# References for injectivity/surjectivity $\pi_i(U(n-k)) \rightarrow \pi_i(U(n))$

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 $$i \le 2(n-k)$$.

and

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?

• 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. – Tyrone Feb 21 at 9:42
• Oh my... I have completely forgotten to look back at the question! Thank you so much. – 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 (The notes you're reading are unpublished and there is a disclaimer at the start about not being proofread, so an off-by-$$1$$ error isn't so bad.)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.