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In mathematics, in the area of
algebraic topology Algebraic topology is a branch of mathematics that 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 classif ...
, 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 In mathematics, in particular in homotopy theory within algebraic topology, the homotopy lifting property (also known as an instance of the right lifting property or the covering homotopy axiom) is a technical condition on a continuous function f ...
that is used to define
fibration The notion of a fibration generalizes the notion of a fiber bundle and plays an important role in algebraic topology, a branch of mathematics. Fibrations are used, for example, in postnikov-systems or obstruction theory. In this article, all map ...
s.


Definition

Let X\,\! be a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
, and let A \subset X. We say that the pair (X,A)\,\! has the homotopy extension property if, given a homotopy f_t\colon A \rightarrow Y and a map \tilde_0\colon X \rightarrow Y such that \left.\tilde_0\_A = f_0, there exists an ''extension'' of f_t to a homotopy \tilde_t\colon X \rightarrow Y such that \left.\tilde_t\_A = f_t.A. Dold, ''Lectures on Algebraic Topology'', pp. 84, Springer That is, the pair (X,A)\,\! has the homotopy extension property if any map G\colon ((X\times \) \cup (A\times I)) \rightarrow Y can be extended to a map G'\colon X\times I \rightarrow Y (i.e. G\,\! and G'\,\! agree on their common domain). If the pair has this property only for a certain
codomain In mathematics, the codomain or set of destination 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 . The term range is sometimes ambiguously used to refer to either ...
Y\,\!, we say that (X,A)\,\! has the homotopy extension property with respect to Y\,\!.


Visualisation

The homotopy extension property is depicted in the following diagram 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 \tilde which makes the diagram commute. By
currying In mathematics and computer science, currying is the technique of translating the evaluation of a function that takes multiple arguments into evaluating a sequence of functions, each with a single argument. For example, currying a function f tha ...
, note that a map \tilde \colon X \to Y^I is the same as a map \tilde \colon X\times I \to Y . Note that this diagram is dual to (opposite to) that of the
homotopy lifting property In mathematics, in particular in homotopy theory within algebraic topology, the homotopy lifting property (also known as an instance of the right lifting property or the covering homotopy axiom) is a technical condition on a continuous function f ...
; this duality is loosely referred to as Eckmann–Hilton duality.


Properties

* If X\,\! is a cell complex and A\,\! is a subcomplex of X\,\!, then the pair (X,A)\,\! has the homotopy extension property. * A pair (X,A)\,\! has the homotopy extension property if and only if (X\times \ \cup A\times I) is a retract of X\times I.


Other

If (X, A) has the homotopy extension property, then the simple inclusion map i\colon A \to X is a cofibration. In fact, if you consider any cofibration i\colon Y \to Z, then we have that \mathbf is homeomorphic to its image under \mathbf. 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

*
Homotopy lifting property In mathematics, in particular in homotopy theory within algebraic topology, the homotopy lifting property (also known as an instance of the right lifting property or the covering homotopy axiom) is a technical condition on a continuous function f ...


References

* * {{planetmath reference, urlname=HomotopyExtensionProperty, title=Homotopy extension property Homotopy theory Algebraic topology