Classifying space for U(n): Difference between revisions

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{{Short description|Exact homotopy case}}

{{DISPLAYTITLE:Classifying space for U(''n'')}}

In [[mathematics]], the '''[[classifying space]] for the [[unitary group]]''' U(''n'') is a space BU(''n'') together with a universal bundle EU(''n'') such that any hermitian bundle on a [[paracompact space]] ''X'' is the pull-back of EU(''n'') by a map ''X'' → BU(''n'') unique up to homotopy. A particular application are [[Principal U(1)-bundle|principal U(1)-bundles]].

This space with its universal fibration may be constructed as either

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== Cohomology of BU(''n'')==

<blockquote> '''Proposition:''': The [[cohomology ring]] of the classifying space ''H*''(<math>\operatorname{BU}(''n''))</math> iswith acoefficients in the [[Ring (mathematics)|ring]] <math>\mathbb{Z}</math> of [[polynomialsInteger|integers]] inis ''n''generated variablesby the [[Chern class|Chern classes]]:<ref>Hatcher 02, Theorem 4D.4.</ref>

''c''₁, ..., ''c_n'' where ''c_p'' is of degree 2''p''.</blockquote>

: <math>H^*(\operatorname{BU}(n);\mathbb{Z})

'''Proof:''' Let us first consider the case ''n'' = 1. In this case, U(1) is the circle '''S'''¹ and the universal bundle is '''S'''^∞ → '''CP'''^∞. It is well known<ref>R. Bott, L. W. Tu-- ''Differential Forms in Algebraic Topology'', Graduate Texts in Mathematics 82, Springer</ref> that the cohomology of '''CP'''^''k'' is isomorphic to <math>\mathbf{R}\lbrack c_1\rbrack/c_1^{k+1}</math>, where ''c''₁ is the [[Euler class]] of the U(1)-bundle '''S'''^2''k''+1 → '''CP'''^''k'', and that the injections '''CP'''^''k'' → '''CP'''^''k''+1, for ''k'' ∈ '''N'''*, are compatible with these presentations of the cohomology of the projective spaces. This proves the Proposition for ''n'' = 1.

=\mathbb{Z}[c_1,\ldots,c_n].</math>

'''Proof:''' Let us first consider the case ''n'' = 1. In this case, U(1) is the circle '''S'''¹ and the universal bundle is '''S'''^∞ → '''CP'''^∞. It is well known<ref>R. Bott, L. W. Tu-- ''Differential Forms in Algebraic Topology'', Graduate Texts in Mathematics 82, Springer</ref> that the cohomology of '''CP'''^''k'' is isomorphic to <math>\mathbfmathbb{RZ}\lbrack c_1\rbrack/c_1^{k+1}</math>, where ''c''₁ is the [[Euler class]] of the U(1)-bundle '''S'''^2''k''+1 → '''CP'''^''k'', and that the injections '''CP'''^''k'' → '''CP'''^''k''+1, for ''k'' ∈ '''N'''*, are compatible with these presentations of the cohomology of the projective spaces. This proves the Proposition for ''n'' = 1.

There are homotopy fiber sequences

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is the [[multinomial coefficient]] and <math>k_1,\dots,k_n</math> contains ''r'' distinct integers, repeated <math>n_1,\dots,n_r</math> times, respectively.

== Infinite classifying space ==

The canonical inclusions <math>\operatorname{U}(n)\hookrightarrow\operatorname{U}(n+1)</math> induce canonical inclusions <math>\operatorname{BU}(n)\hookrightarrow\operatorname{BU}(n+1)</math> on their respective classifying spaces. Their respective colimits are denoted as:

: <math>\operatorname{U}

:=\lim_{n\rightarrow\infty}\operatorname{U}(n);</math>

: <math>\operatorname{BU}

:=\lim_{n\rightarrow\infty}\operatorname{BU}(n).</math>

<math>\operatorname{BU}</math> is indeed the classifying space of <math>\operatorname{U}</math>.

==See also==

* [[Classifying space for O(n)|Classifying space for O(''n'')]]

* [[Classifying space for SO(n)]]

* [[Classifying space for SU(n)]]

* [[Topological K-theory]]

* [[Atiyah–Jänich theorem]]

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|publisher=American Mathematical Society

}}

*{{cite book|last=Hatcher|first=Allen|title=Algebraic topology|publisher=[[Cambridge University Press]]|___location=Cambridge|year=2002|language=en|isbn=0-521-79160-X|url=https://pi.math.cornell.edu/~hatcher/AT/ATpage.html}}

*{{cite book|title=Universal principal bundles and classifying spaces|publisher=|___location=|isbn=|url=https://math.mit.edu/~mbehrens/18.906/prin.pdf|doi=|last=Mitchell|first=Stephen|year=August 2001}}

== External links ==