Math 221a—Topology III Fall 2004

Daniel Ruberman

Time: Tuesday and Friday, 10:30-12
Room: 209 Goldsmith (conference room)

Math 221a will cover characteristic classes and vector bundles. After working through the basic facts about vector bundles and fiber bundles (definitions, constructions, important examples) we will construct the important characteristic cohomology classes of such bundles. These are the Stiefel-Whitney and Pontryagin classes for real bundles, the Chern classes for complex bundles, and the Euler class for oriented bundles. There will be lots of examples, and applications to problems in topology.

The construction and working out of the properties of characteristic classes will be by the Splitting Principle, which uses the Leray-Hirsch theorem of algebraic topology as a fundamental ingredient. This approach is explained nicely in the book(s) of Hatcher, which will be our basic text.

Problem Sets
Homework 1, due Sept 28


Allen Hatcher, Vector Bundles and K-theory, available online from If you are going to print out a copy, please use the doublepage version

Allen Hatcher, Algebraic Topology. Available in the bookstore, or online from

I will put the following classic texts on reserve in the library, and will occasionally make use of them.

Milnor and Stasheff, Characteristic Classes, Princeton U. Press.

Husemoller, Fibre Bundles, Springer-Verlag.

Steenrod, The Topology of Fiber Bundles, Princeton U. Press.