View source for Section (fiber bundle)

{{Short description|Right inverse of a fiber bundle map}}
{{One source|date=July 2022}}
[[Image:Bundle section.pi.sigma.svg|right|thumb|A section <math>\sigma</math> of a bundle <math>\pi \colon E\to B</math>.  A section <math>\sigma</math> allows the base space <math>B</math> to be identified with a subspace <math>\sigma(B)</math> of <math>E</math>.]]
[[Image:Vector field.svg|right|thumb|A vector field on <math>\mathbb{R}^2</math>. A section of a [[tangent vector bundle]] is a vector field.]]
[[File:Vector bundle with section.png|thumb|300px|A vector bundle <math>E</math> over a base <math>M</math> with section <math>s</math>.]]

In the [[mathematical]] field of [[topology]], a '''section''' (or '''cross section''')<ref>{{citation|first=Dale|last=Husemöller|authorlink=Dale Husemoller| title=Fibre Bundles|publisher=Springer Verlag|year=1994|isbn=0-387-94087-1|page=12}}</ref> of a [[fiber bundle]] <math>E</math> is a continuous [[Inverse function#Left and right inverses|right inverse]] of the [[Projection (mathematics)|projection function]] <math>\pi</math>. In other words, if <math>E</math>  is a fiber bundle over a [[topological space|base space]], <math>B</math>:
:<math> \pi \colon E \to B</math>
then a section of that fiber bundle is a [[continuous map]],
:<math> \sigma \colon B \to E </math>
such that
:<math> \pi(\sigma(x)) = x </math> for all <math>x \in B </math>.

A section is an abstract characterization of what it means to be a [[Graph of a function|graph]]. The graph of a function <math> g\colon B \to Y </math> can be identified with a function taking its values in the [[Cartesian product]] <math> E = B \times Y </math>, of <math> B </math> and <math> Y </math>:
:<math>\sigma\colon B\to E, \quad \sigma(x) = (x,g(x)) \in E. </math>

Let <math> \pi\colon E \to B </math> be the projection onto the first factor: <math> \pi(x,y) = x </math>.  Then a graph is any function <math> \sigma </math> for which <math> \pi(\sigma(x)) = x </math>.

The language of fibre bundles allows this notion of a section to be generalized to the case when <math>E</math> is not necessarily a Cartesian product.  If <math> \pi\colon E \to B </math> is a fibre bundle, then a section is a choice of point <math> \sigma(x) </math> in each of the fibres.  The condition <math> \pi(\sigma(x)) = x </math> simply means that the section at a point <math> x </math>  must lie over <math> x </math>.  (See image.)

For example, when <math>E</math> is a [[vector bundle]] a section of <math>E</math> is an element of the vector space <math> E_x </math> lying over each point <math>x \in B</math>. In particular, a [[vector field]] on a [[smooth manifold]] <math>M</math> is a choice of [[tangent vector]] at each point of <math>M</math>: this is a ''section'' of the [[tangent bundle]] of <math>M</math>. Likewise, a [[1-form]] on <math>M</math> is a section of the [[cotangent bundle]].

Sections, particularly of [[Principal bundle|principal bundles]] and vector bundles, are also very important tools in [[differential geometry]]. In this setting, the base space <math>B</math> is a [[smooth manifold]] <math>M</math>, and <math>E</math> is assumed to be a smooth fiber bundle over <math>M</math> (i.e., <math>E</math> is a smooth manifold and <math>\pi\colon E\to M</math> is a [[smooth map]]). In this case, one considers the space of '''smooth sections''' of <math>E</math> over an open set <math>U</math>, denoted <math>C^{\infty}(U,E)</math>. It is also useful in [[geometric analysis]] to consider spaces of sections with intermediate regularity (e.g., <math>C^k</math> sections, or sections with regularity in the sense of [[Hölder condition]]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 <math>S^1</math> with fiber <math>F = \mathbb{R} \setminus \{0\}</math> obtained by taking the [[Möbius strip|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 <math>s \colon U \to E</math> where <math>U</math> is an [[open set]] in <math>B</math> and <math>\pi(s(x))=x</math> for all <math>x</math> in <math>U</math>. If <math>(U, \varphi)</math> is a [[local trivialization]] of <math>E</math>, where <math>\varphi</math> is a homeomorphism from <math>\pi^{-1}(U)</math> to <math>U\times F</math> (where <math>F</math> is the [[fiber bundle|fiber]]), then local sections always exist over <math>U</math> in bijective correspondence with continuous maps from <math>U</math> to <math>F</math>. The (local) sections form a [[sheaf (mathematics)|sheaf]] over <math>B</math> called the '''sheaf of sections''' of <math>E</math>.

The space of continuous sections of a fiber bundle <math>E</math> over <math>U</math> is sometimes denoted <math>C(U,E)</math>, while the space of global sections of <math>E</math> is often denoted <math>\Gamma(E)</math> or <math>\Gamma(B,E)</math>.

=== Extending to global sections ===

Sections are studied in [[homotopy theory]] and [[algebraic topology]], where one of the main goals is to account for the existence or non-existence of '''global sections'''. An [[Obstruction theory|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 class]]es, which are cohomological classes. For example, a [[principal bundle]] has a global section if and only if it is [[trivial bundle|trivial]]. On the other hand, a [[vector bundle]] always has a global section, namely the [[zero section]]. However, it only admits a nowhere vanishing section if its [[Euler class]] is zero.

==== Generalizations ====

Obstructions to extending local sections may be generalized in the following manner: take a [[topological space]] and form a [[Category (mathematics)|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]]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 class]]es generalizes the idea of obstructions to our extensions.

== See also ==
* [[Section (category theory)]]
* [[Fibration]]
* [[Gauge theory (mathematics)]]
* [[Principal bundle]]
* [[Pullback bundle]]
* [[Vector bundle]]

== Notes ==
<references/>

== References ==
* [[Norman Steenrod]], ''The Topology of Fibre Bundles'', Princeton University Press (1951). {{ISBN|0-691-00548-6}}.
* David Bleecker, ''Gauge Theory and Variational Principles'', Addison-Wesley publishing, Reading, Mass (1981). {{ISBN|0-201-10096-7}}.
* {{citation|first=Dale|last=Husemöller|authorlink=Dale Husemoller|title=Fibre Bundles|publisher=Springer Verlag|year=1994|isbn=0-387-94087-1}}

== External links ==
* [https://planetmath.org/fiberbundle Fiber Bundle], PlanetMath
* {{MathWorld|urlname=FiberBundle|title=Fiber Bundle}}

[[Category:Fiber bundles| ]]
[[Category:Differential topology]]
[[Category:Algebraic topology]]
[[Category:Homotopy theory]]