Homotopy extension property
- Homotopy extension property
In mathematics, in the area of algebraic topology, the homotopy extension property indicates when a homotopy can be extended to another one, so that the original homotopy is simply the restriction of the extended homotopy.
Definition
Given , we say that the pair has the homotopy extension property with respect to if the following holds:
Given any continuous , for which there is a homotopy of and , we can extend this to a homotopy of and some , where and .
Other
If has the homotopy extension property independent of , then the simple inclusion map is a cofibration.
In fact, if you consider any cofibration , then we have that is homeomorphic to its image under . This implies that any cofibration can be treated as an inclusion map, and therefore it can be treated as having the homotopy extension property.
ee also
* Homotopy lifting property
References
*
Wikimedia Foundation.
2010.
Look at other dictionaries:
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… … Wikipedia
Homotopy — This article is about topology. For chemistry, see Homotopic groups. The two dashed paths shown above are homotopic relative to their endpoints. The animation represents one possible homotopy. In topology, two continuous functions from one… … Wikipedia
Extension (mathematics) — In mathematics, the word extension has many uses. See:Analysis* Carathéodory s extension theorem * Continuous linear extension * M. Riesz extension theorem * Krein extension theorem * Hahn Banach theoremAlgebra* Abelian extension * Algebraic… … Wikipedia
List of mathematics articles (H) — NOTOC H H cobordism H derivative H index H infinity methods in control theory H relation H space H theorem H tree Haag s theorem Haagerup property Haaland equation Haar measure Haar wavelet Haboush s theorem Hackenbush Hadamard code Hadamard… … Wikipedia
Cofibration — In mathematics, in particular homotopy theory, a continuous mapping , where A and X are topological spaces, is a cofibration if it satisfies the homotopy extension property with respect to all spaces Y. The name is because the dual condition, the … Wikipedia
HEP — or hep can mean: * Hep Records, a jazz record label in Scotland * Habitat Evaluation Procedures, a method used to document the quality and quantity of available habitat for selected wildlife species * Head end power, a method for providing… … Wikipedia
Monodromy — In mathematics, monodromy is the study of how objects from mathematical analysis, algebraic topology and algebraic and differential geometry behave as they run round a singularity. As the name implies, the fundamental meaning of monodromy comes… … Wikipedia
List of algebraic topology topics — This is a list of algebraic topology topics, by Wikipedia page. See also: topology glossary List of topology topics List of general topology topics List of geometric topology topics Publications in topology Topological property Contents 1… … Wikipedia
List of general topology topics — This is a list of general topology topics, by Wikipedia page. Contents 1 Basic concepts 2 Limits 3 Topological properties 3.1 Compactness and countability … Wikipedia
mathematics — /math euh mat iks/, n. 1. (used with a sing. v.) the systematic treatment of magnitude, relationships between figures and forms, and relations between quantities expressed symbolically. 2. (used with a sing. or pl. v.) mathematical procedures,… … Universalium
