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