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Concept
Fibration
Summary
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.
Official source
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