Homotopy lifting property

Homotopy lifting property

In mathematics, in particular in homotopy theory within algebraic topology, the homotopy lifting property (also known as 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 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. Given a map \pi\colon E\to B, and a space X, one says that (X,π) has the homotopy lifting property, or that π has the homotopy lifting property with respect to X, if:

  • for any homotopy f\colon X\times [0,1]\to B, and
  • for any map \tilde f_0\colon X\to E lifting f_0 = f|_{X\times\{0\}} (i.e. so that f_0 = \pi\tilde f_0),

there exists a homotopy \tilde f\colon X\times [0,1]\to E lifting f (i.e. so that f = \pi\tilde f) with \tilde f_0 = \tilde f|_{X\times\{0\}}.

The following diagram visualizes 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 \tilde f corresponds to a dotted arrow making the diagram commute. Also compare this to the visualization of the homotopy extension property.

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

N.B. This is the definition of fibration in the sense of Hurewicz, which is more restrictive than fibration in the sense of Serre, for which homotopy lifting only for X a CW complex is required.

Generalization: The Homotopy Lifting Extension Property

There is a common generalization of the homotopy lifting property and the homotopy extension property. Given a pair of spaces X\supseteq Y, for simplicity we denote T \colon = (X\times\{0\}) \cup (Y\times [0,1]) \ \subseteq \ X\times [0,1]. Given additionally a map \pi\colon E\to B\,, one says that (X,Y,π) has the homotopy lifting extension property if:

  • for any homotopy f\colon X\times [0,1]\to B, and
  • for any lifting \tilde g\colon T\to E of g = f | T,

there exists a homotopy \tilde f\colon X\times [0,1]\to E which extends \tilde g (i.e. such that \tilde f|_T=\tilde g).

The homotopy lifting property of (X,π) is obtained by taking Y = ø, so that T above is simply X\times\{0\}.

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

References


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