What is the relation between the homotopy groups of spheres $S^n$ and the homotopy groups of the special orthogonal groups $SO(n)$ (resp. $O(n)$)?
This question occurred to me in the context of classifying real vector bundles over spheres via homotopy classes of maps. One can show that (see for example Hatcher):
For $k>1$, there is a bijection $[S^{k-1},SO(n)]\longleftrightarrow Vect^n(S^k)$ Here $[S^{k-1},SO(n)]$ denotes the set of homotopy classes of maps $S^{k-1}\to SO(n)$ and $Vect^n(S^k)$ denotes the set of isomorphism classes of real rank $n$ vector bundles over $S^k$. Furthermore one has that the map $\pi_i(SO(n))\longrightarrow [S^i,SO(n)]$ which ignores the basepoint data is a bijection, so one can essentialy classify real vector bundles over spheres via the homotopy groups of $SO(n)$.
I'm also interested in the following:
What is the relation between homotopy groups of spheres and the classification of real vector bundles over spheres?
Any references (as well as examples) would be much appreciated.