![Vector bundle with section](https://upload.wikimedia.org/wikipedia/commons/8/8b/Vector_bundle_with_section.png)
In the
mathematical
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
field of
topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
, a section (or cross section)
of a
fiber bundle is a continuous
right inverse of the
projection function In set theory, a projection is one of two closely related types of functions or operations, namely:
* A set-theoretic operation typified by the ''j''th projection map, written \mathrm_, that takes an element \vec = (x_1,\ \ldots,\ x_j,\ \ldots,\ x ...
. In other words, if
is a fiber bundle over a
base space,
:
:
then a section of that fiber bundle is a
continuous map
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in valu ...
,
:
such that
:
for all
.
A section is an abstract characterization of what it means to be a
graph
Graph may refer to:
Mathematics
*Graph (discrete mathematics), a structure made of vertices and edges
**Graph theory, the study of such graphs and their properties
*Graph (topology), a topological space resembling a graph in the sense of discre ...
. The graph of a function
can be identified with a function taking its values in the
Cartesian product , of
and
:
:
Let
be the projection onto the first factor:
. Then a graph is any function
for which
.
The language of fibre bundles allows this notion of a section to be generalized to the case when
is not necessarily a Cartesian product. If
is a fibre bundle, then a section is a choice of point
in each of the fibres. The condition
simply means that the section at a point
must lie over
. (See image.)
For example, when
is a
vector bundle
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every p ...
a section of
is an element of the vector space
lying over each point
. In particular, a
vector field on a
smooth manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
is a choice of
tangent vector
In mathematics, a tangent vector is a vector that is tangent to a curve or surface at a given point. Tangent vectors are described in the differential geometry of curves in the context of curves in R''n''. More generally, tangent vectors are e ...
at each point of
: this is a ''section'' of the
tangent bundle
In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and of ...
of
. Likewise, a
1-form
In differential geometry, a one-form on a differentiable manifold is a smooth section of the cotangent bundle. Equivalently, a one-form on a manifold M is a smooth mapping of the total space of the tangent bundle of M to \R whose restriction to ...
on
is a section of the
cotangent bundle.
Sections, particularly of
principal bundles
In mathematics, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product X \times G of a space X with a group G. In the same way as with the Cartesian product, a principal bundle P is equi ...
and vector bundles, are also very important tools in
differential geometry. In this setting, the base space
is a
smooth manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
, and
is assumed to be a smooth fiber bundle over
(i.e.,
is a smooth manifold and
is a
smooth map
In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if ...
). In this case, one considers the space of smooth sections of
over an open set
, denoted
. It is also useful in
geometric analysis
Geometric analysis is a mathematical discipline where tools from differential equations, especially elliptic partial differential equations (PDEs), are used to establish new results in differential geometry and differential topology. The use of ...
to consider spaces of sections with intermediate regularity (e.g.,
sections, or sections with regularity in the sense of
Hölder condition
In mathematics, a real or complex-valued function ''f'' on ''d''-dimensional Euclidean space satisfies a Hölder condition, or is Hölder continuous, when there are nonnegative real constants ''C'', α > 0, such that
: , f(x) - f(y) , \leq C\ ...
s or
Sobolev spaces
In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of ''Lp''-norms of the function together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense ...
).
Local and global sections
Fiber bundles do not in general have such ''global'' sections (consider, for example, the fiber bundle over
with fiber
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
where
is an
open set
In mathematics, open sets are a generalization of open intervals in the real line.
In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are su ...
in
and
for all
in
. If
is a
local trivialization
In mathematics, and particularly topology, a fiber bundle (or, in 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 ...
of
, where
is a homeomorphism from
to
(where
is the
fiber
Fiber or fibre (from la, fibra, links=no) is a natural or artificial substance that is significantly longer than it is wide. Fibers are often used in the manufacture of other materials. The strongest engineering materials often incorpora ...
), then local sections always exist over
in bijective correspondence with continuous maps from
to
. The (local) sections form a
sheaf
Sheaf may refer to:
* Sheaf (agriculture), a bundle of harvested cereal stems
* Sheaf (mathematics), a mathematical tool
* Sheaf toss, a Scottish sport
* River Sheaf, a tributary of River Don in England
* ''The Sheaf'', a student-run newspaper se ...
over
called the sheaf of sections of
.
The space of continuous sections of a fiber bundle
over
is sometimes denoted
, while the space of global sections of
is often denoted
or
.
Extending to global sections
Sections are studied in
homotopy theory and
algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...
, where one of the main goals is to account for the existence or non-existence of global sections. An
obstruction
Obstruction may refer to:
Places
* Obstruction Island, in Washington state
* Obstruction Islands, east of New Guinea
Medicine
* Obstructive jaundice
* Obstructive sleep apnea
* Airway obstruction, a respiratory problem
** Recurrent airway o ...
denies the existence of global sections since the space is too "twisted". More precisely, obstructions "obstruct" the possibility of extending a local section to a global section due to the space's "twistedness". Obstructions are indicated by particular
characteristic class
In mathematics, a characteristic class is a way of associating to each principal bundle of ''X'' a cohomology class of ''X''. The cohomology class measures the extent the bundle is "twisted" and whether it possesses sections. Characteristic classes ...
es, which are cohomological classes. For example, a
principal bundle
In mathematics, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product X \times G of a space X with a group G. In the same way as with the Cartesian product, a principal bundle P is equi ...
has a global section if and only if it is
trivial
Trivia is information and data that are considered to be of little value. It can be contrasted with general knowledge and common sense.
Latin Etymology
The ancient Romans used the word ''triviae'' to describe where one road split or forked ...
. On the other hand, a
vector bundle
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every p ...
always has a global section, namely the
zero section
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every po ...
. However, it only admits a nowhere vanishing section if its
Euler class
In mathematics, specifically in algebraic topology, the Euler class is a characteristic class of oriented, real vector bundles. Like other characteristic classes, it measures how "twisted" the vector bundle is. In the case of the tangent bundle o ...
is zero.
Generalizations
Obstructions to extending local sections may be generalized in the following manner: take a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
and form a
category
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization, categories in cognitive science, information science and generally
* Category of being
* ''Categories'' (Aristotle)
* Category (Kant)
* Categories (Peirce) ...
whose objects are open subsets, and morphisms are inclusions. Thus we use a category to generalize a topological space. We generalize the notion of a "local section" using sheaves of
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
s, which assigns to each object an abelian group (analogous to local sections).
There is an important distinction here: intuitively, local sections are like "vector fields" on an open subset of a topological space. So at each point, an element of a ''fixed'' vector space is assigned. However, sheaves can "continuously change" the vector space (or more generally abelian group).
This entire process is really the
global section functor
In mathematics, a sheaf is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the data could ...
, which assigns to each sheaf its global section. Then
sheaf cohomology enables us to consider a similar extension problem while "continuously varying" the abelian group. The theory of
characteristic class
In mathematics, a characteristic class is a way of associating to each principal bundle of ''X'' a cohomology class of ''X''. The cohomology class measures the extent the bundle is "twisted" and whether it possesses sections. Characteristic classes ...
es generalizes the idea of obstructions to our extensions.
See also
*
Fibration
*
Gauge theory
*
Principal bundle
In mathematics, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product X \times G of a space X with a group G. In the same way as with the Cartesian product, a principal bundle P is equi ...
*
Pullback bundle In mathematics, a pullback bundle or induced bundle is the fiber bundle that is induced by a map of its base-space. Given a fiber bundle and a continuous map one can define a "pullback" of by as a bundle over . The fiber of over a point in ...
*
Vector bundle
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every p ...
Notes
References
*
Norman Steenrod
Norman Earl Steenrod (April 22, 1910October 14, 1971) was an American mathematician most widely known for his contributions to the field of algebraic topology.
Life
He was born in Dayton, Ohio, and educated at Miami University and University of ...
, ''The Topology of Fibre Bundles'', Princeton University Press (1951). .
* David Bleecker, ''Gauge Theory and Variational Principles'', Addison-Wesley publishing, Reading, Mass (1981). .
*
External links
Fiber Bundle PlanetMath
* {{MathWorld, urlname=FiberBundle, title=Fiber Bundle
Differential topology
Algebraic topology
Homotopy theory