Fiber bundle

In topology, the terms fiber (German: Faser) and fiber space (gefaserter Raum) appeared for the first time in a paper by Herbert Seifert in 1933,[1][2] but his definitions are limited to a very special case. The main difference from the present day conception of a fiber space, however, was that for Seifert what is now called the base space (topological space) of a fiber (topological) space E was not part of the structure, but derived from it as a quotient space of E. The first definition of fiber space was given by Hassler Whitney in 1935 [3] under the name sphere space, but in 1940 Whitney changed the name to sphere bundle.[4]

The theory of fibered spaces, of which vector bundles, principal bundles, topological fibrations and fibered manifolds are a special case, is attributed to Seifert, Heinz Hopf, Jacques Feldbau,[5] Whitney, Norman Steenrod, Charles Ehresmann,[6][7][8] Jean-Pierre Serre,[9] and others.

Fiber bundles became their own object of study in the period 1935–1940. The first general definition appeared in the works of Whitney.[10]

Whitney came to the general definition of a fiber bundle from his study of a more particular notion of a sphere bundle,[11] that is a fiber bundle whose fiber is a sphere of arbitrary dimension.[12]

A fiber bundle is a structure ${\displaystyle (E,\,B,\,\pi ,\,F),}$ where ${\displaystyle E,B,}$ and ${\displaystyle F}$ are topological spaces and ${\displaystyle \pi :E\to B}$ is a continuous surjection satisfying a local triviality condition outlined below. The space ${\displaystyle B}$ is called the base space of the bundle, ${\displaystyle E}$ the total space, and ${\displaystyle F}$ the fiber. The map ${\displaystyle \pi }$ is called the projection map (or bundle projection). We shall assume in what follows that the base space ${\displaystyle B}$ is connected.

We require that for every ${\displaystyle x\in B}$, there is an open neighborhood ${\displaystyle U\subseteq B}$ of ${\displaystyle x}$ (which will be called a trivializing neighborhood) such that there is a homeomorphism ${\displaystyle \varphi$ :\pi ^{-1}(U)\to U\times F} (where ${\displaystyle \pi ^{-1}(U)}$ is given the subspace topology, and ${\displaystyle U\times F}$ is the product space) in such a way that ${\displaystyle \pi }$ agrees with the projection onto the first factor. That is, the following diagram should commute:

Local triviality condition

where ${\displaystyle \operatorname {proj} _{1}:U\times F\to U}$ is the natural projection and ${\displaystyle \varphi$ :\pi ^{-1}(U)\to U\times F} is a homeomorphism. The set of all ${\displaystyle \left\{\left(U_{i},\,\varphi _{i}\right)\right\}}$ is called a local trivialization of the bundle.

Thus for any ${\displaystyle p\in B}$, the preimage ${\displaystyle \pi ^{-1}(\{p\})}$ is homeomorphic to ${\displaystyle F}$ (since this is true of ${\displaystyle \operatorname {proj} _{1}^{-1}(\{p\})}$) and is called the fiber over ${\displaystyle p}$. Every fiber bundle ${\displaystyle \pi :E\to B}$ is an open map, since projections of products are open maps. Therefore ${\displaystyle B}$ carries the quotient topology determined by the map ${\displaystyle \pi .}$

A fiber bundle ${\displaystyle (E,\,B,\,\pi ,\,F)}$ is often denoted

${\displaystyle {\begin{matrix}{}\\F\longrightarrow E\ {\xrightarrow {\,\ \pi \ }}\ B\\{}\end{matrix}}}$

(1)

that, in analogy with a short exact sequence, indicates which space is the fiber, total space and base space, as well as the map from total to base space.

A smooth fiber bundle is a fiber bundle in the category of smooth manifolds. That is, ${\displaystyle E,B,}$ and ${\displaystyle F}$ are required to be smooth manifolds and all the functions above are required to be smooth maps.

A section (or cross section) of a fiber bundle ${\displaystyle \pi }$ is a continuous map ${\displaystyle f:B\to E}$ such that ${\displaystyle \pi (f(x))=x}$ for all x in B. Since bundles do not in general have globally defined sections, one of the purposes of the theory is to account for their existence. The obstruction to the existence of a section can often be measured by a cohomology class, which leads to the theory of characteristic classes in algebraic topology.

The most well-known example is the hairy ball theorem, where the Euler class is the obstruction to the tangent bundle of the 2-sphere having a nowhere vanishing section.

Often one would like to define sections only locally (especially when global sections do not exist). A local section of a fiber bundle is a continuous map ${\displaystyle f:U\to E}$ where U is an open set in B and ${\displaystyle \pi (f(x))=x}$ for all x in U. If ${\displaystyle (U,\,\varphi )}$ is a local trivialization chart then local sections always exist over U. Such sections are in 1-1 correspondence with continuous maps ${\displaystyle U\to F}$. Sections form a sheaf.

Fiber bundles often come with a group of symmetries that describe the matching conditions between overlapping local trivialization charts. Specifically, let G be a topological group that acts continuously on the fiber space F on the left. We lose nothing if we require G to act faithfully on F so that it may be thought of as a group of homeomorphisms of F. A G-atlas for the bundle ${\displaystyle (E,B,\pi ,F)}$ is a set of local trivialization charts ${\displaystyle \{(U_{k},\,\varphi _{k})\}}$ such that for any ${\displaystyle \varphi _{i},\varphi _{j}}$ for the overlapping charts ${\displaystyle (U_{i},\,\varphi _{i})}$ and ${\displaystyle (U_{j},\,\varphi _{j})}$ the function

$${\displaystyle \varphi _{i}\varphi _{j}^{-1}:\left(U_{i}\cap U_{j}\right)\times F\to \left(U_{i}\cap U_{j}\right)\times F}$$

is given by

$${\displaystyle \varphi _{i}\varphi _{j}^{-1}(x,\,\xi )=\left(x,\,t_{ij}(x)\xi \right)}$$

where ${\displaystyle t_{ij}:U_{i}\cap U_{j}\to G}$ is a continuous map called a transition function. Two G-atlases are equivalent if their union is also a G-atlas. A G-bundle is a fiber bundle with an equivalence class of G-atlases. The group G is called the structure group of the bundle; the analogous term in physics is gauge group.

In the smooth category, a G-bundle is a smooth fiber bundle where G is a Lie group and the corresponding action on F is smooth and the transition functions are all smooth maps.

The transition functions ${\displaystyle t_{ij}}$ satisfy the following conditions

1. ${\displaystyle t_{ii}(x)=1\,}$
2. ${\displaystyle t_{ij}(x)=t_{ji}(x)^{-1}\,}$
3. ${\displaystyle t_{ik}(x)=t_{ij}(x)t_{jk}(x).\,}$

The third condition applies on triple overlaps U_i ∩ U_j ∩ U_k and is called the cocycle condition (see Čech cohomology). The importance of this is that the transition functions determine the fiber bundle (if one assumes the Čech cocycle condition).

A principal G-bundle is a G-bundle where the fiber F is a principal homogeneous space for the left action of G itself (equivalently, one can specify that the action of G on the fiber F is free and transitive, i.e. regular). In this case, it is often a matter of convenience to identify F with G and so obtain a (right) action of G on the principal bundle.

It is useful to have notions of a mapping between two fiber bundles. Suppose that M and N are base spaces, and ${\displaystyle \pi _{E}:E\to M}$ and ${\displaystyle \pi _{F}:F\to N}$ are fiber bundles over M and N, respectively. A bundle map or bundle morphism consists of a pair of continuous[13] functions

$${\displaystyle \varphi :E\to F,\quad f:M\to N}$$

such that ${\displaystyle \pi _{F}\circ \varphi =f\circ \pi _{E}.}$ That is, the following diagram commutes:

For fiber bundles with structure group G and whose total spaces are (right) G-spaces (such as a principal bundle), bundle morphisms are also required to be G-equivariant on the fibers. This means that ${\displaystyle \varphi :E\to F}$ is also G-morphism from one G-space to another, that is, ${\displaystyle \varphi (xs)=\varphi (x)s}$ for all ${\displaystyle x\in E}$ and ${\displaystyle s\in G.}$

In case the base spaces M and N coincide, then a bundle morphism over M from the fiber bundle ${\displaystyle \pi _{E}:E\to M}$ to ${\displaystyle \pi _{F}:F\to M}$ is a map ${\displaystyle \varphi :E\to F}$ such that ${\displaystyle \pi _{E}=\pi _{F}\circ \varphi .}$ This means that the bundle map ${\displaystyle \varphi :E\to F}$ covers the identity of M. That is, ${\displaystyle f\equiv \mathrm {id} _{M}}$ and the diagram commutes

Assume that both ${\displaystyle \pi _{E}:E\to M}$ and ${\displaystyle \pi _{F}:F\to M}$ are defined over the same base space M. A bundle isomorphism is a bundle map ${\displaystyle (\varphi ,\,f)}$ between ${\displaystyle \pi _{E}:E\to M}$ and ${\displaystyle \pi _{F}:F\to M}$ such that ${\displaystyle f\equiv \mathrm {id} _{M}}$ and such that ${\displaystyle \varphi }$ is also a homeomorphism.[14]

In the category of differentiable manifolds, fiber bundles arise naturally as submersions of one manifold to another. Not every (differentiable) submersion ${\displaystyle f:M\to N}$ from a differentiable manifold M to another differentiable manifold N gives rise to a differentiable fiber bundle. For one thing, the map must be surjective, and ${\displaystyle (M,N,f)}$ is called a fibered manifold. However, this necessary condition is not quite sufficient, and there are a variety of sufficient conditions in common use.

If M and N are compact and connected, then any submersion ${\displaystyle f:M\to N}$ gives rise to a fiber bundle in the sense that there is a fiber space F diffeomorphic to each of the fibers such that ${\displaystyle (E,B,\pi ,F)=(M,N,f,F)}$ is a fiber bundle. (Surjectivity of ${\displaystyle f}$ follows by the assumptions already given in this case.) More generally, the assumption of compactness can be relaxed if the submersion ${\displaystyle f:M\to N}$ is assumed to be a surjective proper map, meaning that ${\displaystyle f^{-1}(K)}$ is compact for every compact subset K of N. Another sufficient condition, due to Ehresmann (1951) harvtxt error: no target: CITEREFEhresmann1951 (help), is that if ${\displaystyle f:M\to N}$ is a surjective submersion with M and N differentiable manifolds such that the preimage ${\displaystyle f^{-1}\{x\}}$ is compact and connected for all ${\displaystyle x\in N,}$ then ${\displaystyle f}$ admits a compatible fiber bundle structure (Michor 2008, §17).

• The notion of a bundle applies to many more categories in mathematics, at the expense of appropriately modifying the local triviality condition; cf. principal homogeneous space and torsor (algebraic geometry).
• In topology, a fibration is a mapping ${\displaystyle \pi :E\to B}$ that has certain homotopy-theoretic properties in common with fiber bundles. Specifically, under mild technical assumptions a fiber bundle always has the homotopy lifting property or homotopy covering property (see Steenrod (1951, 11.7) for details). This is the defining property of a fibration.
• A section of a fiber bundle is a "function whose output range is continuously dependent on the input." This property is formally captured in the notion of dependent type.

• Steenrod, Norman (1951), The Topology of Fibre Bundles, Princeton University Press, ISBN 978-0-691-08055-0
• Steenrod, Norman (April 5, 1999). The Topology of Fibre Bundles. Princeton Mathematical Series. Vol. 14. Princeton, N.J.: Princeton University Press. ISBN 978-0-691-00548-5. OCLC 40734875.
• Bleecker, David (1981), Gauge Theory and Variational Principles, Reading, Mass: Addison-Wesley publishing, ISBN 978-0-201-10096-9
• Ehresmann, Charles. "Les connexions infinitésimales dans un espace fibré différentiable". Colloque de Topologie (Espaces fibrés), Bruxelles, 1950. Georges Thone, Liège; Masson et Cie., Paris, 1951. pp. 29–55.
• Husemoller, Dale (1994), Fibre Bundles, Springer Verlag, ISBN 978-0-387-94087-8
• Michor, Peter W. (2008), Topics in Differential Geometry, Graduate Studies in Mathematics, vol. 93, Providence: American Mathematical Society, ISBN 978-0-8218-2003-2
• Voitsekhovskii, M.I. (2001) [1994], "Fibre space", Encyclopedia of Mathematics, EMS Press

• Fiber Bundle, PlanetMath
• Rowland, Todd. "Fiber Bundle". MathWorld.
• Making John Robinson's Symbolic Sculpture `Eternity'
• Sardanashvily, Gennadi, Fibre bundles, jet manifolds and Lagrangian theory. Lectures for theoreticians, arXiv:0908.1886