Homotopy extension property

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.

Definition

Let $X\,\!$ be a topological space, and let $A \subset X$. Given a homotopy $f_t\colon A \rightarrow Y$ and a map $\tilde{f}_0\colon X \rightarrow Y$ such that $\tilde{f}_0 |_A = f_0$, we say that the pair $(X,A)\,\!$ has the homotopy extension property if there exists an extension of $\tilde{f}_0$ to the homotopy $\tilde{f}_t\colon X \rightarrow Y$ such that $\tilde{f}_t|_A = f_t$.

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 $F'\colon X\times I \rightarrow Y$ (i.e. $F\,\!$ and $F'\,\!$ agree on their common domain).

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 $Y\,\!$.

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 makes the diagram commute. By currying, note that a map $\tilde{f} \colon X \to Y^I$ is the same as a map $\tilde{f} \colon X\times I \to Y$.

Also compare this to the visualization of the homotopy lifting property.

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 $X\times I.$

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.

See also

• Homotopy lifting property

References

• Hatcher, Allen (2002). Algebraic Topology. Cambridge University Press. ISBN 0-521-79540-0. http://www.math.cornell.edu/~hatcher/AT/ATpage.html. 

• Homotopy extension property on PlanetMath