Math 603D: Homotopy theory, classifying spaces and group cohomology

Course Information


The instructor for the course is me, Ben Williams

Meeting Time, Place

This has not yet been fixed, but the class is likely to meet MWF 12pm-1pm in ESB 4133 (the PIMS lounge). The first meeting is in ESB 4133 on Wednesday 6 September at 12pm.


There is no general textbook reference for this course, but references will be given over the course of the term for specific aspects. Notes may also be made available.


This is a topics course in the homotopy theory of classifying spaces of groups, and related aspects of homotopy theory. We have three principal aims: first, to explain how (algebraic) group cohomology for discrete groups can be viewed a special case of a homotopy-theoretic study of classifying spaces; second, to give an introduction to the theory of G bundles and characteristic classes; and third, to establish computational tools which are more generally applicable.

A first course in algebraic topology (such as Math 527) will be assumed.


Exercises for this course can be found here. Please submit the answers to 5 questions to me on or before 14 December in order to receive a course grade. The list of exercises may be expanded as the course progresses.

Here follows an overambitious list of topics.

  • Locally trivial fibrations
  • Fibrations
  • Classifying spaces and universal bundles
  • Spectral Sequences: the Serre and Eilenberg--Moore Spectral Sequences
  • Characteristic Classes
  • Sheaf cohomology and group cohomology
  • Cohomology Operations
  • The extra-special 2-groups [time permitting]
  • Vector fields on spheres [time permitting]