The first basic example of characteristic classes are the Stiefel-Whitney classes. Given a (real) ${n}$-dimensional vector bundle ${p: E \rightarrow B}$, the Stiefel-Whitney classes take values in the cohomology ring ${H^*(B, \mathbb{Z}/2)}$. They can be used to show that most projective spaces are not parallelizable.

So how do we get them? One way, as discussed last time, is to compute the ${\mathbb{Z}/2}$ cohomology of the infinite Grassmannian. This is possible by using an explicit cell decomposition into Schubert varieties. On the other hand, it seems more elegant to give the axiomatic formulation. That is, following Milnor-Stasheff, we’re just going to list a bunch of properties that we want the Stiefel-Whitney classes to have.

Let ${p: E \rightarrow B }$ be a bundle. The Stiefel-Whitney classes are characteristic classes ${w_i(E) \in H^i(B, \mathbb{Z}/2)}$ that satisfy the following properties.

First, ${w_i(E) = 0}$ when ${i > \dim E}$. When you compute the cohomology of ${\mathrm{Gr}_n(\mathbb{R}^{\infty})}$, the result is in fact a polynomial ring with ${n}$ generators. Consequently, we should only have ${n}$ characteristic classes of an ${n}$-dimensional vector bundle. In addition, we require that ${w_0 \equiv 1}$ always.

Second, like any characteristic class, the ${w_i}$ are natural: they commute with pulling back. If ${E \rightarrow B}$ is a bundle, ${f: B' \rightarrow B}$ is a map, then ${w_i(f^*E) = f^* w_i(E)}$. Without this, they would not be very interesting. (more…)

I’ve been reading about spectra and stable homotopy theory lately, but don’t feel ready to start talking about them here. Instead, I shall say a few words on characteristic classes. The present post will be quite general and preparatory — the more difficult matter is to actually construct such characteristic classes. Our goal is to see that characteristic classes essentially boil down to computing the cohomology of the infinite Grassmannian.

A lot of problems in mathematics involve the existence of sections to vector bundles. For instance, there is the old question of when the sphere is parallelizable. A quick Euler characteristic argument shows that even-dimensional spheres can’t be—then there would be an everywhere nonzero vector field, whose infinitesimal flows would be homotopic to the identity (and consequently having nonzero Lefschetz number by the even-dimensionality) while having no fixed points. In fact, much more is known. Using the group or group-like structures on ${S^1, S^3, S^7}$ (coming from the complex numbers, quarternions, and octonions), it is easy to see that these manifolds are parallelizable. But in fact no other sphere is.

A characteristic class is a means of assigning some invariant to a vector bundle. Ideally, it should be trivial on trivial bundles, so the characteristic class can be thought of as an “obstruction” to finding large numbers of linearly independent sections.

More formally, let ${p: E \rightarrow B}$ be a vector bundle. A characteristic class assigns to this bundle (of some fixed dimension, say ${n}$) an element of the cohomology ring ${H^*(B)}$ (with coefficients in some ring). To be interesting, the characteristic class has to be natural. That is, if ${f: B' \rightarrow B}$ is a map, then the characteristic class of the pull-back bundle ${f^*E \rightarrow B'}$ should be the pull-back of the characteristic class of ${E \rightarrow B}$. (more…)