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?
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Yes; see problems 14B and 14E in Milnor-Stasheff, Characteristic classes. The main point is to verify this for the top Chern class/SW class by identifying both with the Euler class (mod 2). This in turn follows by comparing the integral and mod 2 Thom classes.
answered May 8, 2014 at 11:54
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1$\begingroup$ Does the problem 14B means $w_{2n}=c_n$ mod $2$? Thanks. $\endgroup$ May 8, 2014 at 12:40