Homotopy extension property

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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 . [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 .


The homotopy extension property is depicted in the following diagram

Homotopy extension property.svg

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.

See also


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