# 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.

## 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, which is equivalent to the conditions above, then there exists a map which makes the diagram commute. By currying, note that a map is the same as a map .

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

## 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

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