I'm trying to understand the relationship between Chern classes and Stiefel-Whitney classes, and I came upon this problem (14-E) in Milnor-Stasheff's Characteristic Classes.
We are asked to define the Stiefel-Whitney classes in the same way as was constructed for Chern classes, using mod 2 coefficients: $w_n(\xi) := e(\xi)$ mod 2 and $w_i(\xi):=(\pi_0^*)^{-1}w_i(\xi_0)$ for $i<n$, where $\pi_0^*: H^{2i}(B)\to H^{2i}(E_0)$ is an isomorphism for $i<n$.
In the problem, it states:
In this approach there is some difficulty in showing that $w_{n-1}(\xi_0)$ belongs to the image of $\pi_0^*.$ It suffices to show that $w_{n-1}(\xi_0)$ restricts to zero in each fiber $F_0$, or equivalently, that the tangent bundle $\tau$ of the $(n-1)$-sphere satisfies $w_{n-1}(\tau)=0$.
This is easy to prove since $e(\tau)=0$ or $2$ depending on odd or even dimensions, so $e(\tau)$ mod $2=0=w_{n-1}(\tau).$ But why does it suffice to prove this? That is, I don't necessarily understand why this proves "$w_{n-1}(E_0)$ belongs to the image of $\pi_0^*$." I'm misunderstanding a step in this logic that is perhaps quite trivial.
Best Answer
The map $\pi_0^*$ fits into a long exact sequence including $$ H^{n-1}(B) \to H^{n-1}(E_0) \to H^n(E, E_0)$$ So, we want to check that $w_{n-1}(\xi_0)$ maps to zero under $H^{n-1}(E_0) \to H^n(E, E_0)$. The class in $H^n(E, E_0)$ is determined by its restrictions to the fibers $H^n(F, F_0)$ which is isomorphic via the corresponding long exact sequence to $H^{n-1}(F_0)$, thus we need to see where $w_{n-1}(\xi_0)$ maps to under the restriction $H^{n-1}(E_0)\to H^{n-1}(F_0)$. At this point, by the definition of $\xi_0$ and naturality, it should be easy to see that $w_{n-1}(\xi_0)$ maps to $w_{n-1}(\tau)$ where $\tau$ is the tangent bundle of the $(n-1)$-sphere.