Fiber bundle

In mathematics, and particularly topology, a fiber bundle ( Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space $E$ and a product space $B\; \backslash times\; F$ is defined using a continuous surjective map, $\backslash pi\; :\; E\; \backslash to\; B,$ that in small regions of $E$ behaves just like a projection from corresponding regions of $B\; \backslash times\; F$ to $B.$ The map $\backslash pi,$ called the projection or submersion of the bundle, is regarded as part of the structure of the bundle. The space $E$ is known as the total space of the fiber bundle, $B$ as the base space, and $F$ the fiber.

In the trivial case, $E$ is just $B\; \backslash times\; F,$ and the map $\backslash pi$ is just the projection from the product space to the first factor. This is called a trivial bundle. Examples of non-trivial fiber bundles include the Möbius strip and Klein bottle, as well as nontrivial . Fiber bundles, such as the tangent bundle of a manifold and other more general , play an important role in differential geometry and differential topology, as do .

Mappings between total spaces of fiber bundles that "commute" with the projection maps are known as , and the class of fiber bundles forms a Category theory with respect to such mappings. A bundle map from the base space itself (with the identity mapping as projection) to $E$ is called a section of $E.$ Fiber bundles can be specialized in a number of ways, the most common of which is requiring that the Transition map between the local trivial patches lie in a certain topological group, known as the structure group, acting on the fiber $F$.

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History 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, "Topologie Dreidimensionaler Gefaserter Räume" on Project Euclid.

H. Seifert (1980). 9780126348507, Academic Press. ISBN 9780126348507

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 under the name sphere space, but in 1940 Whitney changed the name to sphere bundle.

The theory of fibered spaces, of which , , topological and are a special case, is attributed to Herbert Seifert, Heinz Hopf, Jacques Feldbau, Whitney, Norman Steenrod, Charles Ehresmann, Jean-Pierre Serre, 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.See

Whitney came to the general definition of a fiber bundle from his study of a more particular notion of a sphere bundle,In his early works, Whitney referred to the sphere bundles as the "sphere-spaces". See, for example: that is a fiber bundle whose fiber is a sphere of arbitrary dimension.

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Formal definition A fiber bundle is a structure $(E,\backslash ,\; B,\backslash ,\; \backslash pi,\backslash ,\; F),$ where $E,\; B,$ and $F$ are topological spaces and $\backslash pi\; :\; E\; \backslash to\; B$ is a continuous surjection satisfying a local triviality condition outlined below. The space $B$ is called the ' of the bundle, $E$ the ', and $F$ the '. The map $\backslash pi$ is called the ' (or ). We shall assume in what follows that the base space $B$ is Connected space.

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

where $\backslash operatorname\{proj\}\_1\; :\; U\; \backslash times\; F\; \backslash to\; U$ is the natural projection and $\backslash varphi\; :\; \backslash pi^\{-1\}(U)\; \backslash to\; U\; \backslash times\; F$ is a homeomorphism. The set of all $\backslash left\backslash \{\backslash left(U\_i,\backslash ,\; \backslash varphi\_i\backslash right)\backslash right\backslash \}$ is called a of the bundle.

Thus for any $p\; \backslash in\; B$, the preimage $\backslash pi^\{-1\}(\backslash \{p\backslash \})$ is homeomorphic to $F$ (since this is true of $\backslash operatorname\{proj\}\_1^\{-1\}(\backslash \{p\backslash \})$) and is called the fiber over $p$. Every fiber bundle $\backslash pi\; :\; E\; \backslash to\; B$ is an open map, since projections of products are open maps. Therefore $B$ carries the quotient topology determined by the map $\backslash pi.$

A fiber bundle $(E,\backslash ,\; B,\backslash ,\; \backslash pi,\backslash ,\; F)$ is often denoted $$\backslash begin\{matrix\}$$

 {} \\
 F \longrightarrow E\ \xrightarrow{\,\ \pi\ }\ B \\
 {}
     

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

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Examples

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Trivial bundle Let $E\; =\; B\; \backslash times\; F$ and let $\backslash pi\; :\; E\; \backslash to\; B$ be the projection onto the first factor. Then $\backslash pi$ is a fiber bundle (of $F$) over $B.$ Here $E$ is not just locally a product but globally one. Any such fiber bundle is called a . Any fiber bundle over a contractible CW-complex is trivial.

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Nontrivial bundles

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Möbius strip Perhaps the simplest example of a nontrivial bundle $E$ is the Möbius strip. It has the circle that runs lengthwise along the center of the strip as a base $B$ and a line segment for the fiber $F$, so the Möbius strip is a bundle of the line segment over the circle. A neighborhood $U$ of $\backslash pi(x)\; \backslash in\; B$ (where $x\; \backslash in\; E$) is an circular arc; in the picture, this is the length of one of the squares. The preimage $\backslash pi^\{-1\}(U)$ in the picture is a (somewhat twisted) slice of the strip four squares wide and one long (i.e. all the points that project to $U$).

A homeomorphism ($\backslash varphi$ in ) exists that maps the preimage of $U$ (the trivializing neighborhood) to a slice of a cylinder: curved, but not twisted. This pair locally trivializes the strip. The corresponding trivial bundle $B\backslash times\; F$ would be a cylinder, but the Möbius strip has an overall "twist". This twist is visible only globally; locally the Möbius strip and the cylinder are identical (making a single vertical cut in either gives the same space).

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Klein bottle A similar nontrivial bundle is the Klein bottle, which can be viewed as a "twisted" circle bundle over another circle. The corresponding non-twisted (trivial) bundle is the 2-torus, $S^1\; \backslash times\; S^1$.

in three-dimensional space.]]

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Covering map A covering map is a fiber bundle such that the bundle projection is a local homeomorphism. It follows that the fiber is a discrete space.

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Vector and principal bundles A special class of fiber bundles, called , are those whose fibers are (to qualify as a vector bundle the structure group of the bundle—see below—must be a linear group). Important examples of vector bundles include the tangent bundle and cotangent bundle of a smooth manifold. From any vector bundle, one can construct the frame bundle of bases, which is a principal bundle (see below).

Another special class of fiber bundles, called , are bundles on whose fibers a free action and transitive action by a group $G$ is given, so that each fiber is a principal homogeneous space. The bundle is often specified along with the group by referring to it as a principal $G$-bundle. The group $G$ is also the structure group of the bundle. Given a representation $\backslash rho$ of $G$ on a vector space $V$, a vector bundle with $\backslash rho(G)\; \backslash subseteq\; \backslash text\{Aut\}(V)$ as a structure group may be constructed, known as the associated bundle.

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Sphere bundles A sphere bundle is a fiber bundle whose fiber is an hypersphere. Given a vector bundle $E$ with a metric tensor (such as the tangent bundle to a Riemannian manifold) one can construct the associated unit sphere bundle, for which the fiber over a point $x$ is the set of all Unit vector in $E\_x$. When the vector bundle in question is the tangent bundle $TM$, the unit sphere bundle is known as the unit tangent bundle.

A sphere bundle is partially characterized by its Euler class, which is a degree $n\; +\; 1$ cohomology class in the total space of the bundle. In the case $n\; =\; 1$ the sphere bundle is called a circle bundle and the Euler class is equal to the first Chern class, which characterizes the topology of the bundle completely. For any $n$, given the Euler class of a bundle, one can calculate its cohomology using a long exact sequence called the Gysin sequence.

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Mapping tori If $X$ is a topological space and $f\; :\; X\; \backslash to\; X$ is a homeomorphism then the mapping torus $M\_f$ has a natural structure of a fiber bundle over the circle with fiber $X.$ Mapping tori of homeomorphisms of surfaces are of particular importance in 3-manifold topology.

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Quotient spaces If $G$ is a topological group and $H$ is a closed subgroup, then under some circumstances, the quotient space $G/H$ together with the quotient map $\backslash pi\; :\; G\; \backslash to\; G/H$ is a fiber bundle, whose fiber is the topological space $H$. A necessary and sufficient condition for ($G,\backslash ,\; G/H,\backslash ,\; \backslash pi,\backslash ,\; H$) to form a fiber bundle is that the mapping $\backslash pi$ admits local cross-sections .

The most general conditions under which the quotient map will admit local cross-sections are not known, although if $G$ is a Lie group and $H$ a closed subgroup (and thus a Lie subgroup by Cartan's theorem), then the quotient map is a fiber bundle. One example of this is the Hopf fibration, $S^3\; \backslash to\; S^2$, which is a fiber bundle over the sphere $S^2$ whose total space is $S^3$. From the perspective of Lie groups, $S^3$ can be identified with the special unitary group $SU(2)$. The Abelian group subgroup of diagonal matrices is Isomorphic group to the circle group $U(1)$, and the quotient $SU(2)/U(1)$ is Diffeomorphism to the sphere.

More generally, if $G$ is any topological group and $H$ a closed subgroup that also happens to be a Lie group, then $G\; \backslash to\; G/H$ is a fiber bundle.

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Sections A (or cross section) of a fiber bundle $\backslash pi$ is a continuous map $f\; :\; B\; \backslash to\; E$ such that $\backslash pi(f(x))\; =\; x$ 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 $f\; :\; U\; \backslash to\; E$ where U is an open set in B and $\backslash pi(f(x))\; =\; x$ for all x in U. If $(U,\backslash ,\; \backslash varphi)$ is a local trivialization chart then local sections always exist over U. Such sections are in 1-1 correspondence with continuous maps $U\; \backslash to\; F$. Sections form a sheaf.

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Structure groups and transition functions 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 Faithful action on F so that it may be thought of as a group of of F. A G-atlas for the bundle $(E,\; B,\; \backslash pi,\; F)$ is a set of local trivialization charts $\backslash \{(U\_k,\backslash ,\; \backslash varphi\_k)\backslash \}$ such that for any $\backslash varphi\_i,\backslash varphi\_j$ for the overlapping charts $(U\_i,\backslash ,\; \backslash varphi\_i)$ and $(U\_j,\backslash ,\; \backslash varphi\_j)$ the function $$\backslash varphi\_i\backslash varphi\_j^\{-1\}\; :\; \backslash left(U\_i\; \backslash cap\; U\_j\backslash right)\; \backslash times\; F\; \backslash to\; \backslash left(U\_i\; \backslash cap\; U\_j\backslash right)\; \backslash times\; F$$ is given by $$\backslash varphi\_i\backslash varphi\_j^\{-1\}(x,\backslash ,\; \backslash xi)\; =\; \backslash left(x,\backslash ,\; t\_\{ij\}(x)\backslash xi\backslash right)$$ where $t\_\{ij\}\; :\; U\_i\; \backslash cap\; U\_j\; \backslash to\; G$ is a continuous map called a '. 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 ' of the bundle. This is related to the gauge group in physics, which is the group of automorphisms of the principal G-bundle that leave the base space unchanged.

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 $t\_\{ij\}$ satisfy the following conditions

1. $t\_\{ii\}(x)\; =\; 1\backslash ,$
2. $t\_\{ij\}(x)\; =\; t\_\{ji\}(x)^\{-1\}\backslash ,$
3. $t\_\{ik\}(x)\; =\; t\_\{ij\}(x)t\_\{jk\}(x).\backslash ,$

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 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.

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Bundle maps It is useful to have notions of a mapping between two fiber bundles. Suppose that $M$ and $N$ are base spaces, and $\backslash pi\_E\; :\; E\; \backslash to\; M$ and $\backslash pi\_F\; :\; F\; \backslash to\; N$ are fiber bundles over $M$ and $N$, respectively. A ' or ' consists of a pair of continuousDepending on the category of spaces involved, the functions may be assumed to have properties other than continuity. For instance, in the category of differentiable manifolds, the functions are assumed to be smooth. In the category of algebraic varieties, they are regular morphisms. functions $$\backslash varphi\; :\; E\; \backslash to\; F,\backslash quad\; f\; :\; M\; \backslash to\; N$$ such that $\backslash pi\_F\backslash circ\; \backslash varphi\; =\; f\; \backslash circ\; \backslash pi\_E$. That is, the following diagram is commutative:

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

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

Assume that both $\backslash pi\_E\; :\; E\; \backslash to\; M$ and $\backslash pi\_F\; :\; F\; \backslash to\; M$ are defined over the same base space $M$. A bundle isomorphism is a bundle map $(\backslash varphi,\backslash ,\; f)$ between $\backslash pi\_E\; :\; E\; \backslash to\; M$ and $\backslash pi\_F\; :\; F\; \backslash to\; M$ such that $f\; \backslash equiv\; \backslash mathrm\{id\}\_M$ and such that $\backslash varphi$ is also a homeomorphism.Or is, at least, invertible in the appropriate category; e.g., a diffeomorphism.

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Differentiable fiber bundles In the category of differentiable manifolds, fiber bundles arise naturally as submersions of one manifold to another. Not every (differentiable) submersion $f\; :\; M\; \backslash to\; N$ 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 $(M,\; N,\; f)$ 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 space and connected, then any submersion $f\; :\; M\; \backslash to\; N$ gives rise to a fiber bundle in the sense that there is a fiber space F diffeomorphic to each of the fibers such that $(E,\; B,\; \backslash pi,\; F)\; =\; (M,\; N,\; f,\; F)$ is a fiber bundle. (Surjectivity of $f$ follows by the assumptions already given in this case.) More generally, the assumption of compactness can be relaxed if the submersion $f\; :\; M\; \backslash to\; N$ is assumed to be a surjective proper map, meaning that $f^\{-1\}(K)$ is compact for every compact subset K of N. Another sufficient condition, due to , is that if $f\; :\; M\; \backslash to\; N$ is a surjective submersion with M and N differentiable manifolds such that the preimage $f^\{-1\}\backslash \{x\backslash \}$ is compact and connected for all $x\; \backslash in\; N,$ then $f$ admits a compatible fiber bundle structure .

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Generalizations

• 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 $\backslash pi\; :\; E\; \backslash to\; B$ that has certain homotopy theory 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 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.

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See also

• Affine bundle
• Algebra bundle
• Characteristic class
• Covering map
• Equivariant bundle
• Fibered manifold
• Fibration
• Gauge theory
• Hopf bundle
• I-bundle
• Natural bundle
• Principal bundle
• Projective bundle
• Pullback bundle
• Quasifibration
• Universal bundle
• Vector bundle
• Wu–Yang dictionary

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Notes

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External links

• Fiber Bundle, PlanetMath
• Making John Robinson's Symbolic Sculpture `Eternity'
• Sardanashvily, Gennadi, Fibre bundles, jet manifolds and Lagrangian theory. Lectures for theoreticians,

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