Type homotopy of Stiefel variety

Question

Why Stiefel variety $V_{n}(\mathbb{C}^{k})$ of $n$-frame in $\mathbb{C}^{k}$ is $2k-2n$ connected? I know that all homotopy groups of infinite Stiefel variety $V_{n}(\mathbb{C}^{\infty})$ vanish and Stiefel variety has a CW structure. Then how can I use Whitehead theorem to prove that it is contractible?

Answer 1

For the first question, there are two manifolds commonly called the Stiefel manifold. The first is the collection of $n$-tuples of linearly independent vectors in $\mathbb{C}^k$, the second is the collection of $n$-tuples of orthonormal vectors in $\mathbb{C}^k$.

The Gram-Schmidt process gives a deformation retract from the first manifold to the second. Thus, we may as well focus on the second description for answering your question.

The group $U(k)$ acts transitively on the set of all $n$-tuples of orthonormal vectors. For a proof, given any orthonormal $n$-tuple $\{v_1,...,v_n\}$, extend it to an orthonormal basis of $\mathbb{C}^k$. The matrix which has these vectors as columns will be in $U(k)$ and $U(k)\cdot \{e_1,....,e_n\} = \{v_1,...,v_n\}$ (where $e_i$ is the standard orthonormal basis of $\mathbb{C}^k.$)

The stabilizer of this action, at the point $\{e_1,...,e_n\}$ is given by all matrices in $U(k)$ for which the initial $n\times n$ block is the identity. It's easy to see that that such a matrix must be block diagonal, with the second block consisting of an arbitrary element of $U(k-n)$.

This proves $V_n(\mathbb{C^k})$ is diffeomorphic to $U(k)/U(k-n)$. It remains to compute the topology of $U(k)/U(k-n)$.

Since $U(k-n+1)/U(k-n) \cong S^{2(k-n+1)-1} = S^{2k-2n+1}$ is $2k-2n$ connected, this implies the inclusion map $$U(k-n)\rightarrow U(k-n+1)$$ is $2k-2n$ connected as well. A similar argument shows the inclusion $$U(k-n+1)\rightarrow U(k-n+2)$$ is even more connected. So the inclusion map $$U(k-n)\rightarrow U(k-n+2)$$ is $2k-2n$ connected. Continuing, by induction, we see the inclusion $$U(k-n)\rightarrow U(k)$$ is also $2k-2n$ connected, so the quotient $U(k)/U(k-n)$ is at least $2k-2n$ connected as claimed.

For you second question, try applying Whitehead's theorem to the inclusion of a point into $V_n(\mathbb{C}^\infty)$.