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Let $X$ and $Y$ topological spaces such that $X \subset Y$. Let $H_n(X)$ and $H_n(Y)$ the homology groups of X and Y and suppose $H_n(X) \simeq H_n(Y)$.
Is it true that the inclusion $i:X \to Y$ induces a group isomorphism between $H_n(X)$ and $H_n(Y)$?

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  • $\begingroup$ What have you tried? $\endgroup$ Nov 28, 2022 at 17:55

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No. Consider $Y=S^1\vee D^2$, where $S^1$ is the $1$-dimensional sphere, while $D^2$ the $2$-dimensional disk. Let $X=\partial D^2$ which is also $S^1$, but the one inside $D^2$. These are homotopy equivalent.

However the inclusion induces trivial map, since it factors through contractible space.

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