Fibration on arvau

Wikipedia / 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.

In this article, all mappings are continuous mappings between topological spaces.

Formal definitions

Homotopy lifting property

satisfies the homotopy lifting property if:

for every homotopy and
)

The following commutative diagram shows the situation:

Fibration

A fibration (also called Hurewicz fibration is called the base space is called the total space. The fiber over

Serre fibration

A Serre fibration (also called weak fibration satisfying the homotopy lifting property for all CW-complexes.

Every Hurewicz fibration is a Serre fibration.

Quasifibration

is called quasifibration is an isomorphism.

Every Serre fibration is a quasifibration.

Examples

The projection is a fibration. That is, trivial bundles are fibrations.
Every covering
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.
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.

Basic concepts

Fiber homotopy equivalence

with the same base space is a fibration homomorphism if the following diagram commutes: is a fiber homotopy equivalence

Pullback fibration

is the pullback yield the following commutative diagram: 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.

of the pathspace fibration and paths is the unit interval carries the subspace topology and carries the compact-open topology.

is also called the homotopy fiber holds.

and is called loop space.

Properties

are homotopy equivalent for each path component
are fiber homotopy equivalent.
is contractible
is a fiber homotopy equivalence.
and contractible total space, there is a weak homotopy equivalence

Puppe sequence

of the fiber into the homotopy fiber is a homotopy equivalence and so on. This leads to a long sequence: 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

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

there exists a long exact sequence of homotopy groups this is given by:

The homomorphisms

Hopf fibration

Hopf fibrations are a family of fiber bundles whose fiber, total space and base space are spheres:

The long exact sequence yields: is contractible to a point:This short exact sequence splits because of the suspension and there are isomorphisms:

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 where the base space is a path connected CW-complex, and an additive homology theory there exists a spectral sequence:

where base space and fiber are path connected, the fundamental group 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 there exist exact sequences (also called Wang sequences) for homology and cohomology:

Orientability

and a fixed commutative ring with a unit, there exists a contravariant functor from the fundamental groupoid

A fibration is called orientable

Euler characteristic

over the field and path connected base space, the Euler characteristic of the total space is given by:.