Math 227B: Algebraic Topology

Characteristic classes and K-Theory

Winter 2012

The goal of the course will be to study vector bundles and fiber bundles. We will discuss: Here are a few motivating questions which can be answered with the techniques from this course:

Given a smooth manifold M of dimension m, let emb(M) resp. imm(M) be the smallest values of n such that M can be embedded, resp. immersed, in the n-dimensional Euclidean space. Whitney's Theorems say that emb(M) is at most 2m, and imm(M) at most 2m-1. It is also clear that emb(M) and imm(M) are at least m. What stronger lower bounds can we obtain, for particular manifolds?

For what values of n is there a bilinear multiplication on R^n without zero divisors? One is familiar with the values n=1 (real numbers), n=2 (complex numbers), n=4 (quaternions) and n=8 (octonions). In fact these are the only possible values.

How can one distinguish smooth manifolds that are homeomorphic but not diffeomorphic? Milnor (1956) gave the first example of such a pair, consisting of the 7-sphere and an exotic 7-sphere.


Other recommended books: Problem sets: