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 mapping satisfies the homotopy lifting property for a space if:
there exists a (not necessarily unique) homotopy lifting (i.e. ) with
The following commutative diagram shows the situation: [1] : 66
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 [1] : 66
A Serre fibration (also called weak fibration) is a mapping satisfying the homotopy lifting property for all CW-complexes. [2] : 375-376
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. [3] : 241-242
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 is a fiber homotopy equivalence if in addition a fibration homomorphism exists, such that the mappings and are homotopic, by fibration homomorphisms, to the identities and [2] : 405-406
Given a fibration and a mapping , the mapping is a fibration, where is the pullback and the projections of onto and yield the following commutative diagram:
The fibration is called the pullback fibration or induced fibration. [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. 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. [2] : 407-408
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 along . 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 where is a path from to , i.e. the loop space . The inclusion of the fiber of into the homotopy fiber of 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. [2] : 407-409
A fibration with fiber 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. [2] : 412
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 [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 yields:
This sequence splits into short exact sequences, as the fiber in 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 so there exist isomorphisms between and for
Analog the fibers in and in are contractible to a point. Further the short exact sequences split and there are families of isomorphisms: [6] : 111
and
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 where the base space is a path connected CW-complex, and an additive homology theory there exists a spectral sequence: [7] : 242
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 where base space and fiber are path connected, the fundamental group acts trivially on and in addition the conditions for and for hold, an exact sequence exists (also known under the name Serre exact sequence):
[7] : 250
This sequence can be used, for example, to prove Hurewicz's theorem or to compute the homology of loopspaces of the form [8] : 162
For the special case of a fibration where the base space is a -sphere with fiber there exist exact sequences (also called Wang sequences) for homology and cohomology: [1] : 456
For a fibration with fiber and a fixed commutative ring with a unit, there exists a contravariant functor from the fundamental groupoid of to the category of graded -modules, which assigns to the module and to the path class the homomorphism where is a homotopy class in
A fibration is called orientable over if for any closed path in the following holds: [1] : 476
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 . [1] : 481
In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of the topological space. The fundamental group is the first and simplest homotopy group. The fundamental group is a homotopy invariant—topological spaces that are homotopy equivalent have isomorphic fundamental groups. The fundamental group of a topological space is denoted by .
In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted which records information about loops in a space. Intuitively, homotopy groups record information about the basic shape, or holes, of a topological space.
In algebraic topology, a branch of mathematics, a spectrum is an object representing a generalized cohomology theory. Every such cohomology theory is representable, as follows from Brown's representability theorem. This means that, given a cohomology theory
,
In mathematics, specifically algebraic topology, an Eilenberg–MacLane space is a topological space with a single nontrivial homotopy group.
In mathematics, specifically algebraic topology, the mapping cylinder of a continuous function between topological spaces and is the quotient
In mathematics, especially homotopy theory, the mapping cone is a construction of topology, analogous to a quotient space. It is also called the homotopy cofiber, and also notated . Its dual, a fibration, is called the mapping fibre. The mapping cone can be understood to be a mapping cylinder with the initial end of the cylinder collapsed to a point. Thus, mapping cones are frequently applied in the homotopy theory of pointed spaces.
In mathematics, the Serre spectral sequence is an important tool in algebraic topology. It expresses, in the language of homological algebra, the singular (co)homology of the total space X of a (Serre) fibration in terms of the (co)homology of the base space B and the fiber F. The result is due to Jean-Pierre Serre in his doctoral dissertation.
In mathematics, the Puppe sequence is a construction of homotopy theory, so named after Dieter Puppe. It comes in two forms: a long exact sequence, built from the mapping fibre, and a long coexact sequence, built from the mapping cone. Intuitively, the Puppe sequence allows us to think of homology theory as a functor that takes spaces to long-exact sequences of groups. It is also useful as a tool to build long exact sequences of relative homotopy groups.
In mathematics, the EHP spectral sequence is a spectral sequence used for inductively calculating the homotopy groups of spheres localized at some prime p. It is described in more detail in Ravenel and Mahowald (2001). It is related to the EHP long exact sequence of Whitehead (1953); the name "EHP" comes from the fact that George W. Whitehead named 3 of the maps of his sequence "E", "H", and "P".
In mathematics and specifically in topology, rational homotopy theory is a simplified version of homotopy theory for topological spaces, in which all torsion in the homotopy groups is ignored. It was founded by Dennis Sullivan and Daniel Quillen. This simplification of homotopy theory makes certain calculations much easier.
In homotopy theory, a branch of algebraic topology, a Postnikov system is a way of decomposing a topological space's homotopy groups using an inverse system of topological spaces whose homotopy type at degree agrees with the truncated homotopy type of the original space . Postnikov systems were introduced by, and are named after, Mikhail Postnikov.
In mathematics, especially homotopy theory, the homotopy fiber is part of a construction that associates a fibration to an arbitrary continuous function of topological spaces . It acts as a homotopy theoretic kernel of a mapping of topological spaces due to the fact it yields a long exact sequence of homotopy groups
In the mathematical surgery theory the surgery exact sequence is the main technical tool to calculate the surgery structure set of a compact manifold in dimension . The surgery structure set of a compact -dimensional manifold is a pointed set which classifies -dimensional manifolds within the homotopy type of .
In mathematics, an n-group, or n-dimensional higher group, is a special kind of n-category that generalises the concept of group to higher-dimensional algebra. Here, may be any natural number or infinity. The thesis of Alexander Grothendieck's student Hoàng Xuân Sính was an in-depth study of 2-groups under the moniker 'gr-category'.
In the mathematical disciplines of algebraic topology and homotopy theory, Eckmann–Hilton duality in its most basic form, consists of taking a given diagram for a particular concept and reversing the direction of all arrows, much as in category theory with the idea of the opposite category. A significantly deeper form argues that the fact that the dual notion of a limit is a colimit allows us to change the Eilenberg–Steenrod axioms for homology to give axioms for cohomology. It is named after Beno Eckmann and Peter Hilton.
In homotopy theory, a branch of mathematics, the Barratt–Priddy theorem expresses a connection between the homology of the symmetric groups and mapping spaces of spheres. The theorem is also often stated as a relation between the sphere spectrum and the classifying spaces of the symmetric groups via Quillen's plus construction.
In differential geometry, the integration along fibers of a k-form yields a -form where m is the dimension of the fiber, via "integration". It is also called the fiber integration.
In algebra, Quillen's Q-construction associates to an exact category an algebraic K-theory. More precisely, given an exact category C, the construction creates a topological space so that is the Grothendieck group of C and, when C is the category of finitely generated projective modules over a ring R, for , is the i-th K-group of R in the classical sense. One puts
This is a glossary of properties and concepts in algebraic topology in mathematics.
In algebraic topology, the path space fibration over a based space is a fibration of the form