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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''.]]
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| [[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.]]
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| 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'':
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| :''π'' : ''E'' → ''B''
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| then a section of that fiber bundle is a [[continuous map]],
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| :''s'' : ''B'' → ''E''
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| such that
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| :<math>\pi(s(x))=x</math> for all ''x'' in ''B''.
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| 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'':
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| :<math> s(x) = (x,g(x)) \in E,\quad s:B\to E</math>
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| 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''.
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| 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.)
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| 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]].
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| 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]]).
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| == Local and global sections ==
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| 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''.
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| 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'').
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| === Extending to global sections ===
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| 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.
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| ==== Generalizations ====
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| 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).
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| 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).
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| 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.
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| == See also ==
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| * [[Fibration]]
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| * [[Gauge theory]]
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| * [[Principal bundle]]
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| * [[Pullback bundle]]
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| * [[Vector bundle]]
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| == Notes ==
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| <references/>
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| == References ==
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| * [[Norman Steenrod]], ''The Topology of Fibre Bundles'', Princeton University Press (1951). ISBN 0-691-00548-6.
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| * David Bleecker, ''Gauge Theory and Variational Principles'', Addison-Wesley publishing, Reading, Mass (1981). ISBN 0-201-10096-7.
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| * {{citation|first=Dale|last=Husemöller|title=Fibre Bundles|publisher=Springer Verlag|year=1994|isbn=0-387-94087-1}}
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| == External links ==
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| * [http://planetmath.org/encyclopedia/FiberBundle.html Fiber Bundle], PlanetMath
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| * {{MathWorld|urlname=FiberBundle|title=Fiber Bundle}}
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| [[Category:Fiber bundles| ]]
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| [[Category:Differential topology]]
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| [[Category:Algebraic topology]]
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| [[Category:Homotopy theory]]
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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
my blog post :: how to hack clash of clans (click through the following page)