Stiefel-Whitney Classes: Simple Example

Question

I need help finding the Stiefel-Whitney classes $w_k(\eta)$ of the normal bundle of the $n$-sphere.

Now since $H^k (S^n ; \mathbb{Z}/2\mathbb{Z}) =0$ for $k \neq 0,n$,

then $w_k(\eta) =0$ for $k \neq 0,n$.

What about when $k=0$ or $n$? Now I know that $w(\eta) = 1+ w_1(\eta)+ w_2(\eta)+ \cdots=1 $.

Can I use this to say that $w_i (\eta)=0$ for any $i$? Or should it be done using the definition of $w_i$?

I feel like I might be missing something here...

Thank you.

Answer 1

The line bundle $\eta$ has a nowhere vanishing section ($x \mapsto x$ is a section if you use the standard embedding $S^n \hookrightarrow \mathbb{R}^{n+1}$), therefore it is trivial. Stiefel-Whitney classes are invariant under isomorphism of bundles, therefore $w_i(\eta) = 0$ for all $i \geq 1$.