Топологиялық кеңістіктердегі фибрациялар

Concept in algebraic topology
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.

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

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.

A mapping is called quasifibration, if for every and holds that the induced mapping is an isomorphism. Every Serre fibration is a quasifibration.

The projection onto the first factor is a fibration. That is, trivial bundles are fibrations. Every covering is a fibration. Specifically, for every homotopy and every lift there exists a uniquely defined lift with
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. An example of a fibration which is not a fiber bundle is given by the mapping induced by the inclusion where a topological space and 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.

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 of the pathspace fibration for a continuous mapping between topological spaces consists of pairs with and paths with starting point where is the unit interval. The space carries the subspace topology of where describes the space of all mappings and carries the compact open topology. The pathspace fibration is given by the mapping with The fiber is also called the homotopy fiber of and consists of the pairs with 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 consists of all paths in which starts at This space is denoted by and is called path space. The pathspace fibration maps each path to its endpoint, hence the fiber consists of all closed paths. The fiber is denoted by and is called loop space.

For a fibration with fiber and base point the inclusion of the fiber into the homotopy fiber is a homotopy equivalence. The mapping with , where and is a path from to in the base space, is a fibration. Specifically it is the pullback fibration of the pathspace fibration This procedure can now be applied again to the fibration and so on. This leads to a long sequence:

The fiber of over a point consists of the pairs with closed paths and starting point , i. e. the loop space The inclusion is 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.

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

For an orientable fibration over the field with fiber 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 .