In the first part of this course, we will adopt a purely algebraic point of view. We will introduce group homology and cohomology using projective resolutions and develop the main techniques to work with them. These include bar and minimal projective resolutions, cup product in cohomology, induction and coninduction functors, restriction and corestriction, alternative descriptions of low degree cohomology, and the Lyndon-Hochschild-Serre spectral sequence. We will compute the homology and cohomology rings of finite abelian groups.

In the remaining time, we will explore a more advance topic which will be chosen according to the preference and background of the audience. Short presentations by students might constitute part of this exploration into either more algebro-geometric or more topological directions. Possible topics include (but are not limited to):

• The structure of the cohomology ring: Bockstein homomorphism, Steenrod operations, Serre's theorem on characterization of elementary abelian groups in terms of their cohomology, Quillen's stratification theorem.
• Related cohomology theories and actions of groups on spaces: Tate cohomology, Borel construction, equivariant cohomology, Smith theorems (on fixed points under group actions).

Texts: Group cohomology and Representations by D. Benson; The Cohomology of Groups by L. Evens; Introduction to Homological Algebra by C. Weibel.

Prerequisites: Math 504/5/6, a course in homological algebra. It would also be helpful (but not essential) to have some background in topology and/or manifolds, such as Math 544/5/6.