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 {\mathbb{RP}^n}. In particular, we want to compute the Stiefel-Whitney classes of the tangent bundle {T(\mathbb{RP}^n)}. The cohomology ring of {\mathbb{RP}^n} with {\mathbb{Z}/2}-coefficients is very nice: it’s {\mathbb{Z}/2[t]/(t^{n+1})}. We’d like to find what {w(T(\mathbb{RP}^n)) \in  \mathbb{Z}/2[t]/(t^{n+1})} is.

On {\mathbb{RP}^n}, we have a tautological line bundle {\mathcal{L}} such that the fiber over {x \in \mathbb{RP}^n} is the set of vectors that lie in the line represented by {x}. Let’s start by figuring out the Stiefel-Whitney classes of this. I claim that

\displaystyle w(\mathcal{L}) = 1+t \in H^*(\mathbb{RP}^n, \mathbb{Z}/2).

The reason is that, if {\mathbb{RP}^1 \hookrightarrow  \mathbb{RP}^n} is a linear embedding, then {\mathcal{L}} pulls back to the tautological line bundle {\mathcal{L}_1} on {\mathbb{RP}^1}. In particular, by the axioms, we know that {w(\mathcal{L}_1) \neq 1}, and in particular has nonzero {w_1}. This means that {w_1(\mathcal{L}) \neq 0} by the naturality. As a result, {w_1(\mathcal{L})} is forced to be {t}, and there can be nothing in other dimensions since we are working with a 1-dimensional bundle. The claim is thus proved. (more…)