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'deformation retract'
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| Title of object: |
deformation retract |
| Canonical Name: |
DeformationRetraction |
| Type: |
Definition |
| Created on: |
2003-03-23 08:21:09 |
| Modified on: |
2004-07-08 17:17:53 |
| Classification: |
msc:55Q05 |
Revision comment (for changes between this and next version):
| Changes for correction #7419 ('letters'). |
Preamble:
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\usepackage{amssymb}
\usepackage{amsmath}
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%\usepackage{psfrag}
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Content:
Let $X$ and $Y$ be topological spaces such that $Y\subset X$.
A \emph{deformation retract} of $X$ onto $Y$ is a collection of
mappings $f_t:X\rightarrow X$, $t\in [0,1]$ such that
\begin{enumerate}
\item $f_0 = id_X$, the identity mapping on $X$,
\item $f_1(X) = Y$,
\item $f_t\,|_{Y} = \operatorname{id}_Y$ for all $t$,
\item the mapping $X\times I\rightarrow X$, $(x,t)\mapsto f_t(x)$ is
continuous.
\end{enumerate}
\subsubsection*{Properties}
\begin{itemize}
\item 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
\PMlinkname{homotopy}{HomotopyOfMaps} between $\operatorname{id}_X$ and some retraction $r$ of $X$ onto $Y$.
\end{itemize}
\subsubsection*{Examples}
\begin{itemize}
\item 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.
\item Setting $f_t(x,t)=(1-t)x+t \frac{x}{||x||}$, $x\in \mathbb{R}^n\backslash \{0\}$, $n>0$, we obtain a
deformation retraction of
$\mathbb{R}^n\backslash \{0\}$ onto the \PMlinkid{$(n-1)$-sphere }{186} $S^{n-1}$.
\item The \PMlinkid{M\"obius strip}{3278} deformation retracts onto the circle $S^1$.
\item 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.)
\end{itemize} |
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