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.



About deformation retract
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](http://www.math.cornell.edu/~hatcher/AT/ATpage.html).


