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2014-01-09T14:43:53Z

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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 due to the fact that the fibers of a covering map are [[discrete space]]s. 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 <math>\pi\colon E\to B</math>, and a space <math>X\,</math>, one says that <math>(X,\pi)\,</math> has the '''''homotopy lifting property''''',<ref>{{cite book | last = Hu | first = Sze-Tsen |title = Homotopy Theory | year=1959}} page 24</ref><ref>{{cite book | last = Husemoller | first = Dale | title = Fibre Bundles| year=1994 }} page 7</ref> or that <math>\pi\,</math> has the '''''homotopy lifting property''''' with respect to <math>X\,</math>, if:

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

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

The following diagram visualizes this situation.
:[[File: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 <math>\tilde f</math> corresponds to a dotted arrow making the diagram commute. Also compare this to the visualization of the [[Homotopy_extension_property#Visualisation|homotopy extension property]].

If the map <math>\pi\,</math> satisfies the homotopy lifting property with respect to ''all'' spaces ''X'', then <math>\pi\,</math> is called a [[fibration]], or one sometimes simply says that ''<math>\pi\,</math> 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 [[Jean-Pierre Serre|Serre]]'', for which homotopy lifting only for <math>X\,</math> 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 <math>X\supseteq Y</math>, for simplicity we denote <math>T \colon = (X\times\{0\}) \cup (Y\times [0,1]) \ \subseteq \ X\times [0,1]</math>. Given additionally a map <math>\pi\colon E\to B\,</math>, one says that ''<math>(X,Y,\pi)\,</math> has the '''homotopy lifting extension property''' '' if:

*for any [[homotopy]] <math>f\colon X\times [0,1]\to B\,</math>, and
*for any lifting <math>\tilde g\colon T\to E</math> of <math>g=f|_T\,</math>,

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

The homotopy lifting property of <math>(X,\pi)\,</math> is obtained by taking <math>Y=\emptyset</math>, so that <math>T\,</math> above is simply <math>X\times\{0\}</math>.

The homotopy extension property of <math>(X,Y)\,</math> is obtained by taking <math>\pi\,</math> to be a constant map, so that <math>\pi\,</math> is irrelevant in that every map to ''E'' is trivially the lift of a constant map to the image point of <math>\pi\,</math>.

==See also==
* [[Covering space]]
* [[Fiber bundle]]
* [[Fibration]]

==Notes==
{{Reflist}}

==References==
*{{cite book | last = Steenrod | first = Norman | title = The Topology of Fibre Bundles | publisher = Princeton University Press | location = Princeton | year = 1951 | isbn = 0-691-00548-6}}
*{{cite book | last = Hu | first = Sze-Tsen | title = Homotopy Theory | publisher = Academic Press Inc. | edition = Third Printing, 1965 |location = New York | year=1959 | isbn= 0-12-358450-7}}
*{{cite book | last = Husemoller | first = Dale | title = Fibre Bundles | publisher = Springer | edition = Third |location = New York | year=1994 | isbn=978-0-387-94087-8}}
*{{citation| last=Hatcher |first= Allen |title=Algebraic Topology |url=http://www.math.cornell.edu/~hatcher/AT/ATpage.html |year= 2002 |publisher=Cambridge University Press |place=Cambridge |isbn=0-521-79540-0}}.

==External links==
* {{springer|author=A.V. Chernavskii|title=Covering homotopy|id=C/c026940}}
* {{nlab|id=homotopy%20lifting%20property|title=homotopy lifting property}}

[[Category:Homotopy theory]]
[[Category:Algebraic topology]]

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