I'm supposed to show that $V_{n,p}$ is $n - p - 1$-connected. In this case, $V_{n,p}$ is the topological group $O_{n}/O_{n - p}$, where $O_{n}$ is the set of $n\times n$ orthonormal matrices. So this means I need to show that $\pi_{n-p-1}(V_{n,p})$ is the $1$-element group.
I was hoping I could get a push in the right direction? I feel like I don't know how to start here.
I realize I haven't done much considering I'm asking for help. So let me clarify my main issue: I really don't know how high level the solution should be. I don't have much experience computing fundamental groups except in really simple cases. Assuming for the moment that the solution is "low level", let me consider a continuous map
$f:(S^{n-p-1}, e_{n-p-1})\to (O/O_{n-p}, O_{n-p})$ then I need to show that $f$ is homotopic to the constant map $x\mapsto O_{n-p}$. But since I have little insight as to this space I feel completely lost.
Update #2:
Is this the exact sequence you mean?
$0\to O_{n-p}\to O_{n}\to O_{n}/O_{n-p}\to 0$
Oh. I think maybe I see what you mean. (Will have another edit shortly.)