Homotopy lifting property
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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 to do with 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 ''X'', one says that ''<math>(X,\pi)</math> has the '''homotopy lifting property''' '' 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>.
Alternative terminology: "<math>\pi</math> has the '''homotopy lifting property''' with respect to ''X''".
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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 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 <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 ''Y'' = ø, so that ''T'' 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 ''π'' is irrelevant in that every maps to ''E'' is trivially the lift of a constant map to the image point of ''π'').
==References==
* {{springer|author=A.V. Chernavskii|title=Covering homotopy|id=C/c026940}}
[[Category:Homotopy theory]]
[[Category:Algebraic topology]]
[[it:Teorema del sollevamento dell'omotopia]]
[[ru:Накрывающая гомотопия]]