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 rel Y between $\operatorname{id}_X$ and some retraction $r$ of $X$ onto $Y$ .

Examples

• If $x_0\in \mathbb{R}^n$ , then $f_t(x,t)=(1-t)x+tx_0$ , $x\in \mathbb{R}^n$ 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 $(n-1)$ -sphere $S^{n-1}$ by setting $$f_t(x,t)=(1-t)x+t \displaystyle{\frac{x}{||x||}},$$ where $x\in \mathbb{R}^n\backslash \{0\}$ , $n>0$ ,
• The Möbius 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.