Homotopy lifting property

From Academic Kids

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 X to another one, Y. It is designed to support the picture of X 'above' Y, by allowing a homotopy taking place in Y to be moved 'upstairs' to X. 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, 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: AB

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

g: X × [0,1] → B

and map

h: XA

such that

p o h = g|X × 0

there is a homotopy

f : X × [0,1] → A

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.

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