In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, in the area of algebraic topology

Algebraic topology

Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.Although algebraic topology...

, the homotopy extension property indicates when a homotopy

Homotopy

In topology, two continuous functions from one topological space to another are called homotopic if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions...

 defined on a subspace

Subspace topology

In topology and related areas of mathematics, a subspace of a topological space X is a subset S of X which is equipped with a natural topology induced from that of X called the subspace topology .- Definition :Given a topological space and a subset S of X, the...

 can be extended to a homotopy defined on a larger space.

Definition

Let be a topological space

Topological space

Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...

, and let .
Given a homotopy and a map such that ,
we say that the pair has the homotopy extension property if
there exists an extension of to the 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

Codomain

In mathematics, the codomain or target set of a function is the set into which all of the output of the function is constrained to fall. It is the set in the notation...

 , 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

Currying

In mathematics and computer science, currying is the technique of transforming a function that takes multiple arguments in such a way that it can be called as a chain of functions each with a single argument...

, 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
  Deformation retract
  In topology, a branch of mathematics, a retraction , as the name suggests, "retracts" an entire space into a subspace. A deformation retraction is a map which captures the idea of continuously shrinking a space into a subspace.- Retract :...
  
   of

Other

If has the homotopy extension property, then the simple inclusion map is a cofibration

Cofibration

In mathematics, in particular homotopy theory, a continuous mappingi\colon A \to X,where A and X are topological spaces, is a cofibration if it satisfies the homotopy extension property with respect to all spaces Y. The name is because the dual condition, the homotopy lifting property, defines...

.

In fact, if you consider any cofibration

Cofibration

In mathematics, in particular homotopy theory, a continuous mappingi\colon A \to X,where A and X are topological spaces, is a cofibration if it satisfies the homotopy extension property with respect to all spaces Y. The name is because the dual condition, the homotopy lifting property, defines...

 , 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.