PlanetMath: deformation retract

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deformation retract

(Definition)

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

Let and 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 onto ) if and only if it is a homotopy rel Y between $\operatorname{id}_X$ and some retraction of onto .

If $x_0\in \mathbb{R}^n$ , then , $x\in \mathbb{R}^n$ 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.
Setting $f_t(x,t)=(1-t)x+t \frac{x}{\vert\vert x\vert\vert}$ , $x\in \mathbb{R}^n\backslash \{0\}$ , , we obtain a deformation retraction of $\mathbb{R}^n\backslash \{0\}$ onto the -sphere $S^{n-1}$ .
The Möbius strip deformation retracts onto the circle .
The -torus with one point removed deformation retracts onto two copies of 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.

"deformation retract" is owned by mathcam. [ full author list (2) | owner history (1) ]

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