Math 221a Fall 07

Math 221a (Topology III) will be basically a course on vector bundles and characterstic classes.  There does not seem to be a single book that covers the material in the way that I would choose to do it, but I will use Allen Hatcher, Vector Bundles and K-theory, available online.  If you are going to print out a copy, please use the doublepage version as the text.

I will also make use of the classic Milnor-Stasheff Lectures on Characterstic Classes and pick and choose from other sources.

 Prerequisites are 121ab (Topology I&II) and 110a (Geometric Analysis); if you are really eager you can take 221a and 110a concurrently.

The course will meet on Mondays and Wednesdays, 12-1:30. At the moment we are meeting in Room 300, but that may change. 

Outline of the course: 

1. Fiber bundles and vector bundles.  Basic examples: tangent bundles, principal bundles, Hopf bundles.  Constructions such as pull-backs, Whitney sum, tensor product, associated bundles, and projectivization.
2. Classifying spaces and Grassmannians.
3. Basic cohomology theory of fibrations; Leray-Hirsch theorem.  We may do some introductory material on spectral sequences (without proofs) if there is sufficient interest.  Cohomology of the Grassmannians.
4. Construction of Stiefel-Whitney and Chern characteristic classes via Leray-Hirsch theorem.
5. Orientations; Euler class and a little obstruction theory.
6. Applications of characteristic classes; spin and spin^c structures.
7. Selection of additional topics: possibilities include homology with coefficients, more obstruction theory, introduction to K-theory, or possibly cobordism theory.