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.

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

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

Here, H is an infinite-dimensional complex Hilbert space, the ei are vectors in H, and $\delta_{ij}$ 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

For n = 1, one has EU(1) = S, which is 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 classes of circle bundles over a manifold M are in one-to-one correspondence with the homotopy classes of maps from M to CP.

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) × ... × 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 Fn(Ck) be the space of orthonormal families of n vectors in Ck and let Gn(Ck) be the Grassmannian of n-dimensional subvector spaces of Ck. The total space of the universal bundle can be taken to be the direct limit of the Fn(Ck) as k → ∞, while the base space is the direct limit of the Gn(Ck) 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 Fn(Ck) and the quotient is the Grassmannian Gn(Ck). The map

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

is a fibre bundle of fibre Fn−1(Ck−1). Thus because $\pi_p(\mathbf{S}^{2k-1})$ is trivial and because of the long exact sequence of the fibration, we have

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

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

$\pi_p(F_n(\mathbf{C}^k)) = \pi_p(F_{n-1}(\mathbf{C}^{k-1})) = \cdots = \pi_p(F_1(\mathbf{C}^{k+1-n})) = \pi_p(\mathbf{S}^{k-n}).$

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

$EU(n)={\lim_{\to}}\;_{k\to\infty}F_n(\mathbf{C}^k)$

be the direct limit of all the Fn(Ck) (with the induced topology). Let

$G_n(\mathbf{C}^\infty)={\lim_\to}\;_{k\to\infty}G_n(\mathbf{C}^k)$

be the direct limit of all the Gn(Ck) (with the induced topology).

Lemma: The group $\pi_p(EU(n))$ is trivial for all p ≥ 1.

Proof: Let γ : Sp → EU(n), since Sp is compact, there exists k such that γ(Sp) is included in Fn(Ck). 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 Fn(Ck) and Gn(Ck) are CW-complexes. One can find a decomposition of these spaces into CW-complexes such that the decomposition of Fn(Ck), resp. Gn(Ck), is induced by restriction of the one for Fn(Ck+1), resp. Gn(Ck+1). Thus EU(n) (and also Gn(C)) 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 c1, ..., cn 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 SCP. It is well known[1] that the cohomology of CPk is isomorphic to $\mathbf{R}\lbrack c_1\rbrack/c_1^{k+1}$, where c1 is the Euler class of the U(1)-bundle S2k+1CPk, and that the injections CPkCPk+1, for kN*, 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)n. and its cohomology is R[x1, ..., xn], where xi is the Euler class of the tautological bundle over the i-th CP. The Weyl group acts on T by permuting the diagonal entries, hence it acts on (CP)n by permutation of the factors. The induced action on its cohomology is the permutation of the $x_i$'s. We deduce

$H^*(BU(n))=\mathbf{R}\lbrack c_1,\ldots,c_n\rbrack,$

where the $c_i$'s are the symmetric polynomials in the $x_i$'s.$\Box$

In contrast to the above description of $H^*(BU(n))$, many authors allow non-homogeneous elements in the cohomology, leading to the description $H^*(BU(n)) = \mathbb{Z}[[c_1,c_2,...,c_n]]$[2].

K-theory of BU(n)

Let us consider topological complex K-theory as the cohomology theory represented by the spectrum $KU$. In this case, $KU^*(BU(n))\cong \mathbb{Z}[t,t^{-1}][[c_1,...,c_n]]$[3], and $KU_*(BU(n))$ is the free $\mathbb{Z}[t,t^{-1}]$ module on $\beta_0$ and $\beta_{i_1}\ldots\beta_{i_r}$ for $n\geq i_j > 0$ and $r\leq n$.[4]

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); Q) = Q[w], where w is element dual to tautological bundle.

For the n-torus, K0(BTn) is numerical polynomials in n variables. The map K0(BTn) → K0(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.

Notes

1. ^ R. Bott, L. W. Tu-- Differential Forms in Algebraic Topology, Graduate Texts in Mathematics 82, Springer
2. ^ Adams, 1974 p. 49
3. ^ Adams 1974, p. 49
4. ^ Adams 1974, p. 47

References

• J. F. Adams (1974), Stable Homotopy and Generalised Homology, University Of Chicago Press, ISBN 0-226-00524-0 Contains calculation of $KU^*(BU(n))$ and $KU_*(BU(n))$.
• 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 $K_0(BG)$ as a $K_0(K)$-comodule for any compact, connected Lie group.
• L. Schwartz (1983), "K-théorie et homotopie stable", Thesis (Université de Paris–VII) Explicit description of $K_0(BU(n))$
• 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