Characteristic Classes

摘要

The aim of this course is to provide an introduction to characteristic classes for post graduate students who had no previous specialised knowledge (nevertheless, the group who faithfully attended the course was quite heteogeneous including beginners and experts). It was of course impossible to give a whorthwhile elementary survey of this very large area of mathematics in six hours'lecturing time. I thus decided to focus on some particular aspects of the theory and to leave a lot out. To choose what to put in and what to leave out I added a (somewhat artificial but) motivating extra goal to these lectures: the construction of the Milnor exotic sphere. The discovery of such a sphere marked a milestone in the history of differential topology and the key tool in this discovery was the Hirzebruch signature theorem which expresses the signature of a manifold as a combination of characteristic classes. The statement of this result will provide us our target and will guide us in our choices.