The first basic example of characteristic classes are the **Stiefel-Whitney classes.** Given a (real) -dimensional vector bundle , the Stiefel-Whitney classes take values in the cohomology ring . 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 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 be a bundle. The **Stiefel-Whitney classes** are characteristic classes that satisfy the following properties.

First, when . When you compute the cohomology of , the result is in fact a polynomial ring with generators. Consequently, we should only have characteristic classes of an -dimensional vector bundle. In addition, we require that always.

Second, like any characteristic class, the are natural: they commute with pulling back. If is a bundle, is a map, then . Without this, they would not be very interesting. (more…)