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Difference between revisions of "Deformation retract"

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''of a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030730/d0307301.png" />''
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''of a topological space $X$''
  
A subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030730/d0307302.png" /> with the following property: There exists a homotopy of the identity mapping of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030730/d0307303.png" /> to some mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030730/d0307304.png" /> under which all points of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030730/d0307305.png" /> remain fixed. If, under the homotopy, the points from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030730/d0307306.png" /> remain in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030730/d0307307.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030730/d0307308.png" /> is known as a strong deformation retract. A deformation retract of a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030730/d0307309.png" /> has the same [[Homotopy type|homotopy type]] as does <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030730/d03073010.png" />. See also [[Retract|Retract]]; [[Retract of a topological space|Retract of a topological space]].
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A subset $A\subset X$ with the following property: There exists a homotopy of the identity mapping of $X$ to some mapping $X\to A$ under which all points of the set $A$ remain fixed. If, under the homotopy, the points from $X\setminus A$ remain in $X\setminus A$, $A$ is known as a strong deformation retract. A deformation retract of a space $X$ has the same [[Homotopy type|homotopy type]] as does $X$. See also [[Retract|Retract]]; [[Retract of a topological space|Retract of a topological space]].

Revision as of 15:52, 10 July 2014

of a topological space $X$

A subset $A\subset X$ with the following property: There exists a homotopy of the identity mapping of $X$ to some mapping $X\to A$ under which all points of the set $A$ remain fixed. If, under the homotopy, the points from $X\setminus A$ remain in $X\setminus A$, $A$ is known as a strong deformation retract. A deformation retract of a space $X$ has the same homotopy type as does $X$. See also Retract; Retract of a topological space.

How to Cite This Entry:
Deformation retract. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Deformation_retract&oldid=32407
This article was adapted from an original article by E.G. Sklyarenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article