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deformation retract (Definition)

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) = Y$,
  3. $ f_t\,\vert _{Y} = \operatorname{id}_Y$ for all $ t$,
  4. the mapping $ X\times I\rightarrow X$, $ (x,t)\mapsto f_t(x)$ is continuous.

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.
  • Setting $ f_t(x,t)=(1-t)x+t \frac{x}{\vert\vert x\vert\vert}$, $ x\in \mathbb{R}^n\backslash \{0\}$, $ n>0$, we obtain a deformation retraction of $ \mathbb{R}^n\backslash \{0\}$ onto the $ (n-1)$-sphere $ S^{n-1}$.
  • 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.



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See Also: retract


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deformation retract is transitive (Result) by mps
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Cross-references: characters, torus, circle, deformation, point, star-shaped, equivalence relation, retraction, continuous, identity mapping, mappings, collection, onto, topological spaces
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This is version 8 of deformation retract, born on 2003-03-23, modified 2006-07-07.
Object id is 4121, canonical name is DeformationRetraction.
Accessed 4261 times total.

Classification:
AMS MSC55Q05 (Algebraic topology :: Homotopy groups :: Homotopy groups, general; sets of homotopy classes)

Pending Errata and Addenda
1. wrong definition by joking on 2008-11-04 13:05:52
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