Let $F,E,B$ be manifolds (we may assume them to be compact, without boundary if necessary) and $$F\to E\to B$$ be a fibre bundle. Suppose $$w(F), w(B),$$ the Stiefel-Whitney class of $F$ and $B$, are known. I notice that in particular, if the bundle is trivial, then $E=B\times F$ and $w(E)=w(B)w(F)$. 


**Question:** In general, are there any formulas to compute the Stiefel-Whitney class of the total space $E$
$$
w(E)
$$ 
in terms of $w(B)$ and $w(F)$? Or even in terms of the cohomology ring 
$$
H^\*(B;\mathbb{Z}\_2), H^\*(F;\mathbb{Z}\_2)
$$
And other factors?



Stiefel-Whitney class of fibre bundles
Just to get started (a bit long for a comment):


For a fiber bundle $F\rightarrow E\overset{\pi}{\rightarrow}B$ one has an exact sequence of bundles over $E$


$$
0\rightarrow T\_vE\rightarrow TE\rightarrow \pi^{\*}TB\rightarrow 0.
$$


Here $T\_vE:=\ker T\pi$ is the vertical part of the tangent space to $E$, i.e. these are the tangent vectors to the fibers. The choice of a complementary bundle amounts to the choice of a connection, but can be identified with $\pi^\*TB$. So one sees that 


$$
w(TE)=w(T\_vE)\pi^\*(w(TB)).
$$


I think one needs more information if one wants to continue the computation. For example one can go on if $E$ is the projectivization of a vector bundle over $B$.


