On trivialities of Stiefel-Whitney classes of vector bundles over iterated suspension spaces

Abstract

A space $B$ is described as W-trivial if for every vector bundle over $B$, all the Stiefel-Whitney classes vanish. We prove that if $B$ is a 9-fold suspension, then $B$ is W-trivial. We also determine all pairs ($k,n$) of positive integers for which $\Sigma^k F P^n$ is W-trivial, where $F=\mathbb{R}, \mathbb{C}$ or $\mathbb{H}$.