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.

$f_{0}=id_{X}$ , the identity mapping on $X$ ,
2.

$f_{1}(X)\subseteq Y$ ,
3.

$Y$ is a retract of $X$ via $f_{1}$ (that is, $f_{1}$ restricted to $Y$ is the identity on $Y$ )
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
Canonical name	DeformationRetract
Date of creation	2013-03-22 13:31:44
Last modified on	2013-03-22 13:31:44
Owner	mathcam (2727)
Last modified by	mathcam (2727)
Numerical id	14
Author	mathcam (2727)
Entry type	Definition
Classification	msc 55Q05
Related topic	Retract
Defines	strong deformation retract