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 from a topological space E to another one, B. It is designed to support the picture of E "above" B by allowing a homotopy taking place in B to be moved "upstairs" to E.

For example, a covering map has a property of unique local lifting of paths to a given sheet; the uniqueness is because the fibers of a covering map are discrete spaces. The homotopy lifting property will hold in many situations, such as the projection in a vector bundle, fiber bundle or fibration, where there need be no unique way of lifting.

Formal definitionEdit

Assume from now on all maps are continuous functions from one topological space to another. Given a map  , and a space  , one says that   has the homotopy lifting property,[1][2] or that   has the homotopy lifting property with respect to  , if:

  • for any homotopy  , and
  • for any map   lifting   (i.e., so that  ),

there exists a homotopy   lifting   (i.e., so that  ) which also satisfies  .

The following diagram depicts this situation.

 

The outer square (without the dotted arrow) commutes if and only if the hypotheses of the lifting property are true. A lifting   corresponds to a dotted arrow making the diagram commute. This diagram is dual to that of the homotopy extension property; this duality is loosely referred to as Eckmann–Hilton duality.

If the map   satisfies the homotopy lifting property with respect to all spaces X, then   is called a fibration, or one sometimes simply says that   has the homotopy lifting property.

Note that this is the definition of fibration in the sense of Witold Hurewicz, which is more restrictive than fibration in the sense of Jean-Pierre Serre, for which homotopy lifting only for   a CW complex is required.

Generalization: homotopy lifting extension propertyEdit

There is a common generalization of the homotopy lifting property and the homotopy extension property. Given a pair of spaces  , for simplicity we denote  . Given additionally a map  , one says that   has the homotopy lifting extension property if:

  • For any homotopy  , and
  • For any lifting   of  ,

there exists a homotopy   which covers   (i.e., such that  ) and extends   (i.e., such that  ).

The homotopy lifting property of   is obtained by taking  , so that   above is simply  .

The homotopy extension property of   is obtained by taking   to be a constant map, so that   is irrelevant in that every map to E is trivially the lift of a constant map to the image point of  .

See alsoEdit

NotesEdit

  1. ^ Hu, Sze-Tsen (1959). Homotopy Theory. page 24
  2. ^ Husemoller, Dale (1994). Fibre Bundles. page 7

ReferencesEdit

External linksEdit