Stiefel-Whitney class of complex projective spaces Let $T\mathbb{C}P^m$ be the tangent bundle of complex projective space. What is the total Stiefel-Whitney class $w(T\mathbb{C}P^m)$? 
Let $a_m$ be the maximal integer such that the $a_m$-th dual Stiefel-Whitney class $\bar w_{a_m}(T\mathbb{C}P^m)$  is nonzero ($\bar w=1/w$, $\bar w_j$ is the degree-$j$-component of $\bar w$). Can we write $a_m$ in terms of $m$?
I do not know how to solve. I even do not know how to solve the case $m=2$...
 A: A  topological complex  vector bundle $E$ on a manifold $M$  has Chern classes $c_i(E)\in H^{2i}(M;\mathbb Z)$ and its underlying  real vector bundle $E_\mathbb R$ has Stiefel-Whitney classes $w_j(E_\mathbb R)\in H^{j}(M;\mathbb Z/2)$.
The relation betwen both could not be more idyllic:    
$\bullet $ The odd Stiefel-Whitney classes of $E_\mathbb R$ vanish: $w_{2j+1}(E_\mathbb R)=0$.
$\bullet \bullet$ The canonical map  $H^{2i}(M;\mathbb Z)\to H^{2i}(M;\mathbb Z/2)$ sends $c_i(E)$ to $w_{2i}(E_\mathbb R)$.  
In the case of $E=T_{\mathbb P^n(\mathbb C)}$ we have $c_i(T_{\mathbb P^n(\mathbb C)})=\binom{n+1}{i}\in  H^{2i}(P^n(\mathbb C);\mathbb Z)=\mathbb Z$ for $i=0,1,\dots,n$ and $c_i(T_{\mathbb P^n(\mathbb C)})=0$ for  $i\gt n$.
Thus reducing mod. $2$ we get $w_{2i}((T_{\mathbb P^n(\mathbb C)})_\mathbb R)=\binom{n+1}{i} \operatorname{mod. 2}\in  H^{2i}(P^n(\mathbb C);\mathbb Z/2)=\mathbb Z/2$ for $i=0,1,\dots,n$ and all the other Stiefel-Whitney classes of $(T_{\mathbb P^n(\mathbb C)})_\mathbb R$ are zero.  
Bibliography
Milnor-Stasheff,  Theorem 14.10 page 169 and Problem 14-B page 171. 
