# About deformation retract

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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Yes. This is part of Whitehead's theorem. See p. 346 of Hatcher's book, available here: math.cornell.edu/~hatcher/AT/ATpage.html – Theo Buehler Feb 4 '11 at 11:10

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.