## Cohomology of GroupsAs 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. |

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abelian group action of G acyclic algebra analogue arbitrary canonical chain complex chain map cochain coefficient cohomologically trivial commutative complete resolution cup product cyclic deduce defined definition denoted diagonal dimension duality group element Euler characteristics example exercise extension finite dimensional finite group finite homological type finite index finite subgroups finite type finitely generated projective follows formula free module free resolution functor G acts G-action G-complex G-module given group G hence Hint Hom(M homology homomorphism homotopy equivalence induced integer isomorphism isotropy group Let F Let G non-trivial Note obtain p-group periodic cohomology Poincare duality prime projective modules projective resolution proof Proposition prove quotient rank relatively injective resp result ring Serre short exact sequence simplicial simply spectral sequence subcomplex subgroup of finite subgroup of G surjection tensor product theorem theory topological torsion-free subgroup type FP universal cover weak equivalence Z-free