Let $f : X \to Y$ be a map between a connected space $X$ and a space $Y$. If $\pi(f) : \pi\_1(X) \to \pi\_1(Y)$ is an isomorphism, and $H\_n(f) : H\_n(X, G) \to H\_n(Y, G)$ is an isomorphism for all $n \ge 1$ and for any local system of coefficients $G$, then $X$ is weakly equivalent to $Y$. Does anyone have a reference (or proof) for this?



Reference needed: Isomorphism on pi_1 and homology gives weak equivalence
You need to assume either that the spaces involved are simple (I believe Emmanuel Dror-Farjoun generalized that to nilpotent), or that the map f induces an isomorphism in homology with local coefficients. There is an exercise in Hatcher's book that discuss this, in Section 4.2 (Ex. 12). You should also look at Peter May's beautiful paper The Dual Whitehead Theorems, in Peter Hilton's birthday conference proceedings.


