In the
mathematical 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
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 an ...
,
:
:
then a section of that fiber bundle is a
continuous map,
:
such that
:
for all
.
A section is an abstract characterization of what it means to be a
graph. The graph of a function
can be identified with a function taking its values in the
Cartesian product
In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is
: A\t ...
, 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 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 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
In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold. It may be described also as the dual bundle to the tangent bundle. Th ...
.
Sections, particularly of
principal bundles and vector bundles, are also very important tools in
differential geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and mult ...
. In this setting, the base space
is a
smooth manifold , and
is assumed to be a smooth fiber bundle over
(i.e.,
is a smooth manifold and
is a
smooth map). In this case, one considers the space of smooth sections of
over an open set
, denoted
. It is also useful in
geometric analysis 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).
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 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 incorporate ...
), then local sections always exist over
in bijective correspondence with continuous maps from
to
. The (local) sections form a
sheaf 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 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 classes, which are cohomological classes. For example, a
principal bundle 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 fork ...
. 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. 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 ...
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 poin ...
and form a
category 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, 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 classes generalizes the idea of obstructions to our extensions.
See also
*
Fibration
*
Gauge theory
In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations ( Lie grou ...
*
Principal bundle
*
Pullback bundle
*
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, ''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