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{{Short description|Exact homotopy case}}

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.

{{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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The [[total space]] EU(''n'') of the [[universal bundle]] is given by

:<math>EU(n)=\left \{e_1,\ldots,e_n \ : \ (e_i,e_j)=\delta_{ij}, e_i\in \mathcal{H} \right \}.</math>

Here, ''H'' isdenotes an infinite-dimensional complex Hilbert space, the ''e''_''i'' are vectors in ''H'', and <math>\delta_{ij}</math> is the [[Kronecker delta]]. The symbol <math>(\cdot,\cdot)</math> is the [[inner product]] on ''H''. Thus, we have that EU(''n'') is the space of [[orthonormal]] ''n''-frames in ''H''.

The [[Group action (mathematics)|group action]] of U(''n'') on this space is the natural one. The [[base space]] is then

:<math>BU(n)=EU(n)/U(n) </math>

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and is the set of [[Grassmannian]] ''n''-dimensional subspaces (or ''n''-planes) in ''H''. That is,

:<math>BU(n) = \{ V \subset \mathcal{H} \ : \ \dim V = n \}</math>

so that ''V'' is an ''n''-dimensional vector space.

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One also has the relation that

:<math>BU(1)= PU(\mathcal{H}),</math>

that is, BU(1) is the infinite-dimensional [[projective unitary group]]. See that article for additional discussion and properties.

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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>

In'''Proof:''' theLet generalus case,first letconsider the case ''Tn'' be= the1. subgroupIn ofthis diagonalcase, matrices. ItU(1) is a [[maximalthe torus]] incircle U(''n'S').''¹ Itsand classifyingthe spaceuniversal bundle is ('''CPS'''^∞) → '''CP'''^∞. It is well known<ref>R. Bott, L. W. Tu-- ''nDifferential Forms in Algebraic Topology'', Graduate Texts in Mathematics 82, Springer</supref>. andthat itsthe cohomology isof '''RCP'''[<sup>''xk''<sub>1</subsup>, ...,is isomorphic to ''x<submath>n\mathbb{Z}\lbrack c_1\rbrack/c_1^{k+1}</submath>''], where ''xc''_i1'' is the [[Euler class]] of the tautological U(1)-bundle over the ''i'S'-th ''<sup>2'CP'k''^∞+1. The [[Weyl group]] acts on→ ''T'CP'''^''k'', byand permutingthat the diagonal entries,injections hence it acts on ('''CP'''^∞''k'') → '''CP'''^''nk''+1, byfor permutation''k'' of∈ the'''N'''*, factors.are Thecompatible inducedwith actionthese onpresentations itsof the cohomology isof the permutationprojective ofspaces. This proves the <math>x_i</math>'s.Proposition Wefor deduce''n'' = 1.

:<math>H^*(BU(n))=\mathbf{R}\lbrack c_1,\ldots,c_n\rbrack,</math>

where the <math>c_i</math>'s are the [[symmetric polynomials]] in the <math>x_i</math>'s.<math>\Box</math>

There are homotopy fiber sequences

In contrast to the above description of <math>H^*(BU(n))</math>, many authors allow non-homogeneous elements in the cohomology, leading to the description <math>H^*(BU(n)) = \mathbb{Z}[[c_1,c_2,...,c_n]]</math><ref>Adams, 1974 p. 49</ref>.

: <math> \mathbb{S}^{2n-1} \to B U(n-1) \to B U(n) </math>

==K-theory of BU(''n'')==

LetConcretely, usa considerpoint topological complex K-theory asof the cohomologytotal theory represented by the spectrumspace <math>KUBU(n-1)</math>. Inis thisgiven case,by a point of the base space <math>KU^*(BU(n))\cong \mathbb{Z}[t,t^{-1}][[c_1,...,c_n]]</math><ref> Adamsclassifying 1974,a p.complex vector space 49<math>V</refmath>, andtogether with a unit vector <math> KU_*(BU(n))u</math> is the freein <math>\mathbb{Z}[t,t^{-1}]V</math>; moduletogether onthey classify <math> u^\beta_0perp < V </math> andwhile the splitting <math>V = (\beta_mathbb{i_1C} u) \ldotsoplus u^\beta_{i_r}perp </math>, fortrivialized by <math>n\geq i_j > 0u</math>, realizes the andmap <math>r B U(n-1) \leqto B U(n) </math>.<ref> Adamsrepresenting 1974,direct p.sum 47with <math>\mathbb{C}.</refmath>

Applying the [[Gysin sequence]], one has a long exact sequence

{{warning|The description of the K-theory of BU(n) below is incomplete, and unclear. The author of the secion below should clarify what they are computing. It seems more like a computation of <math>KU_*KU</math>}}

: <math> H^p ( BU(n) ) \overset{\smile d_{2n} \eta}{\longrightarrow} H^{p+2n} ( BU(n) ) \overset{j^*}{\longrightarrow} H^{p+2n} (BU(n-1)) \overset{\partial}{\longrightarrow} H^{p+1}(BU(n)) \longrightarrow \cdots </math>

where <math>\eta</math> is the [[fundamental class]] of the fiber <math>\mathbb{S}^{2n-1}</math>. By properties of the Gysin Sequence{{Citation needed|date=May 2016}}, <math>j^*</math> is a multiplicative homomorphism; by induction, <math>H^*BU(n-1)</math> is generated by elements with <math> p < -1 </math>, where <math>\partial</math> must be zero, and hence where <math>j^*</math> must be surjective. It follows that <math>j^*</math> must '''always''' be surjective: by the [[universal property]] of [[polynomial ring]]s, a choice of preimage for each generator induces a multiplicative splitting. Hence, by exactness, <math>\smile d_{2n}\eta </math> must always be '''injective'''. We therefore have [[short exact sequence]]s split by a ring homomorphism

: <math> 0 \to H^p ( BU(n) ) \overset{\smile d_{2n} \eta}{\longrightarrow} H^{p+2n} ( BU(n) ) \overset{j^*}{\longrightarrow} H^{p+2n} (BU(n-1)) \to 0 </math>

Thus we conclude <math>H^*(BU(n)) = H^*(BU(n-1))[c_{2n}]</math> where <math>c_{2n} = d_{2n} \eta</math>. This completes the induction.

==K-theory of BU(''n'')==

{{confusing section|date=August 2022}}

Consider topological complex K-theory as the cohomology theory represented by the spectrum <math>KU</math>. In this case, <math>KU^*(BU(n))\cong \mathbb{Z}[t,t^{-1}][[c_1,...,c_n]]</math>,<ref>Adams 1974, p. 49</ref> and <math> KU_*(BU(n))</math> is the free <math>\mathbb{Z}[t,t^{-1}]</math> module on <math>\beta_0</math> and <math>\beta_{i_1}\ldots\beta_{i_r}</math> for <math>n\geq i_j > 0</math> and <math>r\leq n</math>.<ref>Adams 1974, p. 47</ref> In this description, the product structure on <math> KU_*(BU(n)) </math> comes from the H-space structure of <math>BU</math> given by Whitney sum of vector bundles. This product is called the [[Pontryagin product]].

The [[topological K-theory]] is known explicitly in terms of [[numerical polynomial|numerical]] [[symmetric polynomial]]s.

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For the ''n''-torus, ''K''₀(B''Tⁿ'') is numerical polynomials in ''n'' variables. The map ''K''₀(B''Tⁿ'') → ''K''₀(BU(''n'')) is onto, via a [[splitting principle]], as ''Tⁿ'' is the [[maximal torus]] of U(''n''). The map is the symmetrization map

:<math>f(w_1,\dots,w_n) \mapsto \frac{1}{n!} \sum_{\sigma \in S_n} f(x_{\sigma(1)}, \dots, x_{\sigma(n)})</math>

and the image can be identified as the symmetric polynomials satisfying the integrality condition that

:<math> {n \choose n_1, n_2, \ldots, n_r}f(k_1,\dots,k_n) \in \mathbf{Z}</math>

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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-JänichAtiyah–Jänich theorem]]

== Notes ==

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}} Contains calculation of <math>KU^*(BU(n))</math> and <math>KU_*(BU(n))</math>.

*{{citation

|authorauthor1=S. Ochanine, |author2=L. Schwartz |title=Une remarque sur les générateurs du cobordisme complex

|title=Une remarque sur les générateurs du cobordisme complex

|journal=Math. Z.

|volume=190

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}} Explicit description of <math>K_0(BU(n))</math>

*{{citation

|authorauthor1=A. Baker, |author2=F. Clarke, |author3=N. Ray, |author4=L. Schwartz |title=On the Kummer congruences and the stable homotopy of ''BU''

|title=On the Kummer congruences and the stable homotopy of ''BU''

|journal=Trans. Amer. Math. Soc.

|volume=316

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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 ==

* [[nlab:classifying+space|classifying space]] on [[nLab]]

* [[nlab:BU(n)|BU(n)]] on nLab

[[Category:Homotopy theory]]