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This question, Is there a weak homotopy equivalence between $\mathrm{Sp}(2n,\mathbb{C})/\mathrm{U}(n)$ and $\mathrm{SU}(n)$?, is at the end of a long string of my comments in

Is Sp(2N,$\mathbb{C}$)/U(N) isomorphic to SU(N)?

And perhaps the discussion there is useful for this question. I ask this because I don't know how to show this weak homotopy equivalence. I will continue to try to answer this question on my own while I wait for responses but also thought I would just ask with different, more relevant tags.

Arturo Magidin
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Impossibear
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1 Answers1

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No. $Sp(2N,\mathbb{C})$ 2-connected (you can prove this by induction on $N$; see for instance my answer at $\mathrm{Sp}(4, \mathbb{C})$ is simply connected which also works to show it is 2-connected). So by the long exact sequence in homotopy groups, $\pi_2(Sp(2N,\mathbb{N})/U(N))\cong \pi_1(U(N))\cong\mathbb{Z}$. Since $\pi_2(SU(N))$ is trivial (which can be proved by induction on $N$ in a similar way) this means they are not weak homotopy equivalent.

Eric Wofsey
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  • Thank you for your response. I think I have a new question, which is, is there an isomorphism between homotopy group $\pi_m(A)$ and homotopy group $\pi_m(B)$, where $A$ is some subgroup or subset of (2,ℂ)/() and B is some subgroup of $SU(n)$. – Impossibear Jul 19 '19 at 05:57
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    @Impossibear This new question is way too vague. No quantifier on $m$. And yes, take a singleton $A$ and $B$ trivial subgroup. – YCor Jul 19 '19 at 07:58
  • @Ycor, $m=2$ or $3$, say. And what about nontrivial subgroups, like $S^2$? Thank you for your comments. – Impossibear Jul 19 '19 at 22:55
  • @Impossibear anyway this is not the right place for asking follow-up questions. – YCor Jul 19 '19 at 22:58
  • @Ycor After checking a meta mathexchange post, it seems I should ask a new question and link to this one in this case. I will proceed with that in a few hours unless I hear otherwise. – Impossibear Jul 19 '19 at 23:08
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    Yes. I suggest to ask a precise question, and think about it a little before asking to avoid a trivial answer. – YCor Jul 19 '19 at 23:10