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 e_i 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 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

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

is a fibre bundle of fibre F_n−1(C^k−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 F_n(C^k) (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 G_n(C^k) (with the induced topology).

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

Proof: Let γ : S^p → EU(n), since S^p is 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.$\Box$

In addition, U(n) acts freely on EU(n). The spaces F_n(C^k) and G_n(C^k) are 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 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, ..., 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^∞ → CP^∞. It is well known that the cohomology of CP^k is isomorphic to $\mathbf{R}\lbrack c_1\rbrack/c_1^{k+1}$, where c₁ is the Euler class of the U(1)-bundle S^2k+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.

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, ..., x_n], where x_i 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 $H^*(BU(n)) = \mathbb{Z}[[c_1,c_2,...,c_n]]$.

K-theory of BU(n)

Let us consider topological complex K-theory as the cohomology theory represented by the spectrum $KU$. In this case, $KU^*(BU(n))\cong \mathbb{Z}[t,t^{-1}][[c_1,...,c_n]]$, 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 $r\leq n$. In this description, the product structure on $KU_*(BU(n))$ comes from the H-space structure of $BU$ given by Whitney sum of vector bundles. This product is called the Pontryagin product.

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.

See also

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

References

• Contains calculation of $KU^*(BU(n))$ and $KU_*(BU(n))$.
• Contains a description of $K_0(BG)$ as a $K_0(K)$-comodule for any compact, connected Lie group.
• Explicit description of $K_0(BU(n))$