Stiefel-Whitney Classes of a submanifold

Question

Suppose we have a manifold $M$ that is the product of $k$ copies of real projective space, say $$ M = \prod_{i=1}^k \mathbb{R} P^{n_i}.$$ Then $H^*(M; \mathbb{Z}/2) = \mathbb{Z}/2 [ \alpha_1, \dots, \alpha_k] / (\alpha_1^{n_1 + 1}, \dots, \alpha_k^{n_k + 1})$. Let $\sigma \in H^1(M) = \alpha_1 + \dots + \alpha_k$. Now suppose we have some sub-manifold $N \subseteq M$ with inclusion $i : N \to M$. Let $\nu$ denote the normal bundle of $N$ in $M$. I came across a statement that for a specific sub-manifold $N$ constructed in a certain way, $$ w(\nu) = i^*(1 + \sigma) $$ where $w(\nu)$ is the total Stiefel-Whitney class of $\nu$. Is this true for an arbitrary sub-manifold $N$? If so, how can you see this?

Answer 1

There is of course an obvious counter-example when $N=M$ and $i$ is the identity. In this case, $w(\nu)=1$ (since $\nu$ has rank $0$), but $i^*(1 + \sigma) = 1 + \sigma$.

In the case you are considering, they are likely using the fact that the pullback of the tangent bundle of $M$ splits as $TN \oplus \nu = i^* TM$. Hence, the Whitney product formula and naturality imply that $w(TN)\cdot w(\nu) = i^* w(TM)$. The total Stiefel-Whitney class of $\mathbb{R}P^n$ is given by $(1 + \alpha)^{n+1} \in H^*(\mathbb{R}P^n, \mathbb{Z}_2) = \mathbb{Z}_2[\alpha] / (\alpha^{n+1})$. It follows from the Whitney product formula that $w(M) = \prod_{i=1}^k (1+\alpha_i)^{n_i+1}$. One can then compute $w(\nu)$ from $w(TN)$ and the pullback of the $\alpha_i$.

For a less trivial counter-example than the one above, consider the usual embedding of $N=\mathbb{R}P^{n}$ into $M=\mathbb{R}P^{n+k}$. The generator $\alpha$ of $H^1(\mathbb{R}P^{n+k}, \mathbb{Z}_2)$ pulls back to the generator $\tilde{\alpha}$ of $H^1(\mathbb{R}P^{n}, \mathbb{Z}_2)$. Therefore, $w(TN)\cdot w(\nu) = i^* w(TM)$ becomes $(1+\tilde{\alpha})^{n+1} \cdot w(\nu) = (1+\tilde{\alpha})^{n+k+1}$, and so $w(\nu) = (1+\tilde{\alpha})^k$ which in general does not equal $i^*(1+\sigma) = 1+ \tilde{\alpha}$ in $H^*(\mathbb{R}P^n, \mathbb{Z}_2)$. In fact, even if $n=1$ they can be made non-equal by taking $k$ even. This example easily generalizes to products of projective spaces.