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What are the Stiefel-Whitney classes of complex projective spaces?

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closed as off topic by David White, Benoît Kloeckner, Daniel Moskovich, Ryan Budney, Chris Gerig May 5 '13 at 7:24

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up vote 5 down vote accepted

It follows from the Whitney sum formula that $c(T_{\mathbf{C}P^n}) = (1+z)^{n+1}$, where $z \in H^2(\mathbf{C}P^n,\mathbf{Z})$ is the generator corresponding to the canonical bundle. (I assume you meant the Chern classes; for the Stiefel-Whitney of a complex vector bundle restricted to a real vector bundle, $w_{2i}$ is the mod-$2$ reduction of $c_i$; and for the odd Stiefel-Whitney classes note that $H^{2i+1}(\mathbf{C}P^n) = 0$)

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