Fiber bundle explained
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
and a product space
is defined using a continuous
surjective map,
that in small regions of
behaves just like a projection from corresponding regions of
to
The map
called the
projection or
submersion of the bundle, is regarded as part of the structure of the bundle. The space
is known as the
total space of the fiber bundle,
as the
base space, and
the
fiber.
In the trivial case,
is just
and the map
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
is called a
section of
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
.
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]
Formal definition
A fiber bundle is a structure
where
and
are
topological spaces and
is a continuous
surjection satisfying a
local triviality condition outlined below. The space
is called the
of the bundle,
the
, and
the
. The map
is called the
(or
). We shall assume in what follows that the base space
is
connected.
We require that for every
, there is an open
neighborhood
of
(which will be called a trivializing neighborhood) such that there is a
homeomorphism
(where
is given the
subspace topology, and
is the product space) in such a way that
agrees with the projection onto the first factor. That is, the following diagram should
commute:
Local triviality condition|230px|center
where
\operatorname{proj}1:U x F\toU
is the natural projection and
is a homeomorphism. The
set of all
\left\{\left(Ui,\varphii\right)\right\}
is called a
of the bundle.
Thus for any
, the preimage
is homeomorphic to
(since this is true of
) and is called the
fiber over
Every fiber bundle
is an
open map, since projections of products are open maps. Therefore
carries the
quotient topology determined by the map
A fiber bundle
is often denoted 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 smooth manifolds. That is,
and
are required to be smooth manifolds and all the
functions above are required to be
smooth maps.
Examples
Trivial bundle
Let
and let
be the projection onto the first factor. Then
is a fiber bundle (of
) over
Here
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.
Nontrivial bundles
Möbius strip
Perhaps the simplest example of a nontrivial bundle
is the
Möbius strip. It has the
circle that runs lengthwise along the center of the strip as a base
and a
line segment for the fiber
, so the Möbius strip is a bundle of the line segment over the circle. A
neighborhood
of
(where
) is an
arc; in the picture, this is the
length of one of the squares. The
preimage
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
).
A homeomorphism (
in) exists that maps the preimage of
(the trivializing neighborhood) to a slice of a cylinder: curved, but not twisted. This pair locally trivializes the strip. The corresponding trivial bundle
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,
.
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
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
-bundle. The group
is also the structure group of the bundle. Given a
representation
of
on a vector space
, a vector bundle with
as a structure group may be constructed, known as the
associated bundle.
Sphere bundles
See main article: Sphere bundle. A sphere bundle is a fiber bundle whose fiber is an n-sphere. Given a vector bundle
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
is the set of all
unit vectors in
. When the vector bundle in question is the tangent bundle
, 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
cohomology class in the total space of the bundle. In the case
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
, given the Euler class of a bundle, one can calculate its cohomology using a long exact sequence called the
Gysin sequence.
Mapping tori
If
is a
topological space and
is a
homeomorphism then the
mapping torus
has a natural structure of a fiber bundle over the
circle with fiber
Mapping tori of homeomorphisms of
surfaces are of particular importance in
3-manifold topology.
Quotient spaces
If
is a
topological group and
is a
closed subgroup, then under some circumstances, the
quotient space
together with the quotient map
is a fiber bundle, whose fiber is the topological space
. A
necessary and sufficient condition for (
) to form a fiber bundle is that the mapping
admits local cross-sections .
The most general conditions under which the quotient map will admit local cross-sections are not known, although if
is a
Lie group and
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,
, which is a fiber bundle over the sphere
whose total space is
. From the perspective of Lie groups,
can be identified with the
special unitary group
. The
abelian subgroup of
diagonal matrices is
isomorphic to the
circle group
, and the quotient
is
diffeomorphic to the sphere.
More generally, if
is any topological group and
a closed subgroup that also happens to be a Lie group, then
is a fiber bundle.
Sections
See main article: article and Section (fiber bundle).
A (or cross section) of a fiber bundle
is a continuous map
such that
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
where
U is an
open set in
B and
for all
x in
U. If
is a local trivialization
chart then local sections always exist over
U. Such sections are in
1-1 correspondence with continuous maps
. 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
is a set of local trivialization charts
such that for any
for the overlapping charts
and
the function
is given by
where
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; 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
satisfy the following conditions
The third condition applies on triple overlaps Ui ∩ Uj ∩ Uk 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
See main article: article and Bundle map. It is useful to have notions of a mapping between two fiber bundles. Suppose that M and N are base spaces, and
and
are fiber bundles over
M and
N, respectively. A
or
consists of a pair of continuous
[14] functions
such that
\piF\circ\varphi=f\circ\piE.
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
is also
G-morphism from one
G-space to another, that is,
for all
and
In case the base spaces M and N coincide, then a bundle morphism over M from the fiber bundle
to
is a map
such that
This means that the bundle map
covers the identity of
M. That is,
and the following diagram commutes:
Assume that both
and
are defined over the same base space
M. A bundle
isomorphism is a bundle map
between
and
such that
and such that
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
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
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
gives rise to a fiber bundle in the sense that there is a fiber space
F diffeomorphic to each of the fibers such that
is a fiber bundle. (Surjectivity of
follows by the assumptions already given in this case.) More generally, the assumption of compactness can be relaxed if the submersion
is assumed to be a surjective
proper map, meaning that
is compact for every compact
subset K of
N. Another sufficient condition, due to, is that if
is a surjective
submersion with
M and
N differentiable manifolds such that the preimage
is compact and connected for all
then
admits a compatible fiber bundle structure .
Generalizations
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 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
Notes
- Topologie dreidimensionaler gefaserter Räume. Herbert. Seifert. Herbert Seifert. Acta Mathematica. 60. 1933. 147–238. 10.1007/bf02398271. free.
- https://projecteuclid.org/euclid.acta/1485887992 "Topologie Dreidimensionaler Gefaserter Räume"
- Book: Seifert, H. . Seifert and Threlfall, A textbook of topology . 1980 . Academic Press . W. Threlfall, Joan S. Birman, Julian Eisner . 0-12-634850-2 . New York . 5831391.
- Sphere spaces. Hassler. Whitney. Hassler Whitney. Proceedings of the National Academy of Sciences of the United States of America. 21. 7. 1935. 464–468. 10.1073/pnas.21.7.464. free. 16588001. 1076627. 1935PNAS...21..464W.
- On the theory of sphere bundles. Hassler. Whitney. Hassler Whitney. . 26. 2. 1940. 148–153. 10.1073/pnas.26.2.148. 16588328. 1078023. 1940PNAS...26..148W. free.
- Sur la classification des espaces fibrés. Jacques. Feldbau. Jacques Feldbau. Comptes rendus de l'Académie des Sciences. 208. 1939. 1621–1623.
- Sur la théorie des espaces fibrés. Charles. Ehresmann. Charles Ehresmann . Coll. Top. Alg. Paris. C.N.R.S.. 1947. 3–15.
- Sur les espaces fibrés différentiables. Charles. Ehresmann. Charles Ehresmann . . 224. 1947. 1611–1612.
- Les prolongements d'un espace fibré différentiable. Charles. Ehresmann. Charles Ehresmann . Comptes rendus de l'Académie des Sciences. 240. 1955. 1755–1757.
- Homologie singulière des espaces fibrés. Applications. Jean-Pierre. Serre. Jean-Pierre Serre. . 54. 3. 1951. 425–505. 10.2307/1969485. 1969485.
- See
- In his early works, Whitney referred to the sphere bundles as the "sphere-spaces". See, for example:
- Whitney . Hassler . Hassler Whitney . Sphere spaces . Proc. Natl. Acad. Sci. . 21 . 7 . 462–468 . 1935 . 10.1073/pnas.21.7.464 . 16588001 . 1076627 . 1935PNAS...21..464W . free .
- Whitney . Hassler . Hassler Whitney . Topological properties of differentiable manifolds . Bull. Amer. Math. Soc. . 43 . 12 . 785–805 . 1937 . 10.1090/s0002-9904-1937-06642-0 . free .
- Whitney . Hassler . Hassler Whitney . On the theory of sphere bundles . Proc. Natl. Acad. Sci. . 26 . 2 . 148–153 . 1940 . 10.1073/pnas.26.2.148 . 16588328 . 1078023 . 1940PNAS...26..148W . free .
- 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.
- Or is, at least, invertible in the appropriate category; e.g., a diffeomorphism.
References
- Charles . Ehresmann . Charles Ehresmann. Les connexions infinitésimales dans un espace fibré différentiable . Colloque de Topologie (Espaces fibrés), Bruxelles, 1950 . Georges Thone, Liège; Masson et Cie. . Paris . 1951 . 29–55.
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