djvu References......Page all_19785_to_00314.cpc0303.djvu Notation Index......Page all_19785_to_00314.cpc0309.djvu Index......Page all_19785_to_00314.cpc0311.djvu 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.