MATH 626 Cohomology of Groups

Group cohomology sits at the intersection of algebra and topology: you study a group by resolving it homologically and by realizing it geometrically as a classifying space, then translating questions about extensions, actions, and invariants back and forth between the two pictures. The semester is built around computation, you work through projective resolutions, the bar complex, restriction/transfer, and spectral sequences on homeworks and a term project, with deeper structural results like Serre's, Mislin's, and Quillen's stratification theorems framing what those computations are good for. It assumes you're comfortable with modules and basic algebraic topology, and it's the natural feeder into research on representation theory of finite groups, equivariant topology, and modular invariant theory.