Fibration

In algebraic topology, a branch of mathematics, is meant by a fibration (also Hurewicz fibration ) is a continuous map of topological spaces satisfying the homotopy lifting property with respect to each high topological space. Fibrations play in homotopy theory, a sub-area of algebraic topology, a major role. Roughly speaking, are fibrations pairs of spaces with a picture of each other, which allow you can " retire " any homotopies in the image space along the given image on the original image space.

Definition

Homotopy high elevation property

Denote the unit interval.

A continuous map of topological spaces, satisfies the homotopy lifting property for the high topological space if for all continuous maps

As well as

So that the chart

Commutes, a figure

There, so and.

Hurewicz - fibrations

A fibration (also Hurewicz fibration ) is a continuous map, which satisfies the homotopy lifting property for all high topological spaces.

Is called the total space, the base of the fibration. The inverse image of a point is called over with fiber.

If the base is connected, on the fibers from various points are homotopy equivalent.

Serre- fibrations

A Serre fibration is a continuous map, which satisfies the homotopy lifting property for all high- CW - complexes.

For sufficiently (and equivalent) is that they met with the homotopy lifting property for the high rooms.

Quasifaserungen

A Quasifaserung is a continuous map for which

For each and every is an isomorphism.

If the base is path-connected, all fibers are weak homotopy equivalent.

Each Serre fibration is a Quasifaserung.

Examples

• Be an arbitrary topological space and let

• Each overlay is a fibration.
• General of each fiber bundle is a Serre fibration. In this case, the inverse images of different points are not only homotopy equivalent, but even homeomorphic.
• There are examples of fiber bundles that are not Hurewicz - fiberings. But fiber bundles over paracompact spaces are always Hurewicz - fibrations ( set of Huebsch - Hurewicz ).
• A fibration, which need not be a fiber bundle is the way fibration of a topological space.

Long exact homotopy sequence

For Serre- fibrations ( and more generally for Quasifaserungen ) one has a long exact sequence of homotopy groups

Here, and the fiber.

For example, the Hopf fibration with fiber. As is known to all, it follows for all, especially.

Homology groups of fibrations

The homology groups of Serre- fibrations can often be calculated with the help of spectral sequences.