Given a real Grassmanians $$\textrm{Gr}(n,m)=O(n+m)/O(n)\times O(m)$$ could we determine the following specific homotopy groups: $$\pi_0\textrm{Gr}(n,m)=?$$ $$\pi_n\textrm{Gr}(n,m)=?$$ $$\pi_m\textrm{Gr}(n,m)=?$$ $$\pi_{n+m}\textrm{Gr}(n,m)=?$$
You can provide partial answers. No need to be complete if some of the answers are too harsh. We can boil down the answer by writing in terms of homotopy groups of $O(n+m)$, $O(n)$, $ O(m)$.
We take $n,m \geq 1$.