If I have a compact connected Lie group $G$ and a (relatively nice) simply-connected topological abelian group $A$, when is it the case that a given $f\colon BG \to BA$ loops to a (continuous) homomorphism $G\to A$? This seems like it can't always be true, and also should be some kind of classical result, but I'm not sure where to look.
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asked Jun 21, 2019 at 6:06
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2$\begingroup$ Should that be "when is $BG\to BA$ the delooping of a homomorphism"? $\endgroup$– Mark GrantCommented Jun 21, 2019 at 6:37
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1$\begingroup$ A more usual way of posing the question is: when does an $E_1$-homomorphism come from a map of topological groups? I don't know the answer though, but it is not always $\endgroup$– Denis NardinCommented Jun 21, 2019 at 6:42
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$\begingroup$ @DenisNardin yes, that's another way I was thinking of it. $\endgroup$– David Roberts ♦Commented Jun 21, 2019 at 6:50
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3$\begingroup$ Unfortunately I think that this question is quite sensitive to $A$ and not just to $BA$. For example, $A$ does not have any homomorphisms from $G$ if it does not have any homomorphisms from $S^1$, which requires that it have elements of finite order $(g^n = 1)$; however, any classifying space $BA$ is equivalent to a classifying space $BA'$ where $A'$ has no finite-order elements. $\endgroup$– Tyler LawsonCommented Jun 21, 2019 at 8:00
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1$\begingroup$ I'll point to mathoverflow.net/questions/156408/…, which addresses a related question: what criteria imply that $\mathrm{Hom}(G,H)\to \mathrm{Map}_*(BG,BH)$ is a weak homotpy equivalence? The answer there says nothing about your case I think, but perhaps the methods used could be helpful. I don't know. $\endgroup$– Charles RezkCommented Jul 5, 2019 at 19:38
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1 Answer
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If $G$ is a compact connected topological group and $A$ is a locally compact abelian topological group, then for any map $f:BG\to BA$ the looped map $\Omega f:\Omega BG\to \Omega BA$ is homotopically equivalent to a homomorphism $\phi:G\to A$. This follows from the main result of
Scheffer, Wladimiro, Maps between topological groups that are homotopic to homomorphisms, Proc. Am. Math. Soc. 33, 562-567 (1972). ZBL0236.22008.
I'm not sure if this answers your question, however, which seems to be about when $f$ is in the image of the classifying space functor $B:\mathsf{TopGrp}\to\mathsf{Top}$.
answered Jun 21, 2019 at 7:49
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$\begingroup$ Thanks, you are indeed answering the intended question, but unfortunately I can't assume $A$ is locally compact! It is in my example the geometric realisation of a simplicial abelian Lie group.... But this is a good answer in any case. $\endgroup$– David Roberts ♦Commented Jun 21, 2019 at 7:51
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1$\begingroup$ Reformulation of the question: is $\hom(G,A) \to \hom_{A_\infty}(G,A)$ surjective on $\pi_0$? (The target is $A_\infty$-homomorphisms $G \to A$). One might be able to use Boardman-Vogt obstruction theory to study this, by replacing $G$ with a cofibrant $A_\infty$-space. $\endgroup$ Commented Jun 21, 2019 at 12:28
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$\begingroup$ I will accept this in the absence of any other suggestions, because it shows how subtle the question is, and how much it will depend on the topology of $A$. $\endgroup$– David Roberts ♦Commented Jul 22, 2019 at 12:07