Homotopy extension property

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In mathematics, in the area of algebraic topology, the homotopy extension property indicates when a homotopy defined on a subspace can be extended to a homotopy defined on a larger space.

[edit] Definition

Let $X\,\!$ be a topological space, and let $A \subset X$ . We say that the pair $(X,A)\,\!$ has the homotopy extension property if, given any map $f_0\colon X \rightarrow Y$ and any homotopy $f_t\colon A \rightarrow Y$ , there exists an extension $f_t\colon X \rightarrow Y$ that agrees with $f_0.\,\!$ That is, the pair $(X,A)\,\!$ has the homotopy extension property if any map

$f\colon (X\times \{0\} \cup A\times I) \rightarrow Y$

can be extended to a map $f\colon X\times I \rightarrow Y.$

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

[edit] Visualisation

The homotopy extension property is depicted in the following diagram

if the above diagram (without the dashed map) commutes, which is equivalent to the conditions above, then there exists a map $\tilde{f}$ which fits in and makes the diagram commute. By currying, note that a map $\tilde{f} \colon X \to Y^I$ is the same as a map $\tilde{f} \colon X\times I \to Y$ .

[edit] Properties

If $X\,\!$ is a cell complex and $A\,\!$ is a subcomplex of $X\,\!$ , then the pair $(X,A)\,\!$ has the homotopy extension property.

A pair $(X,A)\,\!$ has the homotopy extension property if and only if $(X\times \{0\} \cup A\times I)$ is a retract of $X\times I.$

[edit] Other

If $\mathbf{\mathit{(X,A)}}$ has the homotopy extension property, then the simple inclusion map $i: A \to X$ is a cofibration.

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