Fibration

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.

satisfies the homotopy lifting property for a space

if: there exists a (not necessarily unique) homotopy

satisfying the homotopy lifting property for all spaces

satisfying the homotopy lifting property for all CW-complexes.

with the same base space is a fibration homomorphism if the following diagram commutes: The mapping

is a fiber homotopy equivalence if in addition a fibration homomorphism

[2]: 405-406 With the pathspace construction, any continuous mapping can be extended to a fibration by enlarging its domain to a homotopy equivalent space.

between topological spaces consists of pairs

For the special case of the inclusion of the base point

, an important example of the pathspace fibration emerges.

maps each path to its endpoint, hence the fiber

is again a homotopy equivalence and iteration yields the sequence:

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.

there exists a long exact sequence of homotopy groups.

[2]: 376 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

This short exact sequence splits because of the suspension homomorphism

Further the short exact sequences split and there are families of isomorphisms:[6]: 111

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.

where the base space is a path connected CW-complex, and an additive homology theory

Fibrations do not yield long exact sequences in homology, as they do in homotopy.

But under certain conditions, fibrations provide exact sequences in homology.

where base space and fiber are path connected, the fundamental group

[7]: 250 This sequence can be used, for example, to prove Hurewicz's theorem or to compute the homology of loopspaces of the form

with a unit, there exists a contravariant functor from the fundamental groupoid of

Here the Euler characteristics of the base space and the fiber are defined over the field