# Stiefel manifolds are $n - p - q$ connected.

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 it $f$ 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.)

-
I would use the long exact sequence of homotopy groups associated to the inclusion $O_{n-p}\to O_n$. – Grumpy Parsnip Mar 13 '12 at 22:22
OK my edits happened while you were saying this but I will take a closer look at this now. Thank you. – Kyle Mar 13 '12 at 22:26
I edited my question with another update. I'm not sure I full understand what you're suggestion says. – Kyle Mar 13 '12 at 22:56
This is the sequence I was referring to: $\cdots \pi_k(O_{n-p})\to \pi_k(O_n)\to \pi_k(O_n,O_{n-p})\to \pi_{k-1}(O_{n-p})\to \cdots$ – Grumpy Parsnip Mar 13 '12 at 22:59
The identity matrix is the most natural choice of basepoint. – Grumpy Parsnip Mar 13 '12 at 23:14
Let us call $St_p(\mathbb R^n)$ the Stiefel manifold of orthonormal $p$-frames in $\mathbb R^n$.
Call $e_n$ the vector $(0,0,\cdots,0,1)$. We then have the fibration over the sphere $S^{n-1}$: $$St_{p-1}(\mathbb R^{n-1}) \to St_p(\mathbb R^n)\to S^{n-1}$$ where the first morphism sends $(v_1,\cdots, v_{n-1})$ to $(v_1,\cdots,v_{n-1},e_n)$ and the second one sends $(v_1,\cdots, v_{n-1},v_n)$ to $v_n.$
$$\pi_i(St_p(\mathbb R^n))=\pi_{i}(St_{p-1}(\mathbb R^{n-1}))=...= \pi_{i}(St_1(\mathbb R^{n-p+1}))= \pi_{i}(S^{n-p})$$
and these homotopy groups are zero if $\pi_{i}(S^{n-p})=0$, which is the case for $i\leq n-p-1$ so that $St_p(\mathbb R^n)$ is indeed $(n-p-1)$- connected.