Vector bundles over iterated suspensions of stunted real projective spaces

TL;DR

This paper characterizes when the total Stiefel-Whitney class is trivial for vector bundles over iterated suspensions of stunted real projective spaces, providing a complete classification based on parameters k, m, n.

Contribution

It completely determines the conditions on k, m, n under which the total Stiefel-Whitney class equals one for all vector bundles over these spaces.

Findings

Identifies all (k,m,n) where w(ξ)=1 for any vector bundle ξ.

Provides a classification of vector bundles with trivial Stiefel-Whitney classes over these spaces.

Enhances understanding of vector bundle topology over iterated suspensions of stunted real projective spaces.

Abstract

Let $X^k_{m,n}=\Sigma^k (\mathbb R\mathbb P^m/\mathbb R\mathbb P^n)$. In this note we completely determine the values of $k,m,n$ for which the total Stiefel-Whitney class $w(\xi)=1$ for any vector bundle $\xi$ over $X^k_{m,n}$.