# deformation retract is transitive

###### Proposition.

Let $Z\subset Y\subset X$ be nested topological spaces. If there exist a deformation retraction (http://planetmath.org/DeformationRetraction) of $X$ onto $Y$ and a deformation retraction of $Y$ onto $Z$, then there also exists a deformation retraction of $X$ onto $Z$. In other words, “being a deformation retract of” is a transitive relation.

###### Proof.

Since $Y$ is a deformation retract of $X$, there is a homotopy $F:I\times X\to X$ between $\operatorname{id}_{X}$ and a retract $r:X\to Y$ of $X$ onto $Y$. Similarly, there is a homotopy $G:I\times Y\to Y$ between $\operatorname{id}_{Y}$ and a retract $s:Y\to Z$ of $Y$ onto $Z$.

First notice that since both $r$ and $s$ fix $Z$, the map $sr:X\to Z$ is a retraction.

Now define a map $\widetilde{G}:I\times X\to X$ by $\widetilde{G}=iG(\operatorname{id}_{I}\times r)$, where $i:Y\hookrightarrow X$ is inclusion. Observe that

• $\widetilde{G}(0,x)=r(x)$ for any $x\in X$;

• $\widetilde{G}(1,x)=sr(x)$ for any $x\in X$; and

• $\widetilde{G}(t,a)=a$ for any $a\in Z$.

Hence $\widetilde{G}$ is a homotopy between the retractions $r$ and $sr$.

Finally we must glue together the homotopies (http://planetmath.org/GluingTogentherContinuousFunctions) $F$ and $\widetilde{G}$ to get a homotopy between $\operatorname{id}_{X}$ and $sr$. To do this, define a function $H:I\times X\to X$ by

 $H(t,x)=\begin{cases}F(2t,x),&0\leq t\leq\frac{1}{2}\\ \widetilde{G}(2t-1,x),&\frac{1}{2}\leq t\leq 1.\end{cases}$

Since $F(1,x)=\widetilde{G}(0,x)=r(x)$, the gluing yieds a continuous map. By construction,

• $H(0,x)=x$ for all $x\in X$;

• $H(1,x)=sr(x)$ for all $x\in X$; and

• $H(t,a)=a$ for any $a\in Z$.

Hence $H$ is a homotopy between the identity map on $X$ and a retraction of $X$ onto $Z$. We conclude that $H$ is a deformation retraction of $X$ onto $Z$. ∎

Title deformation retract is transitive DeformationRetractIsTransitive 2013-03-22 15:43:59 2013-03-22 15:43:59 mps (409) mps (409) 4 mps (409) Result msc 55Q05