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 because 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.
Assume from now on all mappings are continuous functions from a topological space to another. Given a map , and a space , one says that has the homotopy lifting property, or that has the homotopy lifting property with respect to , if:
- for any homotopy , and
- for any map lifting (i.e., so that ),
there exists a homotopy lifting (i.e., so that ) which also satisfies .
The following diagram depicts this situation.
The outer square (without the dotted arrow) commutes if and only if the hypotheses of the lifting property are true. A lifting corresponds to a dotted arrow making the diagram commute. This diagram is dual to that of the homotopy extension property; this duality is loosely referred to as Eckmann–Hilton duality.
If the map satisfies the homotopy lifting property with respect to all spaces X, then is called a fibration, or one sometimes simply says that has the homotopy lifting property.
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 , for simplicity we denote . Given additionally a map , one says that has the homotopy lifting extension property if:
- For any homotopy , and
- For any lifting of ,
there exists a homotopy which covers (i.e., such that ) and extends (i.e., such that ).
The homotopy lifting property of is obtained by taking , so that above is simply .
The homotopy extension property of is obtained by taking to be a constant map, so that is irrelevant in that every map to E is trivially the lift of a constant map to the image point of .
- Hu, Sze-Tsen (1959). Homotopy Theory. page 24
- Husemoller, Dale (1994). Fibre Bundles. page 7
- Steenrod, Norman (1951). The Topology of Fibre Bundles. Princeton: Princeton University Press. ISBN 0-691-00548-6.
- Hu, Sze-Tsen (1959). Homotopy Theory (Third Printing, 1965 ed.). New York: Academic Press Inc. ISBN 0-12-358450-7.
- Husemoller, Dale (1994). Fibre Bundles (Third ed.). New York: Springer. ISBN 978-0-387-94087-8.
- Hatcher, Allen (2002), Algebraic Topology, Cambridge: Cambridge University Press, ISBN 0-521-79540-0.