Homotopy lifting property

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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 definition[edit]

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.

Homotopy lifting property.png

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.

A weaker notion of fibration is Serre fibration, for which homotopy lifting is only required for all CW complexes .

Generalization: homotopy lifting extension property[edit]

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 also[edit]


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


External links[edit]