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In [[mathematics]], the '''[[classifying space]] for the [[unitary group]]''' U(''n'') is a space
This space with its universal fibration may be constructed as either
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:<math>EU(n)=\{e_1,\ldots,e_n : (e_i,e_j)=\delta_{ij}, e_i\in \mathcal{H} \}. \, </math>
Here, ''H'' is an infinite-dimensional complex Hilbert space, the ''e''<
The [[group action]] of U(''n'') on this space is the natural one. The [[base space]] is then
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that is, BU(1) is the infinite-dimensional [[projective unitary group]]. See that article for additional discussion and properties.
For a [[torus]] ''T'', which is abstractly isomorphic to <math>U(1)\times \dots \times U(1)</math>, but need not have a chosen identification, one writes
The [[topological K-theory]] <math>K_0(BT)</math> is given by [[numerical polynomial]]s; more details below.
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be the [[direct limit]] of all the ''G''<sub>''n''</sub>('''C'''<sup>''k''</sup>) (with the induced topology).
<blockquote>'''Lemma:''' The group <math>\pi_p(EU(n))</math> is trivial for all <math>p\ge 1</math>.<
▲The group <math>\pi_p(EU(n))</math> is trivial for all <math>p\ge 1</math>.<br />
'''Proof:''' Let
▲Let <math>\gamma</math> be a map from the sphere <math>S^p</math> to EU(''n''). As <math>S^p</math> is [[compact space|compact]], there exists ''k'' such that <math>\gamma(S^p)</math> is included in ''F''<sub>''n''</sub>('''C'''<sup>''k''</sup>). By taking ''k'' big enough, we see that <math>\gamma</math> is homotopic, with respect to the base point, to the constant map.<math>\Box</math>
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 <math>F_n(\mathbb{C}^{k+1})</math>, resp. <math>G_n(\mathbb{C}^{k+1})</math>. Thus EU(''n'') (and also <math>G_n(\mathbb{C}^\infty)</math>) is a CW-complex. By [[Whitehead theorem|Whitehead Theorem]] and the above Lemma, EU(''n'') is contractible.
== Cohomology of
<blockquote> '''Proposition:''' The [[cohomology]] of the classifying space <math>H^*(BU(n))</math> is a [[Ring (mathematics)|ring]] of [[polynomials]] in ''n'' variables
<math>c_1,\ldots,c_n</math> where <math>c_p</math> is of degree <math>2p</math>.</blockquote>
<blockquote>'''Proof:''' Let us first consider the case ''n'' = 1. In this case, U(1) is the circle <math>S^1</math> and the universal bundle is <math>S^\infty\longrightarrow \mathbb{C}P^\infty</math>. 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 <math>\mathbb{C}P^k</math> is isomorphic to <math>\mathbb{R}\lbrack c_1\rbrack/c_1^{k+1}</math>, where <math>c_1</math> is the [[Euler class]] of the U(1)-bundle <math>S^{2k+1}\longrightarrow \mathbb{C}P^k</math>, and that the injections <math>\mathbb{C}P^k\longrightarrow \mathbb{C}P^{k+1}</math>, for <math>k\in \mathbb{N}^*</math>, are compatible with these presentations of the cohomology of the projective spaces. This proves the Proposition for ''n'' = 1.▼
▲Springer</ref> that the cohomology of <math>\mathbb{C}P^k</math> is isomorphic to <math>\mathbb{R}\lbrack c_1\rbrack/c_1^{k+1}</math>, where <math>c_1</math> is the [[Euler class]] of the U(1)-bundle <math>S^{2k+1}\longrightarrow \mathbb{C}P^k</math>, and that the injections <math>\mathbb{C}P^k\longrightarrow \mathbb{C}P^{k+1}</math>, for <math>k\in \mathbb{N}^*</math>, are compatible with these presentations of the cohomology of the projective spaces. This proves the Proposition for ''n'' = 1.
In the general case, let ''T'' be the subgroup of diagonal matrices. It is a [[maximal torus]] in U(''n''). Its classifying space is <math>(\mathbb{C}P^\infty)^n</math> and its cohomology is <math>\mathbb{R}\lbrack x_1,\ldots,x_n\rbrack</math>, where <math>x_i</math> is the [[Euler class]] of the tautological bundle over the ''i''-th <math>\mathbb{C}P^\infty</math>. The [[Weyl group]] acts on ''T'' by permuting the diagonal entries, hence it acts on <math>(\mathbb{C}P^\infty)^n</math> by permutation of the factors. The induced action on its cohomology is the permutation of the <math>x_i</math>'s. We deduce <br /><math>H^*(BU(n))=\mathbb{R}\lbrack c_1,\ldots,c_n\rbrack,</math><br />where the <math>c_i</math>'s are the [[symmetric polynomials]] in the <math>x_i</math>'s. </blockquote>
==
The [[topological K-theory]] is known explicitly in terms of [[numerical polynomial|numerical]] [[symmetric polynomial]]s.
The K-theory reduces to computing <math>K_0</math>, since K-theory is 2-periodic by the [[Bott periodicity theorem]], and
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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<math>K_0(BU(1))</math> is the ring of [[numerical polynomial]]s in ''w'', regarded as a subring of <math>H_*(BU(1);\mathbf{Q})=\mathbf{Q}[w]</math>, where ''w'' is element dual to tautological bundle.
For the ''n''-torus, <math>K_0(BT^n)</math> is numerical polynomials in ''n'' variables. The map <math>K_0(BT^n) \to K_0(BU(n))</math> is onto, via a [[splitting principle]], as <math>T^n</math> is the [[maximal torus]] of
:<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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