Classifying space for U(n)

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In mathematics, the classifying space for U(n) may be constructed as either

  1. the Grassmannian of n-planes in an infinite-dimensional complex Hilbert space; or,
  2. the direct limit, with the induced topology, of Grassmannians of n planes.

Both constructions are detailed here.

Contents

[edit] Construction 1

The total space EU(n) of the universal bundle is given by

EU(n)=\{e_1,\ldots,e_n : (e_i,e_j)=\delta_{ij}, e_i\in \mathcal{H} \}.

Here, H is an infinite-dimensional complex Hilbert space, the ei are vectors in H, and δij is the Kronecker delta. The symbol (\cdot,\cdot) 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

BU(n) = EU(n) / U(n)

and is the set of Grassmannian n-dimensional subspaces (or n-planes) in H. That is,

BU(n) = \{ V \subset \mathcal{H} : \dim V = n \}

so that V is an n-dimensional vector space.

[edit] Construction 2

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

[edit] Validity of the second construction

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

Let F_n(\mathbb{C}^k) be the space of orthonormal families of n vectors in \mathbb{C}^k. The group U(n) acts freely on F_n(\mathbb{C}^k) and the quotient is the Grassmannian G_n(\mathbb{C}^k) of n-dimensional subvector spaces of \mathbb{C}^k. The map

\begin{align} F_n(\mathbb{C}^k) & \longrightarrow & S^{2k-1} \\ (e_1,\ldots,e_n) & \longmapsto & e_n \end{align}

is a fibre bundle of fibre F_{n-1}(\mathbb{C}^{k-1}). Thus because πp(S2k − 1) is trivial and because of the long exact sequence of the fibration, we have

\pi_p(F_n(\mathbb{C}^k))=\pi_p(F_{n-1}(\mathbb{C}^{k-1}))

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

\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}).

This last group is trivial for k > n + p. Let

EU(n)={\lim_{\rightarrow}}\;_{k\rightarrow\infty}F_n(\mathbb{C}^k)

be the direct limit of all the F_n(\mathbb{C}^k) (with the induced topology). Let

G_n(\mathbb{C}^\infty)={\lim_{\rightarrow}}\;_{k\rightarrow\infty}G_n(\mathbb{C}^k)

be the direct limit of all the G_n(\mathbb{C}^k) (with the induced topology).

Lemma
The group πp(EU(n)) is trivial for all p\ge 1.
Proof Let γ be a map from the sphere Sp to EU(n). As Sp is compact, there exists k such that γ(Sp) is included in F_n(\mathbb{C}^k). By taking k big enough, we see that γ is homotopic, with respect to the base point, to the constant map. \Box

In addition, U(n) acts freely on EU(n). The spaces F_n(\mathbb{C}^k) and G_n(\mathbb{C}^k) are CW-complexes. One can find a decomposition of these spaces into CW-complexes such that the decomposition of F_n(\mathbb{C}^k), resp. G_n(\mathbb{C}^k), is induced by restriction of the one for F_n(\mathbb{C}^{k+1}), resp. G_n(\mathbb{C}^{k+1}). Thus EU(n) (and also G_n(\mathbb{C}^\infty)) is a CW-complex. By Whitehead Theorem and the above Lemma, EU(n) is contractible.

[edit] Case of n = 1 (first construction)

In the case of n = 1, one has

EU(1)= S^\infty.\,

which is a contractible space (see Contractibility of unit sphere in Hilbert space.)

The base space is then

BU(1)= \mathbb{C}P^\infty,\,

the infinite-dimensional complex projective space. Thus, the set of isomorphism classes of circle bundles over a manifold M are in one-to-one correspondence with the homotopy classes of maps from M to \mathbb{C}P^\infty.

One also has the relation that

BU(1)= PU(\mathcal{H}),

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)\times \dots \times U(1), but need not have a chosen identification, one writes BT.

The topological K-theory K0(BT) is given by numerical polynomials; more details below.

[edit] Cohomology of BU(n)

Proposition
The cohomology of the classifying space H * (BU(n)) is a ring of polynomials in n variables c_1,\ldots,c_n where cp is of degree 2p.
Proof Let us first consider the case n = 1. In this case, U(1) is the circle S1 and the universal bundle is S^\infty\longrightarrow \mathbb{C}P^\infty. It is well known[1] that the cohomology of \mathbb{C}P^k is isomorphic to \mathbb{R}\lbrack c_1\rbrack/c_1^{k+1}, where c1 is the Euler class of the U(1)-bundle S^{2k+1}\longrightarrow \mathbb{C}P^k, and that the injections \mathbb{C}P^k\longrightarrow \mathbb{C}P^{k+1}, for k\in \mathbb{N}^*, 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 (\mathbb{C}P^\infty)^n and its cohomology is \mathbb{R}\lbrack x_1,\ldots,x_n\rbrack, where xi is the Euler class of the tautological bundle over the i-th \mathbb{C}P^\infty. The Weyl group acts on T by permuting the diagonal entries, hence it acts on (\mathbb{C}P^\infty)^n by permutation of the factors. The induced action on its cohomology is the permutation of the xi's. We deduce
H^*(BU(n))=\mathbb{R}\lbrack c_1,\ldots,c_n\rbrack,
where the ci's are the symmetric polynomials in the xi's. \Box

[edit] K-theory of BU(n)

The topological K-theory is known explicitly in terms of numerical symmetric polynomials.

The K-theory reduces to computing K0, since K-theory is 2-periodic 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 K_*(X) = \pi_*(K) \otimes K_0(X), where \pi_*(K)=\mathbf{Z}[t,t^{-1}], where t is the Bott generator.

K0(BU(1)) is the ring of numerical polynomials in w, regarded as a subring of H_*(BU(1);\mathbf{Q})=\mathbf{Q}[w], where w is element dual to tautological bundle.

For the n-torus, K0(BTn) is numerical polynomials in n variables. The map K_0(BT^n) \to 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

f(w_1,\dots,w_n) \mapsto \frac{1}{n!} \sum_{\sigma \in S_n} f(x_{\sigma(1)},\dots,x_{\sigma(n)})

and the image can be identified as the symmetric polynomials satisfying the integrality condition that

{n \choose n_1, n_2, \ldots, n_r}f(k_1,\dots,k_n) \in \mathbf{Z}

where

{n \choose k_1, k_2, \ldots, k_m} = \frac{n!}{k_1!\, k_2! \cdots k_m!}

is the multinomial coefficient and k_1,\dots,k_n contains r distinct integers, repeated n_1,\dots,n_r times, respectively.

[edit] References

S. Ochanine, L. Schwartz (1985), “Une remarque sur les générateurs du cobordisme complex”, Math. Z. 190: 543–557  Contains a description of K0(BG) as a K0(K)-comodule for any compact, connected Lie group.

L. Schwartz (1983), K-théorie et homotopie stable, Université de Paris–VII  Explicit description of K0(BU(n))

A. Baker, F. Clarke, N. Ray, L. Schwartz (1989), “On the Kummer congruences and the stable homotopy of BU”, Trans. Amer. Math. Soc. 316 (2): 385–432, <http://links.jstor.org/sici?sici=0002-9947%28198912%29316%3A2%3C385%3AOTKCAT%3E2.0.CO%3B2-F> 

[edit] See also

[edit] References

  1. ^ R. Bott, L. W. Tu -- Differential Forms in Algebraic Topology, Graduate Texts in Mathematics 82, Springer