A classical result states that the quotient $SO(4)/SO(3)$ is homotopy equivalent to $S^3$. In fact, this can be stated in more general terms since $SO(n+1)/SO(n)$ has the homotopy type of $S^n$. What I don't know is what is the homotopy type of $SO(n+1)/SO(n-1)$ or even if it has a general formultation. The concrete case I am interested in is the one for $n=3$, i.e. the homotopy type of $SO(4)/SO(2)$. Or, at least if its first homotopy groups $\pi\_k(SO(4)/SO(2))$ (for $k=1,2,3$) are known. Thanks a lot in advance.



Homotopy type of $SO(4)/SO(2)$
This is the [Stiefel manifold](https://en.wikipedia.org/wiki/Stiefel_manifold) $V\_2(\mathbb{R}^4)$. It fits into a fibration $S^2\to V\_2(\mathbb{R}^4)\to S^3$ and so has trivial $\pi\_1$. By playing around with the long exact sequence of this fibration I was able to show that $\pi\_2(V\_2(\mathbb{R}^4))\cong \mathbb{Z}$ and $\pi\_3(V\_2(\mathbb{R}^4))\cong \mathbb{Z}\oplus\mathbb{Z}$ (but I don't know a reference off the top of my head). 


