Persistence length

2014-01-23T12:15:58Z

5.147.205.155: removed double use and left the specific one for semantical clarity

In [[mathematics]], in the area of [[algebraic topology]], the '''homotopy extension property''' indicates which [[homotopy|homotopies]] defined on a [[subspace topology|subspace]] can be extended to a homotopy defined on a larger space.

==Definition==

Let <math>X\,\!</math> be a [[topological space]], and let <math>A \subset X</math>.
We say that the pair <math>(X,A)\,\!</math> has the '''homotopy extension property''' if, given a homotopy <math>f_t\colon A \rightarrow Y</math> and a map <math>F_0\colon X \rightarrow Y</math> such that <math>F_0 |_A = f_0</math>, there exists an ''extension'' of <math>F_0</math> to a homotopy <math>F_t\colon X \rightarrow Y</math> such that
<math>F_t|_A = f_t</math>. <ref>A. Dold, ''Lectures on Algebraic Topology'', pp. 84, Springer ISBN 3-540-58660-1</ref>

That is, the pair <math>(X,A)\,\!</math> has the homotopy extension property if any map
<math>G\colon (X\times \{0\} \cup A\times I) \rightarrow Y</math>
can be extended to a map <math>G'\colon X\times I \rightarrow Y</math> (i.e. <math>G\,\!</math> and <math>G'\,\!</math> agree on their common domain).

If the pair has this property only for a certain [[codomain]] <math>Y\,\!</math>, we say that <math>(X,A)\,\!</math> has the homotopy extension property with respect to <math>Y\,\!</math>.

==Visualisation==
The homotopy extension property is depicted in the following diagram

[[Image:Homotopy_extension_property.svg|175px|center]]

If the above diagram (without the dashed map) commutes, which is equivalent to the conditions above, then there exists a map <math> \tilde{f}</math> which makes the diagram commute. By [[currying]], note that a map <math> \tilde{f} \colon X \to Y^I</math> is the same as a map <math> \tilde{f} \colon X\times I \to Y </math>.

Also compare this to the visualization of the [[Homotopy_lifting_property#Formal_definition|homotopy lifting property]].

==Properties==
* If <math>X\,\!</math> is a [[cell complex]] and <math>A\,\!</math> is a subcomplex of <math>X\,\!</math>, then the pair <math>(X,A)\,\!</math> has the homotopy extension property.

* A pair <math>(X,A)\,\!</math> has the homotopy extension property if and only if <math>(X\times \{0\} \cup A\times I)</math> is a [[Deformation retract|retract]] of <math>X\times I.</math>

==Other==

If <math>\mathbf{\mathit{(X,A)}}</math> has the homotopy extension property, then the simple inclusion map <math>i: A \to X</math> is a [[cofibration]].

In fact, if you consider any [[cofibration]] <math>i: Y \to Z</math>, then we have that <math>\mathbf{\mathit{Y}}</math> is [[homeomorphic]] to its image under <math>\mathbf{\mathit{i}}</math>. This implies that any cofibration can be treated as an inclusion map, and therefore it can be treated as having the homotopy extension property.

==See also==
* [[Homotopy lifting property]]

==References==
{{Reflist}}

*{{cite book | first = Allen | last = Hatcher | authorlink = Allen Hatcher | year = 2002 | title = Algebraic Topology | publisher = Cambridge University Press | isbn = 0-521-79540-0 | url = http://www.math.cornell.edu/~hatcher/AT/ATpage.html}}

* {{planetmath reference|id=1600|title=Homotopy extension property}}

[[Category:Homotopy theory]]
[[Category:Algebraic topology]]

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