Hi,


this question is related to my question [here](https://mathoverflow.net/questions/67551/weak-homotopy-equivalence-of-h-spaces). 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?



Delooping maps between H-spaces 
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)").


