Hi,

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?

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