Vector bundles over iterated suspensions of stunted real projective spaces

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

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