deformation retract

Let $X$ and $Y$ be topological spaces such that $Y\subset X$.A deformation retract of $X$ onto $Y$ is a collection ofmappings $f_t:X\to X$, $t\in [0,1]$ such that

1. $f_0=i{d}_X$, the identity mapping on $X$,

2. $f_1(X)\subseteq Y$,

3. $Y$ is a retract of $X$ via $f_1$ (that is, $f_1$ restricted to $Y$ is the identity on $Y$)

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

Examples

• If $x_0\in {ℝ}^n$, then$f_t(x)=(1-t)x+t{x}_0$, $x\in {ℝ}^n$ shows that ${ℝ}^n$ deformation retracts onto $\{x_0\}$.Since $\{x_0\}\subset {ℝ}^n$,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 ${ℝ}^n$ onto a point.

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

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

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

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

• The $2$-torus with one point removed deformation retracts ontotwo copies of $S^1$ joined at one point. (The circles can bechosen 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.