On Stiefel-Whitney classes of vector bundles over real Stiefel Manifolds

TL;DR

This paper investigates the possible degrees of nonzero Stiefel-Whitney classes of vector bundles over real Stiefel manifolds, establishing bounds and relations among these classes.

Contribution

It provides bounds on the degrees of nonzero Stiefel-Whitney classes and describes their structure when the first nonzero class occurs at a power of two.

Findings

At most two possible degrees for nonzero classes up to 2(n-k).

If n > k(k+4)/4, the nonzero classes occur at degrees multiple of a power of two.

Relations among classes at degrees that are multiples of 2^q.

Abstract

In this article we show that there are at most two integers up to $2(n-k)$, which can occur as the degrees of nonzero Stiefel-Whitney classes of vector bundles over the Stiefel manifold $V_k(\mathbb{R}^n)$. In the case when $n> k(k+4)/4$, we show that if $w_{2^q}(\xi)$ is the first nonzero Stiefel-Whitney class of a vector bundle $\xi$ over $V_k(\mathbb{R}^n)$ then $w_t(\xi)$ is zero if $t$ is not a multiple of $2^q.$ In addition, we give relations among Stiefel-Whitney classes whose degrees are multiples of $2^q$.