A pair has the homotopy extension property when every homotopy on whose initial map extends to can itself be extended to a homotopy on .
A cofibration is a map with the homotopy extension property. An inclusion of a CW complex into another as a subcomplex is a cofibration; collapsing that subcomplex gives the quotient used in the K-theory six-term exact sequence.
If is contractible and has the homotopy extension property, the quotient mapis a homotopy equivalence. Extend a contraction of to ; its endpoint is constant on and therefore factors through to supply a homotopy inverse.
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The Homotopy Extension Property (HEP) is a fundamental concept in algebraic topology, particularly in the context of topological spaces and homotopy theory. It essentially describes a condition under which homotopies defined on a subspace can be extended to the entire space.