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Let $X\subset Y$ be CW-complexes. Denote $i\colon X\to Y$ be an inclusion map. Is it true that $i$ is deformation retract if and only if $i$ is homotopy equivalence? When I saw some papers about h-cobordism theorem, authors usually checks inclusion map is a homotopy equivalence to prove that inclusion map is deformation retract. |
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If $X$ is a subcomplex of $Y$ and the inclusion map is a homotopy equivalence, then $X$ is a deformation retract of $Y$. See for example proposition 0.16 and corollary 0.20 in Hatcher. |
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