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…)