The first part of this course is an introduction to Characteristic Classes. The last 3 weeks will focus on Differential Forms. The class is intended for first year graduate students.
There will be a midterm and a final. There will also be weekly homework. Homework is posted to this website each Friday and due at the start of class the following Friday. Late homework will not be accepted.Homework is usually taken directly from the book Characteristic Classes by Milnor; the notation x:y means problem y from section x.
Notes on Differential Forms can be downloaded here and will be updated as we go along (notes updated 3/1/2016).
It is expected that students taking the class have taken the fall graduate algebraic topology class before. Thus we will assume material such as fundamental group, homology, cohomology, higher homotopy groups, Eilenberg-MacLane spaces (although we will recall some results as necessary).
Some familiarity with basic smooth topology is assumed (implicit function theorem, Sard's theorem, transversality, partitions of unity), roughly the first few chapters of Milnor's Topology from the Differentiable Viewpoint.
We aim to go through most of the book Characteristic Classes by Milnor and Stasheff.
You might also find it helpful to watch Milnor's Hedrick lectures on Differential Topology from 1965, available from the Simons Foundation here.