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 ${f}_{t}:X\to X$, $t\in [0,1]$ such that

1.
${f}_{0}=i{d}_{X}$, the identity mapping on $X$,

2.
${f}_{1}(X)\subseteq Y$,
 3.

4.
the mapping $X\times I\to X$, $(x,t)\mapsto {f}_{t}(x)$ 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 ${f}_{t}:X\to X$, $t\in [0,1]$ is a deformation retract (of $X$ onto $Y$) if and only if it is a homotopy^{} (http://planetmath.org/HomotopyOfMaps) rel Y ${\mathrm{id}}_{X}$ and some retraction $r$ of $X$ onto $Y$.
Examples

•
If ${x}_{0}\in {\mathbb{R}}^{n}$, then ${f}_{t}(x)=(1t)x+t{x}_{0}$, $x\in {\mathbb{R}}^{n}$ 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 starshaped 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$\mathrm{(}\mathrm{n}\mathrm{}\mathrm{1}\mathrm{)}$sphere ${S}^{n1}$ by setting
$${f}_{t}(x)=(1t)x+t\frac{x}{x},$$ 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  20130322 13:31:44 
Last modified on  20130322 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 