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
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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.