Editing Section (fiber bundle) (section)

== 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.