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Math 283 is a graduate course in the cohomology of finite groups. Group cohomology deals with
obtaining invariants of groups coming from the cohomology of their classifying spaces. These
computations often have a topological and an algebraic interpretation, and have applications both
in topology and group theory. In topology, group cohomology is used to study possible actions of groups, invariants
of spaces with group actions, obstructions and principal bundles, and it is connected to Steenrod operations. In
group theory and algebra, it can be used to classify group extensions, to determine the projectivity of modules
and to study modular representations. This is a subject with deep interconnections of algebra and topology, with
well-known classical results and modern applications.
Formal prerequisites are basic knowledge of group theory (Math 120, for example) and a course in algebraic
topology about the fundamental group, homology and cohomology (such as Math 215b). Knowledge of higher homotopy
groups and related results as in Math 282b is recommended, but not strictly necessary. Apart from formal
prerequisites, I will assume that you are familiar with basic homological algebra.
For a tentative list of topics that may be covered go here. For the list of topics actually
covered, go to the Syllabus page.
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- Group cohomology from the viewpoint of homological algebra: Resolutions, group rings, Ext and Tor.
- Group cohomology from the viewpoint of algebraic topology: Classifying spaces, principal bundles, aspherical spaces.
- Applications of low-dimensional cohomology: Group extensions, crossed modules.
- Restriction, transfer and norm maps: Primary decomposition, Steenrod operations, Kan extensions.
- Free actions of groups on spheres.
- Actions of groups on lattices of subgroups.
- Fusion systems and p-local groups.
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