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| 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.
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| This space with its universal fibration may be constructed as either
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| # the [[Grassmannian]] of ''n''-planes in an infinite-dimensional complex [[Hilbert space]]; or,
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| # the direct limit, with the induced topology, of [[Grassmannian]]s of ''n'' planes.
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| Both constructions are detailed here.
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| ==Construction as an infinite Grassmannian==
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| The [[total space]] EU(''n'') of the [[universal bundle]] is given by
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| :<math>EU(n)=\left \{e_1,\ldots,e_n \ : \ (e_i,e_j)=\delta_{ij}, e_i\in \mathcal{H} \right \}.</math>
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| Here, ''H'' is an infinite-dimensional complex Hilbert space, the ''e''<sub>''i''</sub> 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''.
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| The [[group action]] of U(''n'') on this space is the natural one. The [[base space]] is then
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| :<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, | |
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| :<math>BU(n) = \{ V \subset \mathcal{H} \ : \ \dim V = n \}</math>
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| so that ''V'' is an ''n''-dimensional vector space.
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| === Case of line bundles ===
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| For ''n'' = 1, one has EU(1) = '''S'''<sup>∞</sup>, which is [[Contractibility of unit sphere in Hilbert space|known to be a contractible space]]. The base space is then BU(1) = '''CP'''<sup>∞</sup>, the infinite-dimensional [[complex projective space]]. Thus, the set of [[isomorphism class]]es of [[circle bundle]]s over a [[manifold]] ''M'' are in one-to-one correspondence with the [[homotopy class]]es of maps from ''M'' to '''CP'''<sup>∞</sup>.
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| One also has the relation that
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| :<math>BU(1)= PU(\mathcal{H}),</math>
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| that is, BU(1) is the infinite-dimensional [[projective unitary group]]. See that article for additional discussion and properties.
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| For a [[torus]] ''T'', which is abstractly isomorphic to U(1) × ... × U(1), but need not have a chosen identification, one writes B''T''.
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| The [[topological K-theory]] ''K''<sub>0</sub>(B''T'') is given by [[numerical polynomial]]s; more details below.
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| ==Construction as an inductive limit==
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| Let ''F<sub>n</sub>''('''C'''<sup>''k''</sup>) be the space of orthonormal families of ''n'' vectors in '''C'''<sup>''k''</sup> and let ''G<sub>n</sub>''('''C'''<sup>''k''</sup>) be the Grassmannian of ''n''-dimensional subvector spaces of '''C'''<sup>''k''</sup>. The total space of the universal bundle can be taken to be the direct limit of the ''F<sub>n</sub>''('''C'''<sup>''k''</sup>) as ''k'' → ∞, while the base space is the direct limit of the ''G''<sub>''n''</sub>('''C'''<sup>''k''</sup>) as ''k'' → ∞.
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| ===Validity of the construction===
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| In this section, we will define the topology on EU(''n'') and prove that EU(''n'') is indeed contractible.
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| The group U(''n'') acts freely on ''F''<sub>''n''</sub>('''C'''<sup>''k''</sup>) and the quotient is the Grassmannian ''G''<sub>''n''</sub>('''C'''<sup>''k''</sup>). The map
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| : <math>\begin{align}
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| F_n(\mathbf{C}^k) & \longrightarrow \mathbf{S}^{2k-1} \\
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| (e_1,\ldots,e_n) & \longmapsto e_n
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| \end{align}</math>
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| is a fibre bundle of fibre ''F''<sub>''n''−1</sub>('''C'''<sup>''k''−1</sup>). Thus because <math>\pi_p(\mathbf{S}^{2k-1})</math> is trivial and because of the [[Homotopy group|long exact sequence of the fibration]], we have
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| : <math>\pi_p(F_n(\mathbf{C}^k))=\pi_p(F_{n-1}(\mathbf{C}^{k-1}))</math>
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| whenever <math>p\leq 2k-2</math>. By taking ''k'' big enough, precisely for <math>k>\tfrac{1}{2}p+n-1</math>, we can repeat the process and get
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| : <math>\pi_p(F_n(\mathbf{C}^k)) = \pi_p(F_{n-1}(\mathbf{C}^{k-1})) = \cdots = \pi_p(F_1(\mathbf{C}^{k+1-n})) = \pi_p(\mathbf{S}^{k-n}).</math>
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| This last group is trivial for ''k'' > ''n'' + ''p''. Let
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| : <math>EU(n)={\lim_{\to}}\;_{k\to\infty}F_n(\mathbf{C}^k)</math>
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| be the [[direct limit]] of all the ''F''<sub>''n''</sub>('''C'''<sup>''k''</sup>) (with the induced topology). Let
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| : <math>G_n(\mathbf{C}^\infty)={\lim_\to}\;_{k\to\infty}G_n(\mathbf{C}^k)</math>
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| be the [[direct limit]] of all the ''G''<sub>''n''</sub>('''C'''<sup>''k''</sup>) (with the induced topology).
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| <blockquote>'''Lemma:''' The group <math>\pi_p(EU(n))</math> is trivial for all ''p'' ≥ 1.</blockquote>
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| '''Proof:''' Let γ : '''S'''<sup>''p''</sup> → EU(''n''), since '''S'''<sup>''p''</sup> is [[compact space|compact]], there exists ''k'' such that γ('''S'''<sup>''p''</sup>) is included in ''F''<sub>''n''</sub>('''C'''<sup>''k''</sup>). By taking ''k'' big enough, we see that γ is homotopic, with respect to the base point, to the constant map.<math>\Box</math>
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| In addition, U(''n'') acts freely on EU(''n''). The spaces ''F''<sub>''n''</sub>('''C'''<sup>''k''</sup>) and ''G''<sub>''n''</sub>('''C'''<sup>''k''</sup>) are [[CW complex|CW-complexes]]. One can find a decomposition of these spaces into CW-complexes such that the decomposition of ''F''<sub>''n''</sub>('''C'''<sup>''k''</sup>), resp. ''G''<sub>''n''</sub>('''C'''<sup>''k''</sup>), is induced by restriction of the one for ''F''<sub>''n''</sub>('''C'''<sup>''k''+1</sup>), resp. ''G''<sub>''n''</sub>('''C'''<sup>''k''+1</sup>). Thus EU(''n'') (and also ''G''<sub>''n''</sub>('''C'''<sup>∞</sup>)) is a CW-complex. By [[Whitehead theorem|Whitehead Theorem]] and the above Lemma, EU(''n'') is contractible.
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| == Cohomology of BU(''n'')==
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| <blockquote> '''Proposition:''' The [[cohomology]] of the classifying space ''H*''(BU(''n'')) is a [[Ring (mathematics)|ring]] of [[polynomials]] in ''n'' variables
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| ''c''<sub>1</sub>, ..., ''c<sub>n</sub>'' where ''c<sub>p</sub>'' is of degree 2''p''.</blockquote>
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| '''Proof:''' Let us first consider the case ''n'' = 1. In this case, U(1) is the circle '''S'''<sup>1</sup> and the universal bundle is '''S'''<sup>∞</sup> → '''CP'''<sup>∞</sup>. 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'''<sup>''k''</sup> is isomorphic to <math>\mathbf{R}\lbrack c_1\rbrack/c_1^{k+1}</math>, where ''c''<sub>1</sub> is the [[Euler class]] of the U(1)-bundle '''S'''<sup>2''k''+1</sup> → '''CP'''<sup>''k''</sup>, and that the injections '''CP'''<sup>''k''</sup> → '''CP'''<sup>''k''+1</sup>, for ''k'' ∈ '''N'''*, are compatible with these presentations of the cohomology of the projective spaces. This proves the Proposition for ''n'' = 1.
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| In the general case, let ''T'' be the subgroup of diagonal matrices. It is a [[maximal torus]] in U(''n''). Its classifying space is ('''CP'''<sup>∞</sup>)<sup>''n''</sup>. and its cohomology is '''R'''[''x''<sub>1</sub>, ..., ''x<sub>n</sub>''], where ''x<sub>i</sub>'' is the [[Euler class]] of the tautological bundle over the ''i''-th '''CP'''<sup>∞</sup>. The [[Weyl group]] acts on ''T'' by permuting the diagonal entries, hence it acts on ('''CP'''<sup>∞</sup>)<sup>''n''</sup> by permutation of the factors. The induced action on its cohomology is the permutation of the <math>x_i</math>'s. We deduce
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| :<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>
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| ==K-theory of BU(''n'')==
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| The [[topological K-theory]] is known explicitly in terms of [[numerical polynomial|numerical]] [[symmetric polynomial]]s.
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| The K-theory reduces to computing ''K''<sub>0</sub>, since K-theory is 2-periodic by the [[Bott periodicity theorem]], and BU(''n'') is a limit of complex manifolds, so it has a [[CW-structure]] with only cells in even dimensions, so odd K-theory vanishes.
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| Thus <math>K_*(X) = \pi_*(K) \otimes K_0(X)</math>, where <math>\pi_*(K)=\mathbf{Z}[t,t^{-1}]</math>, where ''t'' is the Bott generator.
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| ''K''<sub>0</sub>(BU(1)) is the ring of [[numerical polynomial]]s in ''w'', regarded as a subring of ''H''<sub>∗</sub>(BU(1); '''Q''') = '''Q'''[''w''], where ''w'' is element dual to tautological bundle.
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| For the ''n''-torus, ''K''<sub>0</sub>(B''T<sup>n</sup>'') is numerical polynomials in ''n'' variables. The map ''K''<sub>0</sub>(B''T<sup>n</sup>'') → ''K''<sub>0</sub>(BU(''n'')) is onto, via a [[splitting principle]], as ''T<sup>n</sup>'' is the [[maximal torus]] of U(''n''). The map is the symmetrization map
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| :<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> | |
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| and the image can be identified as the symmetric polynomials satisfying the integrality condition that
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| :<math> {n \choose n_1, n_2, \ldots, n_r}f(k_1,\dots,k_n) \in \mathbf{Z}</math>
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| where
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| :<math> {n \choose k_1, k_2, \ldots, k_m} = \frac{n!}{k_1!\, k_2! \cdots k_m!}</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.
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| ==See also==
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| * [[Classifying space for O(n)]]
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| * [[Topological K-theory]]
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| * [[Atiyah-Jänich theorem]]
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| == Notes ==
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| <references />
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| ==References==
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| *{{citation
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| |author=S. Ochanine, L. Schwartz
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| |title=Une remarque sur les générateurs du cobordisme complex
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| |journal=Math. Z.
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| |volume=190
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| |year=1985
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| |issue=4
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| |pages=543–557
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| |doi=10.1007/BF01214753
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| }} Contains a description of <math>K_0(BG)</math> as a <math>K_0(K)</math>-comodule for any compact, connected Lie group.
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| *{{citation
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| |author=L. Schwartz
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| |title=K-théorie et homotopie stable
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| |work=Thesis
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| |publisher=Université de Paris–VII
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| |year=1983
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| }} Explicit description of <math>K_0(BU(n))</math>
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| *{{citation
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| |author=A. Baker, F. Clarke, N. Ray, L. Schwartz
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| |title=On the Kummer congruences and the stable homotopy of ''BU''
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| |journal=Trans. Amer. Math. Soc.
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| |volume=316
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| |issue=2
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| |year=1989
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| |pages=385–432
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| |doi=10.2307/2001355
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| |jstor=2001355
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| |publisher=American Mathematical Society
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| }}
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| [[Category:Homotopy theory]]
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