# Can $H_n(A) \cong H_n(X)$, where $(X,A)$ is a simply-connected $CW$ pair, always be induced by the inclusion map?

Let $(X,A)$ be a simply connected $CW$ pair such that $H_n(A)\cong H_n(X)$ for some $n$. I wonder if the isomorphism can be induced by inclusion $i:A\hookrightarrow X$ in this case.

Remark: Note that if this is always true, then by Hatcher's corollary 4.33, i:$A \hookrightarrow X$ is a homotopy equivalence, and thus by his corollary 0.20, $A$ is a deformation retract of $X$. This will generalize whitehead's theorem. So I feel like it isn't true. But I can't come up with an counterexample.

No. For instance, let $X=D^3\vee S^2$ and let $A=\partial D^3\subset D^3\subset X$. Then $A$ and $X$ have isomorphic homology: $A$ is homeomorphic to $S^2$, and $X$ is homotopy equivalent to $S^2$. But the inclusion $A\to X$ is nullhomotopic, and in particular induces the $0$ map on $H_2$.