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| In [[mathematics]], the '''universal bundle''' in the theory of [[fiber bundle]]s with structure group a given [[topological group]] ''G'', is a specific bundle over a [[classifying space]] ''BG'', such that every bundle with the given [[structure group]] ''G'' over ''M'' is a [[pullback bundle|pullback]] by means of a [[continuous map]]
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| :''M'' → ''BG''.
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| ==Existence of a universal bundle==
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| ===In the CW complex category===
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| When the definition of the classifying space takes place within the homotopy [[category (mathematics)|category]] of [[CW complex]]es, existence theorems for universal bundles arise from [[Brown's representability theorem]].
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| ===For compact Lie groups===
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| We will first prove:<br />
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| '''Proposition'''<br />
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| Let <math>G</math> be a compact [[Lie group]].
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| There exists a contractible space <math>EG</math> on which <math>G</math> acts freely. The projection <math>EG\longrightarrow BG</math> is a
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| <math>G</math>-principal fibre bundle.<br />
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| '''Proof'''
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| There exists an injection of <math>G</math> into a [[unitary group]] <math>U(n)</math> for <math>n</math> big enough.<ref>[[J. J. Duistermaat]] and J. A. Kolk,
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| -- ''Lie Groups'', Universitext, Springer. Corollary 4.6.5</ref>
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| If we find <math>EU(n)</math> then we can take <math>EG</math> to be <math>EU(n)</math>.
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| The construction of ''EU(n)'' is given in [[classifying space for U(n)]].
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| <math>\Box</math>
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| The following Theorem is a corollary of the above Proposition.
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| '''Theorem'''<br />
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| If <math>M</math> is a paracompact manifold and <math>P\longrightarrow M</math> is a principal <math>G</math>-bundle, then there exists a map
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| <math>f:M\longrightarrow BG</math>, well defined up to homotopy, such that <math>P</math> is isomorphic to <math>f^*(EG)</math>, the pull-back
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| of the <math>G</math>-bundle <math>EG\longrightarrow BG</math> by <math>f</math>.<br />
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| '''Proof'''
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| On one hand, the pull-back of the bundle <math>\pi:EG\longrightarrow BG</math> by the natural projection <math>P\times_G EG\longrightarrow BG</math> is the bundle <math>P\times EG</math>. On the other hand, the pull-back of the principal <math>G</math>-bundle <math>P\longrightarrow M</math> by the projection
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| <math>p:P\times_G EG\longrightarrow M</math> is also <math>P\times EG</math><br />
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| <br /><math>\begin{align}
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| P & \longleftarrow & P\times EG& \longrightarrow & EG \\
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| \downarrow & & \downarrow & & \downarrow\pi\\
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| M & \longleftarrow^{\!\!\!\!\!\!\!p} & P\times_G EG & \longrightarrow & BG.
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| \end{align}</math><br />
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| Since <math>p</math> is a fibration with contractible fibre <math>EG</math>,
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| sections of <math>p</math> exist.<ref>A.~Dold
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| -- ''Partitions of Unity in the Theory of Fibrations'',Annals of Math., vol. 78, No 2 (1963)</ref> To such a section <math>s</math>
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| we associate the composition with the projection <math>P\times_G EG\longrightarrow BG</math>. The map we get is the <math>f</math> we were
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| looking for.<br />
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| For the uniqueness up to homotopy, notice that there exists a one to one correspondence between maps
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| <math>f:M\longrightarrow BG</math> such that <math>f^*EG\longrightarrow M</math> is isomorphic to <math>P\longrightarrow M</math> and sections of <math>p</math>. We have just seen
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| how to associate a <math>f</math> to a section. Inversely, assume that <math>f</math> is given. Let <math>\Phi</math> be an isomorphism
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| between <math>f^*EG</math> and <math>P</math>
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| <br /><math>\Phi: \{(x,u)\in M\times EG\mid\,f(x)=\pi(u)\} \longrightarrow P</math>.<br />
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| Now, simply define a section by
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| <br /><math>\begin{align}
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| M & \longrightarrow & P\times_G EG \\
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| x & \longrightarrow & \lbrack \Phi(x,u),u\rbrack.
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| \end{align}</math><br />
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| Because all sections of <math>p</math> are homotopic, the homotopy class of <math>f</math> is unique.
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| <math>\Box</math>
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| ==Use in the study of group actions==
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| The total space of a universal bundle is usually written ''EG''. These spaces are of interest in their own right, despite typically being [[contractible]]. For example in defining the '''homotopy quotient''' or '''homotopy orbit space''' of a [[group action]] of ''G'', in cases where the [[orbit space]] is [[Pathological (mathematics)|pathological]] (in the sense of being a non-[[Hausdorff space]], for example). The idea, if ''G'' acts on the space ''X'', is to consider instead the action on
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| :''Y'' = ''X''×''EG'',
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| and corresponding quotient. See [[equivariant cohomology]] for more detailed discussion.
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| If ''EG'' is contractible then ''X'' and ''Y'' are [[homotopy equivalent]] spaces. But the diagonal action on ''Y'', i.e. where ''G'' acts on both ''X'' and ''EG'' coordinates, may be [[well-behaved]] when the action on ''X'' is not.
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| ==Examples==
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| * [[Classifying space for U(n)]]
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| ==See also==
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| * [[Chern class]]
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| ==External links==
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| *[http://planetmath.org/?op=getobj&from=objects&id=3663 PlanetMath page of universal bundle examples]
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| ==Notes==
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| {{reflist}}
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| [[Category:Homotopy theory]]
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| [[Category:Fiber bundles]]
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