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| In [[mathematics]], an ([[orientation (mathematics)|oriented]]) '''circle bundle''' is an oriented [[fiber bundle]] where the fiber is the [[circle]] <math>\scriptstyle \mathbf{S}^1</math>, or, more precisely, a [[principal bundle|principal ''U''(1)-bundle]]. It is homotopically equivalent to a complex [[line bundle]] with removed zero section. In [[physics]], circle bundles are the natural geometric setting for [[electromagnetism]]. A circle bundle is a special case of a [[fiber bundle#Sphere bundles|sphere bundle]].
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| ==As 3-manifolds==
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| Circle bundles over [[surface]]s are an important example of [[3-manifold]]s. A more general class of 3-manifolds is [[Seifert fiber space]]s, which may be viewed as a kind of "singular" circle bundle, or as a circle bundle over a two-dimensional [[orbifold]].
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| ==Relationship to electrodynamics==
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| The [[Maxwell equation]]s correspond to an [[electromagnetic field]] represented by a [[2-form]] ''F'', with <math>\scriptstyle \pi^{\!*}F</math> being [[cohomologous]] to zero. In particular, there always exists a [[1-form]] ''A'' such that
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| :<math>\scriptstyle \pi^{\!*}F = dA.</math>
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| Given a circle bundle ''P'' over ''M'' and its projection
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| :<math>\pi:P\to M</math>
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| one has the [[homomorphism]]
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| :<math>\scriptstyle \pi^*:H^2(M,\mathbb{Z}) \to H^2(P,\mathbb{Z})</math>
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| where <math>\scriptstyle \pi^{\!*}</math> is the [[pullback]]. Each homomorphism corresponds to a [[Dirac monopole]]; the integer [[cohomology group]]s correspond to the quantization of the [[electric charge]]. | |
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| ==Examples==
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| The [[Hopf fibration]]s are examples of non-trivial circle bundles.
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| ==Classification==
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| The [[isomorphism class]]es of circle bundles over a manifold ''M'' are in one-to-one correspondence with the elements of the second [[integral cohomology group]] <math>\scriptstyle H^2(M,\mathbb{Z})</math> of ''M''. This isomorphism is realized by the [[Euler class]].
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| Equivalently, the isomorphism classes correspond to homotopy classes of maps to the infinite-dimensional [[complex projective space]] <math>CP^\infty</math>, which is the classifying space of [[U(1)]]. See [[classifying space for U(n)]].
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| In homotopy theory terms, the circle and the complex plane without its origin are equivalent. Circle bundles are, by the [[associated bundle]] construction, equivalent to smooth complex [[line bundle]]s because the transition functions of both can be made to live in '''C'''*. In this situation, the Euler class of the circle bundle or real two-plane bundle is the same as the first [[Chern class]] of the line bundle.
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| See also: [[Wang sequence]].
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| ==References==
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| *{{MathWorld |title=Circle Bundle|urlname=CircleBundle}}
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| *{{Citation
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| | last=Chern
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| | first=Shiing-shen
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| | author-link=Shiing-shen Chern
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| | contribution=Circle bundles
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| | contribution-url=http://www.springerlink.com/content/yx080222rrn53273/
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| | title=Lecture Notes in Mathematics
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| | volume=597/1977
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| | pages=114–131
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| | isbn=978-3-540-08345-0
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| | year=1977
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| | publisher = [[Springer Science+Business Media|Springer]] Berlin/Heidelberg
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| }}.
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| [[Category:Fiber bundles]]
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| [[Category:K-theory]]
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| [[Category:Circles]]
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