# Homotopy extension property

In mathematics, in the area of algebraic topology, the **homotopy extension property** indicates which homotopies defined on a subspace can be extended to a homotopy defined on a larger space. The homotopy extension property of cofibrations is dual to the homotopy lifting property that is used to define fibrations.

## Definition

Let be a topological space, and let .
We say that the pair has the **homotopy extension property** if, given a homotopy and a map such that , there exists an *extension* of to a homotopy such that
.^{[1]}

That is, the pair has the homotopy extension property if any map can be extended to a map (i.e. and agree on their common domain).

If the pair has this property only for a certain codomain , we say that has the homotopy extension property with respect to .

## Visualisation

The homotopy extension property is depicted in the following diagram

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 which makes the diagram commute. By currying, note that a map is the same as a map .

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.

## Properties

- If is a cell complex and is a subcomplex of , then the pair has the homotopy extension property.
- A pair has the homotopy extension property if and only if is a retract of

## Other

If has the homotopy extension property, then the simple inclusion map is a cofibration.

In fact, if you consider any cofibration , then we have that is homeomorphic to its image under . 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

## References

- ↑ A. Dold,
*Lectures on Algebraic Topology*, pp. 84, Springer ISBN 3-540-58660-1

- Hatcher, Allen (2002).
*Algebraic Topology*. Cambridge University Press. ISBN 0-521-79540-0. - "Homotopy extension property".
*PlanetMath*.