# Math 227B: Algebraic Topology

Characteristic classes and K-Theory

Winter 2012

• Time and Place: MWF 12-12:50 pm in MS 5148
• Instructor: Ciprian Manolescu
• E-mail: cm_at_math.ucla.edu
• Office: MS 6921
• Office Hours: W 10-11am, Th 11am-12pm
The goal of the course will be to study vector bundles and fiber bundles. We will discuss:
• Stiefel-Whitney classes;
• Chern classes;
• Euler and Pontryagin classes;
• K-Theory;
• Bott periodicity;
• Spectra and generalized homology theories.
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.

Textbooks:

• J. Milnor and J. Stasheff, Characteristic classes, Princeton University Press, 1974.
• A. Hatcher, Vector Bundles and K-Theory, available online.
Other recommended books:
• M. F. Atiyah, K-Theory, W. A. Benjamin, 1967.
• R. Bott, Lectures on K(X), W. A. Benjamin, 1969.
• D. Husemoller, Fibre Bundles, Springer Verlag, 1993.
• M. Karoubi, K-Theory: An Introduction, Springer-Verlag, 1978.
• J. P. May, A Concise Course in Algebraic Topology, Chicago Univ. Press, 1999.
• N. Steenrod, Topology of fiber bundles, Princeton Univ. Press, 1951.
Problem sets: