Fibration

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This article is about fibrations in algebraic topology. For fibrations in category theory, as used in descent theory and categorical logic, see Fibred category.

In mathematics, especially algebraic topology, a fibration is a continuous mapping

p:E\to B\,

satisfying the homotopy lifting property with respect to any space. Fiber bundles (over paracompact bases) constitute important examples. In homotopy theory any mapping is 'as good as' a fibration — i.e. any map can be decomposed as a homotopy equivalence into a "mapping path space" followed by a fibration.

A fibration with the homotopy lifting property for CW complexes (or equivalently, just cubes In) is called a Serre fibration, in honor of the part played by the concept in the thesis of Jean-Pierre Serre. This thesis firmly established in algebraic topology the use of spectral sequences, and clearly separated the notions of fiber bundles and fibrations from the notion of sheaf (both concepts together having been implicit in the pioneer treatment of Jean Leray). Because a sheaf (thought of as an étalé space) can be considered a local homeomorphism, the notions were closely interlinked at the time.

The fibers are by definition the subspaces of E that are the inverse images of points b of B. If the base space B is path connected, it is a consequence of the definition that the fibers of two different points b1 and b2 in B are homotopy equivalent. Therefore one usually speaks of "the fiber" F. Fibrations do not necessarily have the local cartesian product structure that defines the more restricted fiber bundle case, but something weaker that still allows "sideways" movement from fiber to fiber. One of the main desirable properties of the Serre spectral sequence is to account for the action of the fundamental group of the base B on the homology of the total space E.

The projection map from a product space is very easily seen to be a fibration. Fiber bundles have local trivializations — such cartesian product structures exist locally on B, and this is usually enough to show that a fiber bundle is a fibration. More precisely, if there are local trivializations over a "numerable open cover" of B , the bundle is a fibration. Any open cover of a paracompact space — for example any metric space, has a numerable refinement, so any bundle over such a space is a fibration. The local triviality also implies the existence of a well-defined fiber (up to homeomorphism), at least on each connected component of B.

[edit] Examples

In the following examples a fibration is denoted

F → E → B,

where the first map is the inclusion of "the" fiber F into the total space E and the second map is the fibration onto the basis B. This is also referred to as a fibration sequence.

[edit] Fibrations in closed model categories

Fibrations of topological spaces fit into a more general framework, the so-called closed model categories. In such categories, there are distinguished classes of morphisms, the so-called fibrations, cofibrations and weak equivalences. Certain axioms, such as stability of fibrations under composition and pullbacks, factorization of every morphism into the composition of an acyclic cofibration followed by a fibration or a cofibration followed by an acyclic fibration, where the word "acyclic" indicates that the corresponding arrow is also a weak equivalence, and other requirements are set up to allow the abstract treatment of homotopy theory. (In the original treatment, due to Daniel Quillen, the word "trivial" was used instead of "acyclic.")

It can be shown that the category of topological spaces is in fact a model category, where (abstract) fibrations are just the fibrations introduced above and weak equivalences are homotopy equivalences. See Dwyer, Spaliński (1995).

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