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.

Contents

Formal definition

Generalization: The Homotopy Lifting Extension Property

References

Formal definition

Assume from now on all mappings are continuous functions from a topological space to another. Given a map $picolon\; E\; o\; B,$ and a space ''X'', one says that ''$(X,pi)$ has the 'homotopy lifting property' '' if:

★ for any homotopy $fcolon\; X\; imes\; [0,1]\; o\; B$, and

★ for any map $ilde\; f\_0colon\; X\; o\; E$ lifting $f\_0\; =\; f|\_\{X\; imes\{0\}\}$ (i.e. so that $f\_0\; =\; pi\; ilde\; f\_0$),
there exists a homotopy $ilde\; fcolon\; X\; imes\; [0,1]\; o\; E$ lifting $f$ (i.e. so that $f\; =\; pi\; ilde\; f$) with $ilde\; f\_0\; =\; ilde\; f|\_\{X\; imes\{0\}\}$.
Alternative terminology: "$pi$ has the 'homotopy lifting property' with respect to ''X''".

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

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 $Xsupseteq\; Y$, for simplicity we denote $T\; colon\; =\; (X\; imes\{0\})\; cup\; (Y\; imes\; [0,1])\; subseteq\; X\; imes\; [0,1]$. Given additionally a map $picolon\; E\; o\; B,$, one says that ''$(X,Y,pi)$ has the 'homotopy lifting extension property' '' if:

★ for any homotopy $fcolon\; X\; imes\; [0,1]\; o\; B$, and

★ for any lifting $ilde\; gcolon\; T\; o\; E$ of $g=f|\_T$,
there exists a homotopy $ilde\; fcolon\; X\; imes\; [0,1]\; o\; E$ which extends $ilde\; g$ (i.e. such that $ilde\; f|\_T=\; ilde\; g$).
The homotopy lifting property of $(X,pi)$ is obtained by taking ''Y'' = ø, so that ''T'' above is simply $X\; imes\{0\}$.
The homotopy extension property of $(X,Y)$ is obtained by taking $pi$ to be a constant map, so that ''π'' is irrelevant in that every maps to ''E'' is trivially the lift of a constant map to the image point of ''π'').

References

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