05-12-2023-1633 - stereographic projection, ternary plot, section, fiber bundle, bundle, graph embedding, rotation system, plane curve, axiom, entropy, stochastic, universe, convex polytope, subset, empty set, clopen set, chordal graph, induced path, connected space, planar graph, topological space, connected set, surjective function, domain, restricting, right inverse, composition, function, image, unique, codomain, sphere bundle, natural geometric setting, electromagnetism, circle bundle, orientable, space, bottle, over, twist, reversal, orientation, loop, base, alexander trick, dimension, vector bundle, fibered manifold, i-bundle, natural bundle, principle bundle, universal bundle, affine bundle, algebra bundle, homotopy lifting property, 1950, homogenous space, torsor, submersion, dependent, fibration, property, output range, continuous, input, notion, connected, proper map, compactness, assumptions, bundle morphism, cocycle condition, cohomology, g-bundle, regular, left action, transitive, obtain right action, structure group, transition function, g-atlas, equivalence, class, gauge group, lie group, smooth category, smooth map, homeomorphism, ebf, local trivialization, chart, overlapping, topological group, acts continuously, fiber space, lose nothing faith, sheaf, hairy ball theorem, euler class, tangent bundle, 2-sphere, algebraic topology, characteristic classes, obstruction, existence, section, measure, cohomology class, cross section, continuous map, general, defined, sections, account, correspondence, cartan theorem, special unitary group, diffeomorphic to the sphere, circle group, quotient, infinite, divisibility, 3-manifold topology, quotient map, local cross-sections, closed subgroup, quotient space, total space, abelian subgroup, diagonal matrix, matrix, isomorphic, mapping torus, spinor, long exact sequence, ring sphere, gysin sequence, chern class, unit tangent bundle, riemannian manifold, n-sphere, metric, mapping tori, representation, association, bundle, transitive action, free, covering space, covering map, discrete space, bundle projection, 2-torus, non-twisted (trivial), preimage, four squares wide one long, line segment, mobius strip, arc, project, curve, contractible CW-complex, category, globally one, first factor, open map, projection, pi, commute, base space of bundle, connected b, fiber space, differential geometry, projection or submersion of the bundle, continuous surjective map, bristles, sheaf of sections, global sections, denoted, ringed space, zero section, homotropy theory, sheaf cohomology, obstruction theory, open set, etc. (draft)

https://en.wikipedia.org/wiki/Section_(fiber_bundle)

https://en.wikipedia.org/wiki/Fiber_bundle

https://en.wikipedia.org/wiki/Sheaf_(mathematics)

https://en.wikipedia.org/wiki/Open_set

https://en.wikipedia.org/wiki/Vector_bundle

https://en.wikipedia.org/wiki/Principal_bundle

https://en.wikipedia.org/wiki/Characteristic_class

https://en.wikipedia.org/wiki/Obstruction_theory

https://en.wikipedia.org/wiki/Homotopy_theory

https://en.wikipedia.org/wiki/Sheaf_cohomology

https://en.wikipedia.org/wiki/Ringed_space

https://en.wikipedia.org/wiki/Fibration

Fiber bundles do not in general have such global sections (consider, for example, the fiber bundle over ${\displaystyle S^1}$ with fiber ${\displaystyle F=\mathbb{R}\setminus \{0\}}$ obtained by taking the Möbius bundle and removing the zero section), so it is also useful to define sections only locally. A local section of a fiber bundle is a continuous map ${\displaystyle s:U\to E}$ where ${\displaystyle U}$ is an open set in ${\displaystyle B}$ and ${\displaystyle \pi (s(x))=x}$ for all ${\displaystyle x}$ in ${\displaystyle U}$. If ${\displaystyle (U,\phi )}$ is a local trivialization of ${\displaystyle E}$, where ${\displaystyle \phi }$ is a homeomorphism from ${\displaystyle {\pi }^{-1}(U)}$ to ${\displaystyle U\times F}$ (where ${\displaystyle F}$ is the fiber), then local sections always exist over ${\displaystyle U}$ in bijective correspondence with continuous maps from ${\displaystyle U}$ to ${\displaystyle F}$. The (local) sections form a sheaf over ${\displaystyle B}$ called the sheaf of sections of ${\displaystyle E}$.

The space of continuous sections of a fiber bundle ${\displaystyle E}$ over ${\displaystyle U}$ is sometimes denoted ${\displaystyle C(U,E)}$, while the space of global sections of ${\displaystyle E}$ is often denoted ${\displaystyle \mathrm{\Gamma }(E)}$ or ${\displaystyle \mathrm{\Gamma }(B,E)}$. 

https://en.wikipedia.org/wiki/Section_(fiber_bundle)

Not to be confused with an optical fiber bundle.

A cylindrical hairbrush showing the intuition behind the term fiber bundle. This hairbrush is like a fiber bundle in which the base space is a cylinder and the fibers (bristles) are line segments. The mapping ${\displaystyle \pi :E\to B}$ would take a point on any bristle and map it to its root on the cylinder.

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

In the trivial case, ${\displaystyle E}$ is just ${\displaystyle B\times F,}$ and the map ${\displaystyle \pi }$ 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 covering spaces. Fiber bundles, such as the tangent bundle of a manifold and other more general vector bundles, play an important role in differential geometry and differential topology, as do principal bundles.

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

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,^[1][2][3] 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 ^[4] under the name sphere space, but in 1940 Whitney changed the name to sphere bundle.^[5]

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,^[6] Whitney, Norman Steenrod, Charles Ehresmann,^[7][8][9] Jean-Pierre Serre,^[10] 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.^[11]

Whitney came to the general definition of a fiber bundle from his study of a more particular notion of a sphere bundle,^[12] that is a fiber bundle whose fiber is a sphere of arbitrary dimension.^[13]

 https://en.wikipedia.org/wiki/Fiber_bundle

Formal definition

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 ${\displaystyle \phi :{\pi }^{-1}(U)\to U\times F}$ (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:

where ${\displaystyle {\mathrm{proj}}_1:U\times F\to U}$ is the natural projection and ${\displaystyle \phi :{\pi }^{-1}(U)\to U\times F}$ is a homeomorphism. The set of all ${\displaystyle \left\{\left(U_i,{\phi }_i\right)\right\}}$ 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 ${\displaystyle {\mathrm{proj}}_1^{-1}(\{p\})}$) 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

${\displaystyle \begin{array}{c}\\ F⟶E\stackrel{\pi }{\to }B\\ \end{array}}$

(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.

Examples

Trivial bundle

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

Nontrivial bundles

Möbius strip

The Möbius strip is a nontrivial bundle over the circle.

Perhaps the simplest example of a nontrivial bundle ${\displaystyle E}$ is the Möbius strip. It has the circle that runs lengthwise along the center of the strip as a base ${\displaystyle B}$ and a line segment for the fiber ${\displaystyle F}$, so the Möbius strip is a bundle of the line segment over the circle. A neighborhood ${\displaystyle U}$ of ${\displaystyle \pi (x)\in B}$ (where ${\displaystyle x\in E}$) is an arc; in the picture, this is the length of one of the squares. The preimage ${\displaystyle {\pi }^{-1}(U)}$ 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 ${\displaystyle U}$).

A homeomorphism (${\displaystyle \phi }$ in § Formal definition) exists that maps the preimage of ${\displaystyle U}$ (the trivializing neighborhood) to a slice of a cylinder: curved, but not twisted. This pair locally trivializes the strip. The corresponding trivial bundle ${\displaystyle B\times F}$ 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).

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, ${\displaystyle S^1\times S^1}$.

The Klein bottle immersed in three-dimensional space.

A torus.

Covering map

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

Vector and principal bundles

A special class of fiber bundles, called vector bundles, are those whose fibers are vector spaces (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 principal bundles, are bundles on whose fibers a free and transitive action by a group ${\displaystyle G}$ 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 ${\displaystyle G}$-bundle. The group ${\displaystyle G}$ is also the structure group of the bundle. Given a representation ${\displaystyle \rho }$ of ${\displaystyle G}$ on a vector space ${\displaystyle V}$, a vector bundle with ${\displaystyle \rho (G)\subseteq \text{Aut}(V)}$ as a structure group may be constructed, known as the associated bundle.

Sphere bundles

Main article: Sphere bundle

A sphere bundle is a fiber bundle whose fiber is an n-sphere. Given a vector bundle ${\displaystyle E}$ with a metric (such as the tangent bundle to a Riemannian manifold) one can construct the associated unit sphere bundle, for which the fiber over a point ${\displaystyle x}$ is the set of all unit vectors in ${\displaystyle E_x}$. When the vector bundle in question is the tangent bundle ${\displaystyle TM}$, 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 ${\displaystyle n+1}$ cohomology class in the total space of the bundle. In the case ${\displaystyle n=1}$ 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 ${\displaystyle n}$, given the Euler class of a bundle, one can calculate its cohomology using a long exact sequence called the Gysin sequence.

See also: Wang sequence

Mapping tori

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

Quotient spaces

If ${\displaystyle G}$ is a topological group and ${\displaystyle H}$ is a closed subgroup, then under some circumstances, the quotient space ${\displaystyle G/H}$ together with the quotient map ${\displaystyle \pi :G\to G/H}$ is a fiber bundle, whose fiber is the topological space ${\displaystyle H}$. A necessary and sufficient condition for (${\displaystyle G,G/H,\pi ,H}$) to form a fiber bundle is that the mapping ${\displaystyle \pi }$ admits local cross-sections (Steenrod 1951, §7).

The most general conditions under which the quotient map will admit local cross-sections are not known, although if ${\displaystyle G}$ is a Lie group and ${\displaystyle H}$ 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, ${\displaystyle S^3\to S^2}$, which is a fiber bundle over the sphere ${\displaystyle S^2}$ whose total space is ${\displaystyle S^3}$. From the perspective of Lie groups, ${\displaystyle S^3}$ can be identified with the special unitary group ${\displaystyle SU(2)}$. The abelian subgroup of diagonal matrices is isomorphic to the circle group ${\displaystyle U(1)}$, and the quotient ${\displaystyle SU(2)/U(1)}$ is diffeomorphic to the sphere.

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

Sections

Main article: Section (fiber bundle)

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 ${\displaystyle (U,\phi )}$ 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.

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 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 ${\displaystyle \{(U_k,{\phi }_k)\}}$ such that for any ${\displaystyle {\phi }_i,{\phi }_j}$ for the overlapping charts ${\displaystyle (U_i,{\phi }_i)}$ and ${\displaystyle (U_j,{\phi }_j)}$ the function

$${\displaystyle {\phi }_i{\phi }_j^{-1}:\left(U_i\cap U_j\right)\times F\to \left(U_i\cap U_j\right)\times F}$$

is given by

$${\displaystyle {\phi }_i{\phi }_j^{-1}(x,\xi )=\left(x,t_{ij}(x)\xi \right)}$$

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.

Bundle maps

Main article: Bundle map

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^[14] functions

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

such that ${\displaystyle {\pi }_F\circ \phi =f\circ {\pi }_E.}$ 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 morphisms are also required to be G-equivariant on the fibers. This means that ${\displaystyle \phi :E\to F}$ is also G-morphism from one G-space to another, that is, ${\displaystyle \phi (xs)=\phi (x)s}$ 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 ${\displaystyle \phi :E\to F}$ such that ${\displaystyle {\pi }_E={\pi }_F\circ \phi .}$ This means that the bundle map ${\displaystyle \phi :E\to F}$ covers the identity of M. That is, ${\displaystyle f\equiv {\mathrm{i}\mathrm{d}}_M}$ and the following 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 ${\displaystyle (\phi ,f)}$ between ${\displaystyle {\pi }_E:E\to M}$ and ${\displaystyle {\pi }_F:F\to M}$ such that ${\displaystyle f\equiv {\mathrm{i}\mathrm{d}}_M}$ and such that ${\displaystyle \phi }$ is also a homeomorphism.^[15]

Differentiable fiber bundles

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), 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).

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 ${\displaystyle \pi :E\to B}$ 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.

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

Notes

1. 
   

Seifert, Herbert (1933). "Topologie dreidimensionaler gefaserter Räume". Acta Mathematica. 60: 147–238. doi:10.1007/bf02398271.

"Topologie Dreidimensionaler Gefaserter Räume" on Project Euclid.

Seifert, H. (1980). Seifert and Threlfall, A textbook of topology. W. Threlfall, Joan S. Birman, Julian Eisner. New York: Academic Press. ISBN 0-12-634850-2. OCLC 5831391.

Whitney, Hassler (1935). "Sphere spaces". Proceedings of the National Academy of Sciences of the United States of America. 21 (7): 464–468. Bibcode:1935PNAS...21..464W. doi:10.1073/pnas.21.7.464. PMC 1076627. PMID 16588001.

Whitney, Hassler (1940). "On the theory of sphere bundles". Proceedings of the National Academy of Sciences of the United States of America. 26 (2): 148–153. Bibcode:1940PNAS...26..148W. doi:10.1073/pnas.26.2.148. PMC 1078023. PMID 16588328.

Feldbau, Jacques (1939). "Sur la classification des espaces fibrés". Comptes rendus de l'Académie des Sciences. 208: 1621–1623.

Ehresmann, Charles (1947). "Sur la théorie des espaces fibrés". Coll. Top. Alg. Paris. C.N.R.S.: 3–15.

Ehresmann, Charles (1947). "Sur les espaces fibrés différentiables". Comptes rendus de l'Académie des Sciences. 224: 1611–1612.

Ehresmann, Charles (1955). "Les prolongements d'un espace fibré différentiable". Comptes rendus de l'Académie des Sciences. 240: 1755–1757.

Serre, Jean-Pierre (1951). "Homologie singulière des espaces fibrés. Applications". Annals of Mathematics. 54 (3): 425–505. doi:10.2307/1969485. JSTOR 1969485.

See Steenrod (1951, Preface)

In his early works, Whitney referred to the sphere bundles as the "sphere-spaces". See, for example:

• Whitney, Hassler (1935). "Sphere spaces". Proc. Natl. Acad. Sci. 21 (7): 462–468. Bibcode:1935PNAS...21..464W. doi:10.1073/pnas.21.7.464. PMC 1076627. PMID 16588001.
• Whitney, Hassler (1937). "Topological properties of differentiable manifolds". Bull. Amer. Math. Soc. 43 (12): 785–805. doi:10.1090/s0002-9904-1937-06642-0.

Whitney, Hassler (1940). "On the theory of sphere bundles" (PDF). Proc. Natl. Acad. Sci. 26 (2): 148–153. Bibcode:1940PNAS...26..148W. doi:10.1073/pnas.26.2.148. PMC 1078023. PMID 16588328.

Depending 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.

15. Or is, at least, invertible in the appropriate category; e.g., a diffeomorphism.

References

• 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 (1951). "Les connexions infinitésimales dans un espace fibré différentiable". Colloque de Topologie (Espaces fibrés), Bruxelles, 1950. Paris: Georges Thone, Liège; Masson et Cie. 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

External links

• 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

v t e Tensors

v t e Manifolds (Glossary)

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Category:

• Fiber bundles

 https://en.wikipedia.org/wiki/Fiber_bundle

 

In the mathematical field of topology, a sphere bundle is a fiber bundle in which the fibers are spheres ${\displaystyle S^n}$ of some dimension n.^[1] Similarly, in a disk bundle, the fibers are disks ${\displaystyle D^n}$. From a topological perspective, there is no difference between sphere bundles and disk bundles: this is a consequence of the Alexander trick, which implies ${\displaystyle \mathrm{BTop}(D^{n+1})\simeq \mathrm{BTop}(S^n).}$

An example of a sphere bundle is the torus, which is orientable and has ${\displaystyle S^1}$ fibers over an ${\displaystyle S^1}$ base space. The non-orientable Klein bottle also has ${\displaystyle S^1}$ fibers over an ${\displaystyle S^1}$ base space, but has a twist that produces a reversal of orientation as one follows the loop around the base space.^[1]

A circle bundle is a special case of a sphere bundle. 

https://en.wikipedia.org/wiki/Sphere_bundle

In mathematics, a circle bundle is a fiber bundle where the fiber is the circle ${\displaystyle S^1}$.

Oriented circle bundles are also known as principal U(1)-bundles. In physics, circle bundles are the natural geometric setting for electromagnetism. A circle bundle is a special case of a sphere bundle. 

https://en.wikipedia.org/wiki/Circle_bundle

 

 

In mathematics, a surjective function (also known as surjection, or onto function /ˈɒn.tuː/) is a function f such that every element y can be mapped from element x so that f(x) = y. In other words, every element of the function's codomain is the image of at least one element of its domain.^[1][2] It is not required that x be unique; the function f may map one or more elements of X to the same element of Y.

The term surjective and the related terms injective and bijective were introduced by Nicolas Bourbaki,^[3][4] a group of mainly French 20th-century mathematicians who, under this pseudonym, wrote a series of books presenting an exposition of modern advanced mathematics, beginning in 1935. The French word sur means over or above, and relates to the fact that the image of the domain of a surjective function completely covers the function's codomain.

Any function induces a surjection by restricting its codomain to the image of its domain. Every surjective function has a right inverse assuming the axiom of choice, and every function with a right inverse is necessarily a surjection. The composition of surjective functions is always surjective. Any function can be decomposed into a surjection and an injection.

https://en.wikipedia.org/wiki/Surjective_function

In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets. Connectedness is one of the principal topological properties that are used to distinguish topological spaces.

A subset of a topological space ${\displaystyle X}$ is a connected set if it is a connected space when viewed as a subspace of ${\displaystyle X}$.

Some related but stronger conditions are path connected, simply connected, and ${\displaystyle n}$-connected. Another related notion is locally connected, which neither implies nor follows from connectedness. 

https://en.wikipedia.org/wiki/Connected_space

https://en.wikipedia.org/wiki/Topological_space 

https://en.wikipedia.org/wiki/Planar_graph

https://en.wikipedia.org/wiki/Chordal_graph

https://en.wikipedia.org/wiki/Induced_path

https://en.wikipedia.org/wiki/Connected_space

https://en.wikipedia.org/wiki/Section_(fiber_bundle)

https://en.wikipedia.org/wiki/Ternary_plot

https://en.wikipedia.org/wiki/Stereographic_projection

https://en.wikipedia.org/wiki/Rotation_system

https://en.wikipedia.org/wiki/Plane_curve

https://en.wikipedia.org/wiki/Graph_embedding

https://en.wikipedia.org/wiki/Axiom#Non-logical_axioms

https://en.wikipedia.org/wiki/Convex_polytope

https://en.wikipedia.org/wiki/Subset

https://en.wikipedia.org/wiki/Empty_set

https://en.wikipedia.org/wiki/Clopen_set