I have read (Theorem 29.3, line 7, pg 144) the following statement:

$\pi_i(U(n-k)) \rightarrow \pi_i(U(n))$ is surjective for $i \le 2(n-k)+1$ and injective for $i \le 2(n-k)$.


Therefore $\pi_i(St_k(U(n))=0$ for $i \le 2(n-k)$.

What results are used here and where can I find a reference of this?

  • 1
    $\begingroup$ This follows from my answer to your question math.stackexchange.com/questions/3095144/…, since $Gl(\mathbb{C}^n)$ deformation retracts onto $U_n$ by polar decomposition of matrices. $\endgroup$ – Tyrone Feb 21 at 9:42
  • $\begingroup$ Oh my... I have completely forgotten to look back at the question! Thank you so much. $\endgroup$ – CL. Feb 21 at 10:38

I think they have made an off-by-$1$ error, they might have been thinking about the map $BU(n-k) \to BU(n)$. (The notes you're reading are unpublished and there is a disclaimer at the start about not being proofread, so an off-by-$1$ error isn't so bad.)

A great reference for a lot of results like this is Mimura and Toda's "Topology of Lie Groups " I and II. They compute a lot of homotopy and homology groups for classical Lie groups and their classifying spaces. For this particular example, look at Corollary 3.17 on page 68. The statement includes:

... For $i < 2n$, $\pi_i U(n) \cong \pi_i U(n+1)$, ...

Changing variables we see $\pi_iU(n-1)\cong \pi_i U(n)$ for $i < 2(n-1)$, and their argument actually also shows surjectivity in degree $2(n-1)$.

Typically you use the long exact sequence of the fibration

$$ U(n - 1) \to U(n) \to S^{2n - 1}$$

to compute the connectivity of $U(n - 1)\to U(n)$. The degree $i = 2n - 2$ is the last degree where $\pi_i S^{2n-1} = 0$ so our map induces surjective homomorphisms in all degrees $i \leq 2(n-1)$, and moreover they are injective if $i < 2(n-1)$. The (correct) connectivity of your map, and hence the Stiefel manifold, follows.


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