Characteristic classes

Goal:Understand the basic theory of characteristic classes, classifying
spaces and classifying bundles.

Supervisor: Calin Lazaroiu

0. Prerequisites (slides)

Scheme:

A. Basic theory of topological principal bundles.

1. The concept of a principal bundle
2. Isomorphism of principal bundles
3. Basic operations on principal bundles (concept of product on principal bundles)
4. Fibre bundle associated with a principal bundle (e.g. vector bundles)

B. Basic theory of classifying spaces.

C. General theory of characteristic classes.

D. Characteristic classes of principal bundles with semisimple structure group.

E. Applications: Characteristic classes of principal bundles with structure groups U(n) and SU(n).Topological classification of real and complex vector bundles; the splitting principle.

Historical: 50 mins presentation on the history of the subject and the main people who contributed to it.

References:

Milnor & Stasheff : "Characteristic classes", Princeton U.P., 1974

Hatcher: "Vector bundles and K-theory"

More references here.