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Concept
Fibration
Résumé
The notion of a fibration generalizes the notion of a fiber bundle and plays an important role in algebraic topology, a branch of mathematics.
Fibrations are used, for example, in Postnikov systems or obstruction theory.
In this article, all mappings are continuous mappings between topological spaces.
A mapping satisfies the homotopy lifting property for a space if:
for every homotopy and
for every mapping (also called lift) lifting (i.e. )
there exists a (not necessarily unique) homotopy lifting (i.e. ) with
The following commutative diagram shows the situation:
A fibration (also called Hurewicz fibration) is a mapping satisfying the homotopy lifting property for all spaces The space is called base space and the space is called total space. The fiber over is the subspace
A Serre fibration (also called weak fibration) is a mapping satisfying the homotopy lifting property for all CW-complexes.
Every Hurewicz fibration is a Serre fibration.
A mapping is called quasifibration, if for every and holds that the induced mapping is an isomorphism.
Every Serre fibration is a quasifibration.
The projection onto the first factor is a fibration. That is, trivial bundles are fibrations.
Every covering satisfies the homotopy lifting property for all spaces. Specifically, for every homotopy and every lift there exists a uniquely defined lift with
Every fiber bundle satisfies the homotopy lifting property for every CW-complex.
A fiber bundle with a paracompact and Hausdorff base space satisfies the homotopy lifting property for all spaces.
An example for a fibration, which is not a fiber bundle, is given by the mapping induced by the inclusion where a topological space and is the space of all continuous mappings with the compact-open topology.
The Hopf fibration is a non trivial fiber bundle and specifically a Serre fibration.
Source officielle
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Concepts associés (24)
Groupe d'homotopie
En mathématiques, et plus particulièrement en topologie algébrique, les groupes d'homotopie sont des invariants qui généralisent la notion de groupe fondamental aux dimensions supérieures. Il y a plusieurs définitions équivalentes possibles. Première définition Soit X un espace topologique et un point de X. Soit la boule unité de dimension i de l'espace euclidien . Son bord est la sphère unité de dimension . Le i-ième groupe d'homotopie supérieur est l'ensemble des classes d'homotopie relative à d'applications continues telle que : .
Théorie de l'homotopie
La théorie de l'homotopie est une branche des mathématiques issue de la topologie algébrique dans laquelle les espaces et applications sont considérés à homotopie près. La notion topologique de déformation est étendue à des contextes algébriques notamment via les structures de complexe différentiel puis d’algèbre A. Étant donné deux équivalences d’homotopie f : X′ → X et g : Y → Y′, l’ensemble des classes d'homotopie des applications continues entre X et Y s’identifie à celui des applications entre X′ et Y′ par composition avec f et g.
Homotopy fiber
In mathematics, especially homotopy theory, the homotopy fiber (sometimes called the mapping fiber) is part of a construction that associates a fibration to an arbitrary continuous function of topological spaces . It acts as a homotopy theoretic kernel of a mapping of topological spaces due to the fact it yields a long exact sequence of homotopy groupsMoreover, the homotopy fiber can be found in other contexts, such as homological algebra, where the distinguished trianglegives a long exact sequence analogous to the long exact sequence of homotopy groups.