Math 697: Fiber bundles and classifying spaces

Time: T Th 12-1:15

Location: Math 215

Textbooks:

As we go along, I will try to write up rough lecture notes. They will be attached at the bottom of this page. Note that the exercises have no set due date and are not required. However, feel free to talk to me about them in office hours.

You may also find Cohen's The Topology of Fiber Bundles Lecture Notes, Davis and Kirk's Lecture Notes in Algebraic Topology, Hatcher's Algebraic Topology, Ebert and Randal-Williams' Semi-simplicial spaces, and Milnor and Stasheff's Characteristic Classes to be relevant.

Office Hours: Math 710, Th 11-11:45 and by appointment.

Course Summary:

In this course, we will discuss various types of fiber bundles such as vector bundles, covers, and principal bundles. We will prove that the set of isomorphism classes of bundles is in bijection with the set of homotopy classes of maps to a space called the classifying space. We will describe ways of constructing classifying spaces, including the simplicial bar construction. The homology of classifying spaces of discrete groups is called group homology. We will compute the group homology of cyclic groups and use this to show that a group with torsion cannot act freely on finite dimensional Euclidean space. An emphasis will be placed on geometric models of classifying spaces such as Grassmannians and configuration spaces. Time permitting, we will discuss characteristic classes, Eilenberg-MacLane spaces, fibrations, and the Serre spectral sequence.