Homotopy extension property

Summary

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. The homotopy extension property of cofibrations is dual to the homotopy lifting property that is used to define fibrations.

Definition edit

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

 
then 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 edit

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   which makes the diagram commute. By currying, note that homotopies expressed as maps   are in natural bijection with expressions as maps  .

Note that this diagram is dual to (opposite to) that of the homotopy lifting property; this duality is loosely referred to as Eckmann–Hilton duality.

Properties edit

  • 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 edit

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 edit

References edit

  1. ^ A. Dold, Lectures on Algebraic Topology, pp. 84, Springer ISBN 3-540-58660-1