This course covers the classic book of Milnor and Stasheff on Characteristic Classes. Suppose you have a smooth manifold M. In order to understand M, we derive some invariants associated to the tangent bundle of M (or other bundles). These invariants provide a great deal of surprising information about the original manifold. The most elementary example of this kind of thing is the Euler characteristic, which you may remember is given for a polyhedron in space by the formula
\chi(M) = V – E + F
where V is the number of vertices of the polyhedron, E is the number of edges, and F is the number of faces.
A more sophisticated version of this formula would be the alternating sum of the 0th, 1st, and 2nd Betti numbers (ranks of the homology groups with coefficients in a field) of the two dimensional manifold M. An even more sophisticated view of \chi is the Poincare theorem, which asserts that \chi(M) counts the number of zeros of vector fields on M (in a certain way). We can write \chi(M) as the integral of an n-form over the n-dimensional manifold (actually, this form is given by the Gauss curvature), so \chi(M) could also be thought of as an n-dimensional cohomology class of M. In this form, \chi is an example of a Stiefel-Whitney class.
In fact, there are a set of n Stiefel-Whitney classes on an n-dimensional manifold, which together encode a surprising amount of information about the manifold. For example, they tell you that every (compact, orientable) 3-manifold has a set of 3 linearly independent nowhere vanishing vector fields on it. And they can tell you precisely when an n-manifold is the boundary of an n+1-manifold (again, a compact, orientable n+1 manifold).
Past the Stiefel-Whitney classes are new constructions called the Chern and Pontrjagin classes. These classes are defined not just for smooth manifolds, but for triangulated manifolds and even for topological manifolds. These invariants were the keys to unravelling some of the differences between these three different categories for the study of the topology of manifolds.
This course covers the Stiefel-Whitney, Chern, and Pontrjagin classes, following the book pretty closely with a few important geometric digressions thrown in for good measure. It should be a lot of fun!
Course Notes
I intend to scan and post my lecture notes before class and encourage you to read them over in advance. I also encourage you to try to read ahead in the book. Courses like this are hard, and you shouldn’t expect to get everything out of the lectures.
- Introduction to Characteristic Classes
- The structure of smooth manifolds (Chapter 1)
- Vector Bundles (Chapter 2)
- Operations on Vector Bundles (Chapter 3)
- Stiefel-Whitney Classes (Chapter 4)
- Grassmannians and the Universal Bundle (Chapter 5)
- A cell structure for Grassmannians (Chapter 6)
- The cohomology of Grassmannians (Chapter 7)
- The existence of Stiefel-Whitney classes (Chapter 8)
- Oriented Bundles and the Euler Class (Chapter 9)
- (We skipped Chapter 10, The Thom Isomorphism Theorem in order to save time)
- Computations on Smooth Manifolds (Chapter 11) (3 lectures)
- (We skipped Chapter 12, Obstruction theory and Stiefel-Whitney Classes in order to save time)
- Complex Vector Bundles and Chern Classes (Part I) (Chapters 13-14)
- Chern Classes (Part II) (Chapters 13-14)
- Pontrjagin Classes (Chapters 15-16)
Other course materials
Details
The course meets 11-12:15 in Boyd 326 on Tuesdays and Thursdays.
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