en>Noix07: /* Introductory example */

2014-10-13T15:27:32Z

Introductory example

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	~~[[Image:Bundle section.svg\|right\|thumb\|A section ''s'' of a bundle ''p'' : ''E'' → ''B''. A section ''s'' allows the base space ''B'' to be identified with a subspace ''s''(''B'') of ''E''.]]~~		Gabrielle Straub is what the person can call me although it's not the quite a few feminine of names. Fish keeping is what I follow every week. Managing people is my time job now. My house is now inside of South Carolina. Go to my web to find out more: http://circuspartypanama.com<br><br>my blog post :: how to hack clash of clans ([http://circuspartypanama.com click through the following page])
	~~[[Image:Vector field.svg\|right\|thumb\|A vector field in '''''R'''''<sup>2</sup>. A section of a [[tangent vector bundle]]~~ is ~~a vector field.]]~~

	In the [[mathematical]] field of [[topology]], a '''section''' (or '''cross section''')<ref>{{citation\|first=Dale\|last=Husemöller\|title=Fibre Bundles\|publisher=Springer Verlag\|year=1994\|isbn=0-387-94087-1\|page=12}}</ref> of a [[fiber bundle]] π is a continuous [[Inverse function#Left and right inverses\|right inverse]] of the function π. In other words, if ''π'' is a fiber bundle over a [[topological space\|base space]], ''B'':
	~~:''π'' : ''E'' → ''B''~~

	~~then a section of that fiber bundle is a [[continuous map]],~~
	~~:''s'' : ''B'' → ''E''~~

	~~such that~~
	~~:<math>\pi(s(x))=x</math> for all ''x'' in ''B''.~~

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

	~~Let π : ''E'' → ''X'' be the projection onto~~ the ~~first factor: π(''x'',''y'') = ''x''. Then a graph is any function ''s'' for which π(''s''(''x''))=''x''.~~

	The language of fibre bundles allows this notion of a section to be generalized to the case when ''E'' is not necessarily a Cartesian product. If π : ''E'' → ''B'' is a fibre bundle, then a section is a ~~choice of point ''s''(''x'') in each~~ of ~~the fibres~~. ~~The condition π(''s''(''x'')) = ''x'' simply means that the section at a point ''x'' must lie over ''x''. (See image.)~~

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

	~~Sections, particularly~~ of ~~principal bundles and vector bundles, are also very important tools in [[differential geometry]]~~. ~~In this setting, the base space ''B'' is a [[smooth manifold]] ''M'', and ''E'' is assumed~~ to ~~be a smooth fiber bundle over ''M'' (i.e., ''E'' is a smooth manifold and ''π''~~: ~~''E'' → ''M'' is a [[smooth map]]). In this case, one considers the space of '''smooth sections''' of ''E'' over an open set ''U'', denoted ''C''<sup>∞<~~/~~sup>(''U'',''E''). It is also useful in [[geometric analysis]] to consider spaces of sections with intermediate regularity (e.g., ''C''<sup>''k''<~~/~~sup> 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, so it is also useful to define sections only locally. A '''local section''' of a fiber bundle is a continuous map ''s'' : ''U'' → ''E'' where ''U'' is an [[open set]] in ''B'' and ''π''(''s''(''x'')) = ''x'' for all ''x'' in ''U''. ~~If (''U'', ''φ'') is a [[local trivialization]] of ''E'', where ''φ'' is a homeomorphism from ''π''~~<~~sup~~>−1<~~/sup~~>(''U'') to ''U'' × ''F'' (where ''F'' is the [[fiber bundle\|fiber]]), then local sections always exist over ''U'' in bijective correspondence with continuous maps from ''U'' to ''F''. The (local) sections form a [[sheaf (mathematics)\|sheaf]] over ''B'' called the '''sheaf of sections''' of ''E''.

	~~The space of continuous sections of a fiber bundle ''E'' over ''U'' is sometimes denoted ''C''(''U'',''E''), while the space of global sections of ''E'' is often denoted Γ(''E'') or Γ(''B'',''E'').~~

	~~=== 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 ==~~
	* [[Fibration]]
	* [[Gauge theory]]
	* [[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\|title=Fibre Bundles\|publisher=Springer Verlag\|year=1994\|isbn=0-387-94087-1}}

	~~== External links ==~~
	* [http://~~planetmath~~.~~org/encyclopedia/FiberBundle.html Fiber Bundle], PlanetMath~~
	* {{MathWorld\|urlname=FiberBundle\|title=Fiber Bundle}}

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

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2013-04-05T22:21:01Z

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en>Keithbowden at 12:07, 2 September 2012

2012-09-02T12:07:55Z

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en>Noix07: /* Introductory example */

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en>Keithbowden at 12:07, 2 September 2012