Classifying space for U(n): Difference between revisions

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In [[mathematics]], the '''[[classifying space]] for the [[unitary group]]''' U(''U(n)'') is a space '''B'''(U(''U(n)'')) together with a universal bundle '''E'''(''U(''n)'')) such that any [[hermitian bundle]] on a [[paracompact space]] ''X'' is the pull-back of '''E''' by a map ''X → B'' unique up to homotopy.

This space with its universal fibration may be constructed as either

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==Construction as an infinite Grassmannian==

The [[total space]] <math>EU(''n'')</math> of the [[universal bundle]] is given by

:<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 <math>e_i</math> 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>

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=== Case of line bundles ===

In the case of <math> ''n'' = 1 </math>, one has

:<math>EU(1)= S^\infty.\,</math>

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:<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]] <math>''M</math>'' are in one-to-one correspondence with the [[homotopy class]]es of maps from <math>''M</math>'' to '''CP'''<mathsup>\mathbb{C}P^\infty&infin;</mathsup>.

One also has the relation that

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

that is, <math>BU(1)</math> 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 <math>BT</math>.

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==Construction as an inductive limit==

Let ''F''<mathsub>''n''</sub>F_n(\mathbb{'''C}^'''<sup>''k)''</mathsup>) be the space of orthonormal families of <math>''n</math>'' vectors in '''C'''<mathsup>\mathbb{C}^''k''</mathsup> and let ''G''<mathsub>''n''</sub>G_n(\mathbb{'''C}^'''<sup>''k)''</mathsup>) be the Grassmannian of <math>''n</math>''-dimensional subvector spaces of '''C'''<mathsup>\mathbb{C}^''k''</mathsup>. The total space of the universal bundle can be taken to be the direct limit of the ''F''<mathsub>''n''</sub>F_n(\mathbb{'''C}^'''<sup>''k)''</mathsup>) as <math>''k</math>'' goes to infinity, while the base space is the direct limit of the ''G''<mathsub>''n''</sub>G_n(\mathbb{'''C}^'''<sup>''k)''</mathsup>) as <math>''k</math>'' goes to infinity.

Let <math>F_n(\mathbb{C}^k)</math> be the space of orthonormal families of <math>n</math> vectors in <math>\mathbb{C}^k</math> and let <math>G_n(\mathbb{C}^k)</math> be the Grassmannian of <math>n</math>-dimensional subvector spaces of <math>\mathbb{C}^k</math>. The total space of the universal bundle can be taken to be the direct limit of the <math>F_n(\mathbb{C}^k)</math> as <math>k</math> goes to infinity, while the base space is the direct limit of the <math>G_n(\mathbb{C}^k)</math> as <math>k</math> goes to infinity.

===Validity of the construction===

In this section, we will define the topology on ''EU(n)'' and prove that ''EU(n)'' is indeed contractible.

Let ''F''<mathsub>''n''</sub>F_n(\mathbb{'''C}^'''<sup>''k)''</mathsup>) be the space of orthonormal families of <math>''n</math>'' vectors in '''C'''<mathsup>\mathbb{C}^''k''</mathsup>. The group U(''n'') acts freely on ''F''<mathsub>''n''</sub>U('''C'''^''k'') and the quotient is the Grassmannian ''G''_''n''('''C'''^''k'') of ''n''-dimensional subvector spaces of '''C'''<sup>''k''</mathsup>. actsThe map

freely on <math>F_n(\mathbb{C}^k)</math> and the quotient is the Grassmannian <math>G_n(\mathbb{C}^k)</math> of <math>n</math>-dimensional subvector spaces of <math>\mathbb{C}^k</math>. The map

: <math>\begin{align}

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\end{align}</math>

is a fibre bundle of fibre ''F''<mathsub>F_{''n'' - 1}</sub>(\mathbb{'''C}^{'''<sup>''k'' - 1})</mathsup>). Thus because <math>\pi_p(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(\mathbb{C}^k))=\pi_p(F_{n-1}(\mathbb{C}^{k-1}))</math>

whenever <math>p\leq 2k-2</math>. By taking <math>''k</math>'' big enough, precisely for <math>k>\frac{1}{2}p+n-1</math>, we can repeat the process and get

: <math>\pi_p(F_n(\mathbb{C}^k)) = \pi_p(F_{n-1}(\mathbb{C}^{k-1})) = \cdots = \pi_p(F_1(\mathbb{C}^{k+1-n})) = \pi_p(S^{k-n}).</math>

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: <math>EU(n)={\lim_{\rightarrow}}\;_{k\rightarrow\infty}F_n(\mathbb{C}^k)</math>

be the [[direct limit]] of all the ''F''<mathsub>''n''</sub>F_n(\mathbb{'''C}^'''<sup>''k)''</mathsup>) (with the induced topology). Let

: <math>G_n(\mathbb{C}^\infty)={\lim_{\rightarrow}}\;_{k\rightarrow\infty}G_n(\mathbb{C}^k)</math>

be the [[direct limit]] of all the ''G''<mathsub>''n''</sub>G_n(\mathbb{'''C}^'''<sup>''k)''</mathsup>) (with the induced topology).

'''Lemma'''<br />

The group <math>\pi_p(EU(n))</math> is trivial for all <math>p\ge 1</math>.<br />

'''Proof'''

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''_''n''('''C'''^''k''). 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>

there exists <math>k</math> such that <math>\gamma(S^p)</math> is included in <math>F_n(\mathbb{C}^k)</math>. By taking <math>k</math> 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''_''n''('''C'''^''k'') and ''G''_''n''('''C'''^''k'') are [[CW complex|CW-complexes]]. One can find a decomposition of these spaces into CW-complexes such that the decomposition of ''F''_''n''('''C'''^''k''), resp. ''G''_''n''('''C'''^''k''), 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.

In addition, <math>U(n)</math> acts freely on <math>EU(n)</math>. The spaces <math>F_n(\mathbb{C}^k)</math> and <math>G_n(\mathbb{C}^k)</math> are [[CW complex|CW-complexes]]. One can

find a decomposition of these spaces into CW-complexes such that the decomposition of <math>F_n(\mathbb{C}^k)</math>, resp.

<math>G_n(\mathbb{C}^k)</math>, 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 <math>EU(n)</math> (and also <math>G_n(\mathbb{C}^\infty)</math>) is a CW-complex. By

[[Whitehead theorem|Whitehead Theorem]] and the above Lemma, <math>EU(n)</math> is contractible.

== Cohomology of <math>BU(n)</math> ==

<blockquote> '''Proposition:''' The [[cohomology]] of the classifying space <math>H^*(BU(n))</math> is a [[Ring (mathematics)|ring]] of [[polynomials]] in ''n'' variables

'''Proposition'''<br />

The<math>c_1,\ldots,c_n</math> [[cohomology]] of the classifying spacewhere <math>H^*(BU(n))c_p</math> is a [[Ring (mathematics)|ring]] of [[polynomials]] indegree <math>n2p</math> variables.</blockquote>

<math>c_1,\ldots,c_n</math> where <math>c_p</math> is of degree <math>2p</math>.

<blockquote>'''Proof:''' Let us first consider the case <math>''n'' = 1</math>. In this case, <math>U(1)</math> is the circle <math>S^1</math> and the universal bundle

<br />'''Proof'''

Let us first consider the case <math>n=1</math>. In this case, <math>U(1)</math> 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 <math>U(1)</math>-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 <math>n=1</math>.

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>

In the general case, let <math>T</math> be the subgroup of diagonal matrices. It is a [[maximal torus]] in <math>U(n)</math>. 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 <math>T</math> 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.

<math>\Box</math>

== K-theory of <math>BU(n)</math> ==

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 <math>BU(n)</math> is a limit of complex manifolds, so it has a [[CW-structure]] with only cells in even dimensions, so odd K-theory vanishes.

<math>BU(n)</math> 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.

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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>U(n)</math>. The map is the symmetrization map

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>U(n)</math>. 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>

where

:<math> {n \choose k_1, k_2, \ldots, k_m}

:<math> {n \choose k_1, k_2, \ldots, k_m} = \frac{n!}{k_1!\, k_2! \cdots k_m!}</math>

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.