# Constructing vector bundles of specific stiefel whitney classes?

Is it possible to construct a vector bundle over a given base $X$ such that the $n$th stiefel whitney class vanishes for a given $n?$ What about for some set of integers? Can we make vector bundles of arbitrary stiefel whitney classes?

• Well, the trivial bundle always has vanishing $n$-th Stiefel-Whitney class for any $n$. If you're asking whether we can freely prescribe them, that's not always possible. (For example, they have to be compatible with the Wu formula) – Achim Krause May 7 '15 at 23:31
• Problem $8$-$B$ from Milnor & Stasheff provides a restriction. Namely, if $w(\xi) \neq 1$, then the smallest $i$ such that $w_i(\xi) \neq 0$ is a power of $2$. – Michael Albanese May 8 '15 at 2:18

A rank $n$ vector bundle on $X$ is precisely given by a homotopy class of maps $X\rightarrow BO(n)$. The Stiefel-Whitney classes of the vector bundle describe how that map acts on cohomology with $\mathbb{F}_2$ coefficients.