I hope this is not trivial.


Let $B$ be a nice topological space (paracompact, CW-complex or whatever you think is nice)


For $i=1,\ldots,n$ let $x\_i \in H^i(B,\mathbb{Z}\_2)$ be certain cohomology classes.



> 
> Does there exists a vectorbundle $E$ of rank $n$ with $w\_i=x\_i$?
> 
> 
> 


Of course $w\_i$ is the $i$-th Stiefel Whitney class of $E$.



Vector bundle for prescribed Stiefel-Whitney classes
No, because the Wu formulae express $\mathrm{Sq}^j(w\_i)$ in terms of $w\_k$'s, so if the $x\_i$ you choose don't satisfy this formula, they cannot possibly arise as Stiefel--Whitney classes.


