Hi,

this question is related to my question <a href="http://mathoverflow.net/questions/67551/weak-homotopy-equivalence-of-h-spaces">here</a>. Suppose, I have a topological group $G$ and an $H$-space $H$, which is a CW-complex. Furthermore, I have a map $\varphi \colon G \to H$, that induces an isomorphism $[X, G] \to [X,H]$ 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 $H$-spaces nowadays? Or for Segals $\Gamma$-spaces?