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{{Short description|Exact homotopy case}}

In [[mathematics]], the '''[[classifying space]] for [[unitary group|U(''n'')]]''' may be constructed as either

{{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

# the [[Grassmannian]] of ''n''-planes in an infinite-dimensional complex [[Hilbert space]]; or,

# the direct limit, with the induced topology, of [[Grassmannian|Grassmannians]]s of ''n'' planes.

Both constructions are detailed here.

==Construction 1as an infinite Grassmannian==

The [[total space]] <math>EU(''n'')</math> 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''<mathsub>e_i''i''</mathsub> 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>

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 ===

==Construction 2==

For ''n'' = 1, one has EU(1) = '''S'''^∞, which is [[Contractibility of unit sphere in Hilbert space|known to be a contractible space]]. The base space is then BU(1) = '''CP'''^∞, 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'''^∞.

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.

One also has the relation that

==Validity of the second construction==

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

:<math>BU(1)= PU(H),</math>

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 U(1) × ... × U(1), but need not have a chosen identification, one writes B''T''.

The [[topological K-theory]] ''K''₀(B''T'') is given by [[numerical polynomial]]s; more details below.

==Construction as an inductive limit==

Let ''F_n''('''C'''^''k'') be the space of orthonormal families of ''n'' vectors in '''C'''^''k'' and let ''G_n''('''C'''^''k'') be the Grassmannian of ''n''-dimensional subvector spaces of '''C'''^''k''. The total space of the universal bundle can be taken to be the direct limit of the ''F_n''('''C'''^''k'') as ''k'' → ∞, while the base space is the direct limit of the ''G''_''n''('''C'''^''k'') as ''k'' → ∞.

===Validity of the construction===

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

The group U(''n'') acts freely on ''F''_''n''('''C'''^''k'') and the quotient is the Grassmannian ''G''_''n''('''C'''^''k''). The map

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>. The group <math>U(n)</math> acts

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}

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''<mathsub>F_{''n-1}''−1</sub>(\mathbb{'''C}^{'''<sup>''k-1})''−1</mathsup>). 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 <math>''k</math>'' 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''&nbsp;>&nbsp;''n''&nbsp;+&nbsp;''p''. Let

: <math>EU(n)={\lim_{\rightarrowto}}\;_{k\rightarrowto\infty}F_n(\mathbbmathbf{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(\mathbbmathbf{C}^\infty)={\lim_{\rightarrow}to}\;_{k\rightarrowto\infty}G_n(\mathbbmathbf{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).

<blockquote>'''Lemma:''' The group <math>\pi_p(EU(n))</math> is trivial for all ''p'' ≥ 1.</blockquote>

'''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 <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>

'''Proof:''' Let γ : '''S'''^''p'' → EU(''n''), since '''S'''^''p'' is [[compact space|compact]], there exists ''k'' such that γ('''S'''^''p'') is included in ''F''_''n''('''C'''^''k''). 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, <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.

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 ''F''_''n''('''C'''^''k''+1), resp. ''G''_''n''('''C'''^''k''+1). Thus EU(''n'') (and also ''G''_''n''('''C'''^∞)) is a CW-complex. By [[Whitehead theorem|Whitehead Theorem]] and the above Lemma, EU(''n'') is contractible.

== Case of <math> n = 1 </math> (first construction)==

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

== Cohomology of BU(''n'')==

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

'''Proposition''': The [[cohomology ring]] of <math>\operatorname{BU}(n)</math> with coefficients in the [[Ring (mathematics)|ring]] <math>\mathbb{Z}</math> of [[Integer|integers]] is generated by the [[Chern class|Chern classes]]:<ref>Hatcher 02, Theorem 4D.4.</ref>

: <math>H^*(\operatorname{BU}(n);\mathbb{Z})

[[Contractibility of unit sphere in Hilbert space|known to be a contractible space]].

=\mathbb{Z}[c_1,\ldots,c_n].</math>

'''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>\mathbb{Z}\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.

The base space is then

There are homotopy fiber sequences

:<math>BU(1)= \mathbb{C}P^\infty,\,</math>

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

Concretely, a point of the total space <math>BU(n-1)</math> is given by a point of the base space <math>BU(n)</math> classifying a complex vector space <math>V</math>, together with a unit vector <math>u</math> in <math>V</math>; together they classify <math> u^\perp < V </math> while the splitting <math>V = (\mathbb{C} u) \oplus u^\perp </math>, trivialized by <math>u</math>, realizes the map <math> B U(n-1) \to B U(n) </math> representing direct sum with <math>\mathbb{C}.</math>

One also has the relation that

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

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

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

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

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>.

: <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>

The [[topological K-theory]] <math>K_0(BT)</math> is given by [[numerical polynomial]]s; more details below.

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.

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

'''Proposition'''<br />

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

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

<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 <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>.

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

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

{{confusing section|date=August 2022}}

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

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]].

<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><br />

== 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 ''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.

<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.

''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''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

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

:<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.

== 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–Jänich theorem]]

== Notes ==

Line 141 ⟶ 147:

==References==

*{{citation

|author=SJ. Ochanine, LF. SchwartzAdams

|title=Stable Homotopy and Generalised Homology

|title=Une remarque sur les générateurs du cobordisme complex

|publisher=University Of Chicago Press

|year=1974

|isbn=0-226-00524-0

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

*{{citation

|author1=S. Ochanine |author2=L. Schwartz |title=Une remarque sur les générateurs du cobordisme complex

|journal=Math. Z.

|volume=190

|year=1985

|issue=4

|pages=543–557

|doi=10.1007/BF01214753

Line 157 ⟶ 170:

}} 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

Line 164 ⟶ 176:

|year=1989

|pages=385–432

|url=http://links.jstor.org/sici?sici=0002-9947%28198912%29316%3A2%3C385%3AOTKCAT%3E2.0.CO%3B2-F

|doi=10.2307/2001355

|jstor=10.2307/2001355

|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]]