Relation between Stiefel-Whitney class and Chern class

A complex vector bundle of rank n can be viewed as a real vector bundle of rank 2n. From nLab, we have that the second Stiefel-Whitney class of the real vector bundle is given by the first Chern class of the complex vector bundle mod 2: $w_2=c_1$ mod 2. Do we have similar relations for other Stiefel-Whitney classes, such as $w_{2n}=c_n$ mod 2?

Solution 1:

The following is from Lecture notes of Professor Farrell. A hopefully working link is here(page 95): https://www.dropbox.com/s/80n4wd6xctpe6yr/Characteristic%20Classes%20%28Sparkie%20%E7%9A%84%E5%86%B2%E7%AA%81%E5%89%AF%E6%9C%AC%202014-02-19%29.pdf

We now reach proposition 7. Let $E$ be a complex vector bundle, then the mod 2 reduction of the total Chern class of $E$ is the total Stiefel-Whitney class of $E$. Since there is no Chern classes in odd dimensions, we have there is no Stiefel-Whitney classes in odd dimension as well.

$\textbf{Proof}$

Suppose $E$ is a line bundle. Then we know $w_{0}(L)=1, w_{1}(L)=0,w_{2}(L)=e_{\mathbb{Z}_{2}}(L)=\phi(e(L))$. On the other hand we know $c_{0}(L)=1, c_{1}(L)=e(L)$. So this verified it for line bundles.

For sum of line bundles we have $$ E=\oplus^{n}_{i=1}L_{i} $$ So the total Chern class is $$ c(E)=c(L_{1}) \cup \cdots \cup c(L_{n})\mapsto_{\phi}\omega(E)=\omega(L_{1}) \cup \cdots \cup \omega(L_{n}) $$ We are now going to use splitting principle. But there is a subtle point. We now discuss it.

By the splitting principle there exist $$ f:\mathcal{B}\rightarrow B $$ such that $$ f^{*}(E)=\oplus_{i=1}^{n}L_{i} $$ and $$ f^{*}:H^{*}(B,R)\rightarrow H^{*}(\mathcal{B},R) $$ is monic.

Therefore by naturality and the fact $f^{*}$ is monic with respect to $\mathbb{Z}_{2}$ coefficients. $$ \phi(f^{*}(c(E)))=f^{*}(\phi(c(E)))=f^{*}(\omega(E)) $$