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