Homotopy extension property

Let $X\backslash ,\backslash !$ be a topological space, and let $A\; \backslash subset\; X$. We say that the pair $(X,A)\backslash ,\backslash !$ has the homotopy extension property if, given a homotopy $f\_\backslash bullet\backslash colon\; A\; \backslash rightarrow\; Y^I$ and a map $\backslash tilde\{f\}\_0\backslash colon\; X\; \backslash rightarrow\; Y$ such that $$\backslash tilde\{f\}\_0\backslash circ\; \backslash iota\; =\; \backslash left.\backslash tilde\{f\}\_0\backslash right|\_A\; =\; f\_0\; =\; \backslash pi\_0\; \backslash circ\; f\_\backslash bullet,$$ then there exists an extension of $f\_\backslash bullet$ to a homotopy $\backslash tilde\{f\}\_\backslash bullet\backslash colon\; X\; \backslash rightarrow\; Y^I$ such that $\backslash tilde\{f\}\_\backslash bullet\backslash circ\; \backslash iota\; =\; \backslash left.\backslash tilde\{f\}\_\backslash bullet\backslash right|\_A\; =\; f\_\backslash bullet$.A. Dold, Lectures on Algebraic Topology, pp. 84, Springer {{ISBN|3-540-58660-1}}

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

If the pair has this property only for a certain codomain $Y\backslash ,\backslash !$, we say that $(X,A)\backslash ,\backslash !$ has the homotopy extension property with respect to $Y\backslash ,\backslash !$.

The homotopy extension property is depicted in the following diagram

File:Homotopy extension property rotated.svg

If the above diagram (without the dashed map) commutes (this is equivalent to the conditions above), then pair (X,A) has the homotopy extension property if there exists a map $\backslash tilde\{f\}\_\backslash bullet$ which makes the diagram commute. By currying, note that homotopies expressed as maps $\backslash tilde\{f\}\_\backslash bullet\; \backslash colon\; X\; \backslash to\; Y^I$ are in natural bijection with expressions as maps $\backslash tilde\{f\}\_\backslash bullet\; \backslash colon\; X\backslash times\; I\; \backslash to\; Y$.

Note that this diagram is dual to (opposite to) that of the homotopy lifting property; this duality is loosely referred to as Eckmann–Hilton duality.

• Any CW pair $(X,A)\backslash ,\backslash !$ (that is, $X\backslash ,\backslash !$ is a cell complex and $A\backslash ,\backslash !$ is a subcomplex of $X\backslash ,\backslash !$) has the homotopy extension property.{{cite book | first = Allen | last = Hatcher | authorlink = Allen Hatcher | year = 2002 | title = Algebraic Topology | publisher = Cambridge University Press | isbn = 0-521-79540-0 | page=15 | url = http://pi.math.cornell.edu/~hatcher/AT/ATpage.html}} Proposition 0.16

• A pair $(X,A)\backslash ,\backslash !$ has the homotopy extension property if and only if $(X\backslash times\; \backslash \{0\backslash \}\; \backslash cup\; A\backslash times\; I)$ is a retract of $X\backslash times\; I.${{cite book | first = Allen | last = Hatcher | authorlink = Allen Hatcher | year = 2002 | title = Algebraic Topology | publisher = Cambridge University Press | isbn = 0-521-79540-0 | page=14 | url = http://pi.math.cornell.edu/~hatcher/AT/ATpage.html}}

If $(X,\; A)$ has the homotopy extension property, then the simple inclusion map $\backslash iota\backslash colon\; A\; \backslash to\; X$ is a cofibration.

In fact, if $\backslash iota\backslash colon\; Y\; \backslash to\; Z$ is a cofibration, then $\backslash mathbf\{\backslash mathit\{Y\}\}$ is homeomorphic to its image under $\backslash iota$. This implies that any cofibration can be treated as an inclusion map, and therefore it can be treated as having the homotopy extension property.

• Homotopy lifting property

{{Reflist}}

• {{planetmath reference|urlname=HomotopyExtensionProperty|title=Homotopy extension property}}

Category:Homotopy theory

Category:Algebraic topology