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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'' 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''.
The [[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>
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.
=== Case of line bundles ===
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>. ▼
In the case of ''n'' = 1, one has
:<math>EU(1)= S^\infty.\,</math>
[[Contractibility of unit sphere in Hilbert space|known to be a contractible space]].
The base space is then
:<math>BU(1)= \mathbb{C}P^\infty,\,</math>
▲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>.
One also has the relation that
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 B''BTT''.
The [[topological K-theory]] ''K''<mathsub>K_0(BT)0</mathsub>(B''T'') is given by [[numerical polynomial]]s; more details below.
==Construction as an inductive limit==
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'' goes→ to infinity∞, while the base space is the direct limit of the ''G''<sub>''n''</sub>('''C'''<sup>''k''</sup>) as ''k'' goes to→ infinity∞.
===Validity of the construction===
: <math>\begin{align}
F_n(\mathbbmathbf{C}^k) & \longrightarrow \mathbf{S}^{2k-1} \\
(e_1,\ldots,e_n) & \longmapsto e_n
\end{align}</math>
is a fibre bundle of fibre ''F''<sub>''n'' - 1−1</sub>('''C'''<sup>''k'' - 1−1</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
: <math>\pi_p(F_n(\mathbbmathbf{C}^k))=\pi_p(F_{n-1}(\mathbbmathbf{C}^{k-1}))</math>
whenever <math>p\leq 2k-2</math>. By taking ''k'' big enough, precisely for <math>k>\fractfrac{1}{2}p+n-1</math>, we can repeat the process and get
: <math>\pi_p(F_n(\mathbbmathbf{C}^k)) = \pi_p(F_{n-1}(\mathbbmathbf{C}^{k-1})) = \cdots = \pi_p(F_1(\mathbbmathbf{C}^{k+1-n})) = \pi_p(\mathbf{S}^{k-n}).</math>
This last group is trivial for ''k'' > ''n'' + ''p''. Let
: <math>EU(n)={\lim_{\rightarrowto}}\;_{k\to\infty}F_n(\mathbbmathbf{C}^k)</math>
be the [[direct limit]] of all the ''F''<sub>''n''</sub>('''C'''<sup>''k''</sup>) (with the induced topology). Let
: <math>G_n(\mathbbmathbf{C}^\infty)={\lim_\rightarrowto}\;_{k\to\infty}G_n(\mathbbmathbf{C}^k)</math>
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>.</blockquote>
'''Proof:''' Let γ be a map from the sphere: '''S'''<mathsup>S^''p''</mathsup> to→ EU(''n'')., Assince '''S'''<mathsup>S^''p''</mathsup> is [[compact space|compact]], there exists ''k'' such that <math>\gammaγ('''S^'''<sup>''p)''</mathsup>) 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>
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''<mathsub>''n''</sub>F_n(\mathbb{'''C}^{'''<sup>''k''+1})</mathsup>), resp. ''G''<mathsub>''n''</sub>G_n(\mathbb{'''C}^{'''<sup>''k''+1})</mathsup>). Thus EU(''n'') (and also ''G''<mathsub>''n''</sub>G_n(\mathbb{'''C}^\infty)'''<sup>∞</mathsup>)) is a CW-complex. By [[Whitehead theorem|Whitehead Theorem]] and the above Lemma, EU(''n'') is contractible.
== Cohomology of BU(''n'')==
<blockquote> '''Proposition:''' The [[cohomology]] of the classifying space <math>''H^*''(BU(''n''))</math> is a [[Ring (mathematics)|ring]] of [[polynomials]] in ''n'' variables
''c''<mathsub>1</sub>c_1,\ldots ...,c_n ''c<sub>n</mathsub>'' where ''c<mathsub>c_pp</mathsub>'' is of degree <math>2p</math>2''p''.</blockquote>
<blockquote>'''Proof:''' Let us first consider the case ''n'' = 1. In this case, U(1) is the circle '''S'''<mathsup>S^1</mathsup> and the universal bundle is '''S'''<mathsup>∞</sup>S^\infty\longrightarrow \mathbb{C}P^\infty→ '''CP'''<sup>∞</mathsup>. 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'''<mathsup>\mathbb{C}P^''k''</mathsup> is isomorphic to <math>\mathbbmathbf{R}\lbrack c_1\rbrack/c_1^{k+1}</math>, where ''c''<mathsub>c_11</mathsub> is the [[Euler class]] of the U(1)-bundle '''S'''<mathsup>S^{2k2''k''+1}\longrightarrow</sup> \mathbb{C}P^→ '''CP'''<sup>''k''</mathsup>, and that the injections '''CP'''<mathsup>\mathbb{C}P^''k\longrightarrow''</sup> \mathbb{C}P^{→ '''CP'''<sup>''k''+1}</mathsup>, 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 ('''CP'''<mathsup>∞</sup>(\mathbb{C}P^\infty)^<sup>''n''</mathsup>. and its cohomology is '''R'''[''x''<mathsub>\mathbb{R}\lbrack1</sub>, x_1...,\ldots,x_n\rbrack ''x<sub>n</mathsub>''], where ''x<mathsub>x_ii</mathsub>'' is the [[Euler class]] of the tautological bundle over the ''i''-th '''CP'''<mathsup>\mathbb{C}P^\infty∞</mathsup>. The [[Weyl group]] acts on ''T'' by permuting the diagonal entries, hence it acts on ('''CP'''<mathsup>∞</sup>(\mathbb{C}P^\infty)^<sup>''n''</mathsup> 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))=\mathbbmathbf{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. <math>\Box</blockquotemath>
==K-theory of BU(''n'')==
The [[topological K-theory]] is known explicitly in terms of [[numerical polynomial|numerical]] [[symmetric polynomial]]s.
The K-theory reduces to computing ''K''<mathsub>K_00</mathsub>, 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.
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.
''K''<mathsub>0</sub>K_0(BU(1))</math> is the ring of [[numerical polynomial]]s in ''w'', regarded as a subring of ''H''<mathsub>∗</sub>H_*(BU(1);\mathbf{ '''Q}''') =\mathbf{ '''Q}'''[''w'']</math>, where ''w'' is element dual to tautological bundle.
For the ''n''-torus, ''K''<mathsub>0</sub>K_0(BT^B''T<sup>n)</mathsup>'') is numerical polynomials in ''n'' variables. The map ''K''<mathsub>0</sub>K_0(BT^B''T<sup>n</sup>'') \to→ K_0''K''<sub>0</sub>(BU(''n''))</math> is onto, via a [[splitting principle]], as ''T<mathsup>T^n</mathsup>'' 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>
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