Vector bundles over iterated suspensions of stunted real projective   spaces
  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 $X^k_{m,n}$. 
Aniruddha C. Naolekar and Ajay Singh Thakur
math.AT