Classifying space for U(n)

In mathematics, the classifying space for the unitary group U(n) is a space B(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

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.

Construction as an infinite Grassmannian

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 e_i are vectors in H, and δ_ij 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.

Case of line bundles

In the case of n = 1, one has

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

known to be a contractible 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.

Construction as an inductive limit

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.

Validity of the construction

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

Let $F_n(\mathbb{C}^k)$ 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 π_p(S^{2k − 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 π_p(EU(n)) is trivial for all $p\ge 1$.
Proof Let γ be a map from the sphere S^p to EU(n). As S^p is compact, there exists k such that γ(S^p) 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.

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 c_p 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 x_i 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 x_i's. We deduce
$H^*(BU(n))=\mathbb{R}\lbrack c_1,\ldots,c_n\rbrack,$
where the c_i's are the symmetric polynomials in the x_i's. $\Box$

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 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 $K_*(X) = \pi_*(K) \otimes K_0(X)$, 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.

See also

• Classifying space for O(n)
• Topological K-theory
• Atiyah-Jänich theorem

Notes

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

References

• S. Ochanine, L. Schwartz (1985), "Une remarque sur les générateurs du cobordisme complex", Math. Z. 190 (4): 543–557, doi:10.1007/BF01214753  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", Thesis (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. (American Mathematical Society) 316 (2): 385–432, doi:10.2307/2001355, JSTOR 2001355