Homotopy lifting property

In mathematics, in particular in homotopy theory within algebraic topology, the homotopy lifting property (also known as the right lifting property) 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 to do with the fact that 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=

Assume from now on all mappings are continuous functions from a topological space to another. One says that

: p : E → B

has the homotopy lifting property with respect to a space X if for any Homotopy

: g : X × [0,1] → B

and map

: h : X → E

such that

: p o h = g|X × 0

there is a Homotopy

: f : X × [0,1] → E

such that

: p o f = g

and f|X × 0 equals h .

If a map satisfies the homotopy lifting property with respect to all spaces X , one sometimes simply says that it satisfies the homotopy lifting property. Such a map is called a Fibration. This is the definition of fibration in the sense of Hurewicz , which is more restrictive than the fibration in the sense of Serre , for which homotopy lifting only for X a CW complex is required.

=Generalizations=

There is also the more general concept of the homotopy lifting property with respect to a pair ( X , Y ). Here one requires that given a homotopy

: X × [0,1] → B ,

a lift of that map on X × 0, and a lift on Y × [0,1] such that the two lifts agree on Y × 0, the lift can be extended to a lift of the Homotopy. The homotopy lifting property is obtained by taking Y = ø.