Let $Gr_n=BO(n)$ be the real infinite grassmanian. I want to know wether $\pi_1(Gr_n)=\mathbb{Z}/2$ acts nontrivially on the higher homotopy groups of $Gr_n$. It is known that the action can be computed by inspection of the action of $O(n)$ on $\pi_*(O(n),\text{id})$ by conjugation. But I don't know whether that helps. The case $n=1$ is clear since the universal cover of $Gr_1=\mathbb{R}P^\infty$, which is given by $S^\infty$, is contractible. I want to know this to see whether $\pi_{m+1}(Gr_n,*)=\pi_m(O(n),\text{id})$ computes the isomorphism classes of $n$-dimensional vector bundles on $S^{m+1}$.
Can the question be rephrased as: Does flipping the orientation of an oriented vector bundle on $S^m$ change the oriented isomorphism type of the oriented vector bundle? Note that $\pi_{m}(Gr_n, *)=\pi_m(\tilde{Gr}_n, *)=[S^m, \tilde{Gr}_n]$ classifies orientable vector bundles on $S^m$. Does the deck transformation group act by flipping the orientation of the vector bundle here?
