I am currently reading a paper where the author needs to show that a map $i: X \hookrightarrow Y$ between 2 $CW$- complexes $X$,$Y$ (and $X$ is contained in $Y$) is a deformation retraction.

He claims it suffice to show that for every $\psi : D^n \hookrightarrow Y$ such that $\psi(\partial D) \subset X$, we can produce a deformation retraction of $\psi(D)$ to a disk contained in $X$ while fixing $\psi(\partial D)$.

Why is this equivalent to showing that there is a deformation retraction from $Y$ to $X$ ?

  • $\begingroup$ Is that all the paper assumes about the map $i : X \hookrightarrow Y$? There are no further conditions on $i$? For example, I presume $i$ should be continuous, but there need to be more conditions as well or else this statement is false. $\endgroup$ – Lee Mosher Jan 30 at 12:44
  • $\begingroup$ $i$ is the inclusion map which is continuous. But other than that there are no other additional conditions. $\endgroup$ – cr1t1cal Jan 30 at 14:23
  • $\begingroup$ Also $\psi$ is continuous as well. These are the maps that give Y it’s CW complex structure. $\endgroup$ – cr1t1cal Jan 30 at 14:32
  • $\begingroup$ Do you mean deformation retraction or strong deformation retraction? $\endgroup$ – Paul Frost Jan 30 at 17:33

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