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.

Formal definitions

Homotopy lifting property

A mapping satisfies the homotopy lifting property for a space X if: there exists a (not necessarily unique) homotopy lifting h (i.e. ) with The following commutative diagram shows the situation:

Fibration

A fibration (also called Hurewicz fibration) is a mapping satisfying the homotopy lifting property for all spaces X. The space B is called base space and the space E is called total space. The fiber over b \in B is the subspace

Serre fibration

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.

Quasifibration

A mapping is called quasifibration, if for every b \in B, and i \geq 0 holds that the induced mapping is an isomorphism. Every Serre fibration is a quasifibration.

Examples

Basic concepts

Fiber homotopy equivalence

A mapping between total spaces of two fibrations and with the same base space is a fibration homomorphism if the following diagram commutes: The mapping f is a fiber homotopy equivalence if in addition a fibration homomorphism exists, such that the mappings f \circ g and g \circ f are homotopic, by fibration homomorphisms, to the identities and

Pullback fibration

Given a fibration and a mapping, the mapping is a fibration, where is the pullback and the projections of f^*(E) onto A and E yield the following commutative diagram: The fibration p_f is called the pullback fibration or induced fibration.

Pathspace fibration

With the pathspace construction, any continuous mapping can be extended to a fibration by enlarging its domain to a homotopy equivalent space. This fibration is called pathspace fibration. The total space E_f of the pathspace fibration for a continuous mapping between topological spaces consists of pairs (a, \gamma) with a \in A and paths with starting point where I = [0, 1] is the unit interval. The space carries the subspace topology of where B^I describes the space of all mappings I \to B and carries the compact-open topology. The pathspace fibration is given by the mapping with The fiber F_f is also called the homotopy fiber of f and consists of the pairs (a, \gamma) with a \in A and paths where and holds. For the special case of the inclusion of the base point, an important example of the pathspace fibration emerges. The total space E_i consists of all paths in B which starts at b_0. This space is denoted by PB and is called path space. The pathspace fibration maps each path to its endpoint, hence the fiber p^{-1}(b_0) consists of all closed paths. The fiber is denoted by \Omega B and is called loop space.

Properties

Puppe sequence

For a fibration with fiber F and base point b_0 \in B the inclusion of the fiber into the homotopy fiber is a homotopy equivalence. The mapping with, where e \in E and is a path from p(e) to b_0 in the base space, is a fibration. Specifically it is the pullback fibration of the pathspace fibration PB \to B along p. This procedure can now be applied again to the fibration i and so on. This leads to a long sequence: "" The fiber of i over a point consists of the pairs where \gamma is a path from to b_0, i.e. the loop space \Omega B. The inclusion of the fiber of i into the homotopy fiber of i is again a homotopy equivalence and iteration yields the sequence:""Due to the duality of fibration and cofibration, there also exists a sequence of cofibrations. These two sequences are known as the Puppe sequences or the sequences of fibrations and cofibrations.

Principal fibration

A fibration with fiber F is called principal, if there exists a commutative diagram: The bottom row is a sequence of fibrations and the vertical mappings are weak homotopy equivalences. Principal fibrations play an important role in Postnikov towers.

Long exact sequence of homotopy groups

For a Serre fibration there exists a long exact sequence of homotopy groups. For base points b_0 \in B and this is given by: The homomorphisms and are the induced homomorphisms of the inclusion and the projection

Hopf fibration

Hopf fibrations are a family of fiber bundles whose fiber, total space and base space are spheres: The long exact sequence of homotopy groups of the hopf fibration yields:"" This sequence splits into short exact sequences, as the fiber S^1 in S^3 is contractible to a point:""This short exact sequence splits because of the suspension homomorphism and there are isomorphisms:""The homotopy groups are trivial for i \geq 3, so there exist isomorphisms between \pi_i(S^2) and \pi_i(S^3) for i \geq 3. Analog the fibers S^3 in S^7 and S^7 in S^{15} are contractible to a point. Further the short exact sequences split and there are families of isomorphisms: and

Spectral sequence

Spectral sequences are important tools in algebraic topology for computing (co-)homology groups. The Leray-Serre spectral sequence connects the (co-)homology of the total space and the fiber with the (co-)homology of the base space of a fibration. For a fibration with fiber F, where the base space is a path connected CW-complex, and an additive homology theory G_* there exists a spectral sequence: Fibrations do not yield long exact sequences in homology, as they do in homotopy. But under certain conditions, fibrations provide exact sequences in homology. For a fibration with fiber F, where base space and fiber are path connected, the fundamental group \pi_1(B) acts trivially on H_*(F) and in addition the conditions H_p(B) = 0 for 0<p<m and H_q(F) = 0 for 0<q<n hold, an exact sequence exists (also known under the name Serre exact sequence):""This sequence can be used, for example, to prove Hurewicz's theorem or to compute the homology of loopspaces of the form \Omega S^n: For the special case of a fibration where the base space is a n-sphere with fiber F, there exist exact sequences (also called Wang sequences) for homology and cohomology:

Orientability

For a fibration with fiber F and a fixed commutative ring R with a unit, there exists a contravariant functor from the fundamental groupoid of B to the category of graded R-modules, which assigns to b \in B the module H_*(F_b, R) and to the path class [\omega] the homomorphism where h[\omega] is a homotopy class in A fibration is called orientable over R if for any closed path \omega in B the following holds:

Euler characteristic

For an orientable fibration over the field \mathbb{K} with fiber F and path connected base space, the Euler characteristic of the total space is given by:""Here the Euler characteristics of the base space and the fiber are defined over the field \mathbb{K}.