# Proof of the "Whitehead theorem for homology"

In the Wikipedia article about the Whitehead theorem, it is said that

Combining [the Whitehead theorem for homotopy] with the Hurewicz theorem yields a useful corollary: a continuous map $$f\colon X\to Y$$ between simply connected CW complexes that induces an isomorphism on all integral homology groups is a homotopy equivalence.

However, I have trouble understanding how to arrive at this corollary: The Hurewicz theorem only makes a statement about $$(n-1)$$-connected CW complexes, i.e. complexes $$X$$ with $$\pi_i(X)=0$$ for all $$i\leq n-1$$.

• You need the more general version of the Hurewicz theorem that applies to pairs. That version is sometimes called the "relative Hurewicz theorem", whereas the one you're mentioning is called the "absolute Hurewicz theorem". Dec 30, 2023 at 22:49
• @Thorgott I have seen that version, but how is it used? Dec 30, 2023 at 22:53
• Dec 30, 2023 at 23:31

As noted in Weak Homotopy Equivalence Induces Isomorphisms on Homology, we can replace $$Y$$ by the mapping cylinder of $$f$$; the effect is that we can assume that $$f$$ is an inclusion. Then we have long exact sequences for the pair $$(Y, X)$$ in homotopy and homology. If $$f$$ induces an isomorphism on homology in all dimensions, then $$H_n(Y,X) =0$$ for all $$n$$. Combined with simple connectivity and the relative Hurewicz theorem, we see that $$\pi_n(Y,X)=0$$ for all $$n$$, which means that $$f$$ induces an isomorphism on homotopy groups, so it is a homotopy equivalence.