Last time we gave the axiomatic description of the Stiefel-Whitney classes. Today, following Milnor-Stasheff, we want to look at what happens in the particular case of real projective space . In particular, we want to compute the Stiefel-Whitney classes of the tangent bundle . The cohomology ring of with -coefficients is very nice: it’s . We’d like to find what is.

On , we have a tautological line bundle such that the fiber over is the set of vectors that lie in the line represented by . Let’s start by figuring out the Stiefel-Whitney classes of this. I claim that

The reason is that, if is a linear embedding, then pulls back to the tautological line bundle on . In particular, by the axioms, we know that , and in particular has nonzero . This means that by the naturality. As a result, is forced to be , and there can be nothing in other dimensions since we are working with a 1-dimensional bundle. The claim is thus proved. (more…)