# deformation retract

Let $X$ and $Y$ be topological spaces such that $Y\subset X$. A deformation retract of $X$ onto $Y$ is a collection of mappings $f_{t}:X\rightarrow X$, $t\in[0,1]$ such that

1. 1.

$f_{0}=id_{X}$, the identity mapping on $X$,

2. 2.

$f_{1}(X)\subseteq Y$,

3. 3.

$Y$ is a retract of $X$ via $f_{1}$ (that is, $f_{1}$ restricted to $Y$ is the identity on $Y$)

4. 4.

the mapping $X\times I\rightarrow X$, $(x,t)\mapsto f_{t}(x)$ is continuous, where the topology on $X\times I$ is the product topology.

Of course, by condition 3, condition 2 can be improved: $f_{1}(X)=Y$.

A deformation retract is called a strong deformation retract if condition 3 above is replaced by a stronger form: $Y$ is a retract of $X$ via $f_{t}$ for every $t\in[0,1]$.

## Properties

• Let $X$ and $Y$ be as in the above definition. Then a collection of mappings $f_{t}:X\rightarrow X$, $t\in[0,1]$ is a deformation retract (of $X$ onto $Y$) if and only if it is a homotopy (http://planetmath.org/HomotopyOfMaps) rel Y $\operatorname{id}_{X}$ and some retraction $r$ of $X$ onto $Y$.

## Examples

• If $x_{0}\in\mathbb{R}^{n}$, then $f_{t}(x)=(1-t)x+tx_{0}$, $x\in\mathbb{R}^{n}$ shows that $\mathbb{R}^{n}$ deformation retracts onto $\{x_{0}\}$. Since $\{x_{0}\}\subset\mathbb{R}^{n}$, it follows that deformation retract is not an equivalence relation.

• The same map as in the previous example can be used to deformation retract any star-shaped set in $\mathbb{R}^{n}$ onto a point.

• we obtain a deformation retraction of $\mathbb{R}^{n}\backslash\{0\}$ onto the http://planetmath.org/node/186$(n-1)$-sphere $S^{n-1}$ by setting

 $f_{t}(x)=(1-t)x+t\displaystyle{\frac{x}{||x||}},$

where $x\in\mathbb{R}^{n}\backslash\{0\}$, $n>0$,

• The http://planetmath.org/node/3278Möbius strip deformation retracts onto the circle $S^{1}$.

• The $2$-torus with one point removed deformation retracts onto two copies of $S^{1}$ joined at one point. (The circles can be chosen to be longitudinal and latitudinal circles of the torus.)

• The characters E,F,H,K,L,M,N, and T all deformation retract onto the charachter I, while the letter Q deformation retracts onto the letter O.

Title deformation retract DeformationRetract 2013-03-22 13:31:44 2013-03-22 13:31:44 mathcam (2727) mathcam (2727) 14 mathcam (2727) Definition msc 55Q05 Retract strong deformation retract