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

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, ${\text{s}}^{k}D(m,n)$, is $W$-trivial or not.