TITLE: Reference needed: Isomorphism on pi_1 and homology gives weak equivalence
QUESTION [8 upvotes]: 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?

REPLY [8 votes]: The proof in the simply connected case is well-known and is a consequence of the relative Hurewicz theorem. See Corollary 1, page 79 of Mosher and Tangora's book, Cohomology Operations and Applications in Homotopy Theory for this case.
If $X$ and $Y$ aren't 1-connected, then $f$ lifts to a map of universal covers
$\tilde f: \tilde X \to \tilde Y$ and your assumption about local coefficients implies that $\tilde f$ is a homology isomorphism. We can therefore apply the previous paragraph to show that $\tilde f$ is a weak equivalence. This implies that $f$ is since $f$ is a $\pi_1$-isomorphism.