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this question is related to my question here. Suppose, I have a topological group $G$ and an $A_{\infty}$-space $H$, which is a CW-complex. Furthermore, I have a map $\varphi \colon G \to H$, that induces an isomorphism of groups $[X, G] \to [X,H]$ for finite CW-complexes $X$.

Is this enough to deloop $\varphi$, i.e. does there exist a map $B\varphi > \colon BG \to BH$. Or can I at least deduce a weak equivalence between $BG$ and $BH$ from this?

btw.: What is the "standard" reference for $A_{\infty}$-spaces and $H$-spaces nowadays? Or for Segals $\Gamma$-spaces?

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$H$-spaces may not have classifying spaces. – Torsten Ekedahl Jul 26 '11 at 8:44
I think you should emphasize that the isomorphism $[X, G] \to [X, H]$ is an isomorphism of groups, not just sets. This then yields the condition that $\pi_0(H)$ is a group and that $H$ has a classifying space. – S. Carnahan Jul 26 '11 at 8:58
@Scott: Thank you. I corrected this. @Torsten: Hmm, in the case I have in mind, H is actually a Gamma-space, so it should have a classifying space. What are the conditions for an H-space to have a classifying space? – Ulrich Pennig Jul 26 '11 at 9:10
It should have a given $A_\infty$-structure. Note that the actual choice of structure affects what the classifying space would be so you must specify it. A $\Gamma$-structure is stronger but again the choice of $\Gamma$-structure will affect what classifying space you are talking about. – Torsten Ekedahl Jul 26 '11 at 9:46
changed $H$-space to $A_{\infty}$-space according to Torsten's comment – Ulrich Pennig Jul 26 '11 at 12:42
up vote 9 down vote accepted

No, it is not: $S^3$ admits uncountably many loop space structures (c.f. Rector "Loop structures on the homotopy type of $S^3$"), but only $12 (= \vert \pi_6(S^3) \vert)$ H-space structures (c.f. James "Multiplication on spheres (II)").

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