BUNDLES AND CHARACTERISTIC CLASSES

Math 541: Topics in Topology Fall 2002

**Professor Tim Cochran**

**MWF 1 HB 423**

This course will be an introduction to fiber bundles (especially vector bundles and principal bundles) and to characteristic classes. It will assume Math 444 and math 445, as well as some familiarity with manifolds and a little differential topology. However, a student without these prerequisites might get a lot out of auditing the class. There may be time to add some related topics such as higher-homotopy groups, or classification of manifolds via surgery theory.

We will try to stress examples a lot, especially low-dimensional examples. Applications that will be discussed include obstructions to embedding manifolds, obstructions to existence of vector fields, cobordism theory and exotic differentiable structures on spheres. Our primary approach will be topological but probably we will also discuss the approach to characteristic classes via connections and differential forms.

The primary text will be:

__Characteristic Classes__ by J. Milnor and J. Stashef ,
Princeton University Press,

but this will be supplemented because it does not discuss fiber bundles. This book has NOT been ordered by the bookstore so students who wish to own it need to order it now

from Amazon.com. The other book that I like is __The
Topology of Fiber Bundles__ by N. Steenrod.

The grade will
be based on problem sets (**several problems per week**) and **class
participation.**