Computation on characteristic classes

Question

I am organizing a reading seminar on characteristic classes. The audience in the seminar is interested in symplectic and contact manifolds. I work in categorification and would like to compute some invariants in Lawson-Lipshitz-Sarkar spectra of Khovanov homology which is a cubical complex. We read first seven chapters of Milnor's characteristic class last semester.

This semester I want to focus on the computational aspects of characteristic classes. Could anyone please suggest me some papers that mostly deal with computational techniques of Chern and Stiefel-Whitney classes? It should explore different ways of computations. Thank you for your suggestions.

Answer 1

I wrote a blog post that turned into quite a nice exercise in characteristic classes. The goal was to compute the cohomology of a smooth hypersurface of degree $d$ in $\mathbb{CP}^3$, as a ring. This computation turns out to involve computations of Euler classes, Chern classes, Pontryagin classes, Wu classes, and Stiefel-Whitney classes. The techniques aren't difficult (mostly everything follows from the computation of the Chern classes of $\mathbb{CP}^n$) but it's fun to see them work together to produce a concrete result like this.

Answer 2

There is a great theorem proved by Halperlin - Toledo, which says that if you have a triangulated manifold $M$, you can represent combinatorially every Stiefel - Whitney class $w^k(M) \in H^k(M, \mathbf Z_2)$ as follows: take the barycentric subdivision, and assign 1 to every $(n-k)$-simplex in it. This defines a cycle in $H_{n-k}(M, \mathbf Z_2)$ that is Poincaré dual to $w^k$. Ta-dah!

Answer 3

The papers "Characteristic Classes and Homogeneous Spaces, I, II" by A. Borel and F. Hirzebruch is a classical resource. In general, homogeneous spaces form a rich class of examples to compute and play with.

It is useful to try to construct vector bundles with prescribed characteristic classes. The simplest example is that any class in $H^2(B)$ is the first Chern class of a complex line bundle over $B$, and any class in $H^1(B;\mathbb Z_2)$ is the first Stiefel-Whitney class of a real line bundle over $B$. This can be found in Husemoller's "Fiber bundles".

If you wish to have similar results for other vector bundles, you need to proceed by obstruction theory from maps from the base $B$ to the classifying space $BSO(n)$. Since the rational cohomology algebra of $BSO(n)$ is a polynomial algebra over appropriate Euler and Pontryagin classes, $BSO(n)$ is rationally a product $\Pi$ of Eilenberg-MacLane spaces with each factor corresponding to a characteristic class. Then we try to lift a map $B\to\Pi$ to $BSO(n)$. The conclusion is that after multiplying suitable classes in $H^*(B)$ by appropriate positive integers, one can realize them as Euler and Pontryagin classes of an oriented vector bundle over $B$. Also, these classes determine a vector bundle up to finite ambiguity (among all oriented vector bundles of a given fiber dimension). The only reference I know is to papers of mine (sorry!), see appendix B and appendix A even though this is standard material.