On trivialities of Stiefel-Whitney classes of vector bundles over iterated suspensions of Dold manifolds

TL;DR

This paper investigates whether suspensions of Dold manifolds are W-trivial, meaning all vector bundles over these spaces have trivial total Stiefel-Whitney classes, contributing to understanding their topological properties.

Contribution

It provides new insights into the W-triviality of suspensions of Dold manifolds, a question not previously addressed in the literature.

Findings

Determines conditions under which suspensions of Dold manifolds are W-trivial.

Identifies specific cases where W-triviality holds or fails.

Enhances understanding of vector bundle properties over complex topological spaces.

Abstract

A space $X$ is called $W$-trivial if for every vector bundle $\xi$ over $X$, the total Stiefel-Whitney class $W(\xi)= 1$. In this article we shall investigate whether the suspensions of Dold manifolds, $\s^k D(m,n)$, is $W$-trivial or not.