##  [Homotopy Extension Property](/homotopy-extension-property-0) 

 Definition

A property of an inclusion i:A→X (or of the pair (X,A)) that any homotopy H_t defined on A can be extended to a homotopy ˜H_t on X starting from any map f:X→Y restricting to the given homotopy on A; equivalently a recognition criterion for cofibrations in many settings.

 

 

 

 

 

 





## Principle

Principle

Control extensions by local deformation data: if A sits in X as a cofibration (or an NDR-pair), homotopies on A admit extensions to X because one can thicken A and push the homotopy outwards along a cylinder or deformation retraction.

 

 

 

 

 





## Demonstration

Demonstration

In CW complexes the inclusion of a subcomplex A↪X has the HEP: a homotopy on A extends cell-by-cell to X using the cellular attaching maps and the fact cells are attached by cofibrations, allowing induction over skeleta.

 

 

 

 

## Misapplication

Misapplication

Assuming arbitrary subspace inclusions have HEP leads to false extensions (for instance pathological embeddings in bad point-set topologies); conflating HEP with homotopy lifting property (HLP) for fibrations is another common mistake.

 

 

 

 

 





## Consequence

Consequence

When a pair has HEP one can construct extensions of maps and homotopies, identify cofibrations, form homotopy pushouts, and apply obstruction-theoretic arguments for extending structures stepwise.

 

 

 

 

## Reversal

Reversal

The dual or opposite situation is the homotopy lifting property (HLP) for fibrations, where homotopies lift along a map rather than extend from a subspace; or simply inclusions that obstruct extension so that local homotopies cannot be globally extended.

 

 

 

 

 





## Boundary

Boundary

Pertains to inclusions of subspaces and topological pairs; it presumes reasonable categories (CW complexes, compactly generated spaces, or spaces where cofibrations are well behaved) and excludes arbitrary maps that are not inclusions or settings without cylinders.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Tension exists between HEP and HLP (extension versus lifting), and between the intuitive idea that subspaces extend homotopies and counterexamples in pathological point-set contexts; one must distinguish cofibration hypotheses from weaker inclusion properties.

 

 

 

 

 





## Synthesis

Synthesis

HEP identifies inclusions that allow homotopies defined on a subspace to be extended to the whole space; in practice it characterizes cofibrations and underpins inductive and obstruction-theoretic constructions in homotopy theory.