MATH 661: Geometric Topology (Fall 2007)

Instructor: Allen Hatcher

Meeting Time & Room

Here are two possible topics for the course:

1. Basic topology of 3-manifolds. Perelman's recent work verifies a conjectural picture of the classification of 3-manifolds that arose around 25 years ago, and the primary aim of the course would be to present this picture. The main prerequisite for the course would be basic algebraic topology as in 651 (fundamental group, covering spaces, homology). Some prior exposure to differentiable manifolds would also be helpful, although what's needed here is minimal and could be picked up as one goes along.
2. Bott Periodicity and its applications such as the nonexistence of real division algebras in dimensions other than 1, 2, 4, and 8. The framework here is vector bundles and topological K-theory. Prerequisite: basic algebraic topology as in 651. The source for the fiirst part of the course would be the set of notes on my webpage, and a goal for the latter part of the course would be to extend these notes to cover Bott periodicity for the real case as well as the easier complex case. This involves the very nice topic of Clifford algebras, generalizing quaternions.

The choice between the two topics will depend on the interests of the audience.