Computing $\pi_3(\mathrm{Gr}_2(\mathbb{R}^4))$ How can one go about computing the 3rd homotopy group of the Grassmannian manifold of 2-planes through the origin in $\mathbb{R}^4$?
I don't want to be more general in the question, because:
1) I think this is related to homotopy groups of spheres which is currently intractable.
2) This particular group pops up in my thoughts on vector bundles.

$Gr_2(\mathbb R^4)$ is the quotient $O_4 / O_2 \times O_2$, which is double covered by $SO_4 / SO_2 \times SO_2$.  $SO_4$ is double-covered by $S^3 \times S^3$.   So $Gr_2(\mathbb R^4)$ has $(S^3 / SO_2)^2$ as a covering space, and this is just $(S^2)^2$.   So $\pi_3 Gr_2(\mathbb R^4)$ is the same as $\pi_3 (S^2 \times S^2)$ which is a free abelian group of rank two. 
