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Hatcher's ALgebraic Topology contains the following corollary.

A map $f: X \to Y$ between simply-connected $CW$ complexes is a homotopy equivalence if $f_* : H_n(X) \to H_n(Y)$ is an isomorphism for each $n$>

Proof: After replacing $Y$ with the mapping cylinder $M_f$ we take $f$ to be a inclusion $f: X \hookrightarrow Y$. Since $X$ and $Y$ are simply-connected we have that $\pi_1(X,Y)=0$. This relative Hurewicz theorem then says that the first non-trivial homotopy group is isomorphic to the first non-trivial homology group.

He then goes on to claim that $H_n(Y,X)$ is trivial for all $n$. Why is this true??

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The map $H_n(X) \to H_n(Y)$ in the relative long exact sequence is an isomorphism. So by exactness $H_n(Y,X)$ must vanish.

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