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

${\displaystyle EU(n)=\{e_{1},\ldots ,e_{n}:(e_{i},e_{j})=\delta _{ij},e_{i}\in {\mathcal {H}}\}.\,}$ 

Here, H is an infinite-dimensional complex Hilbert space, the ${\displaystyle e_{i}}$  are vectors in H, and ${\displaystyle \delta _{ij}}$  is the Kronecker delta. The symbol ${\displaystyle (\cdot ,\cdot )}$  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

${\displaystyle BU(n)=EU(n)/U(n)\,}$ 

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

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

so that V is an n-dimensional vector space.

Case of line bundles

In the case of n = 1, one has

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

known to be a contractible space.

The base space is then

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

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 CP^∞.

One also has the relation that

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

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

The topological K-theory ${\displaystyle K_{0}(BT)}$  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 goes to infinity, while the base space is the direct limit of the G_n(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(C^k) be the space of orthonormal families of n vectors in C^k. The group U(n) acts freely on F_n(C^k) and the quotient is the Grassmannian G_n(C^k) of n-dimensional subvector spaces of C^k. The map

${\displaystyle {\begin{aligned}F_{n}(\mathbb {C} ^{k})&\longrightarrow S^{2k-1}\\(e_{1},\ldots ,e_{n})&\longmapsto e_{n}\end{aligned}}}$ 

is a fibre bundle of fibre F_{n - 1}(C^{k - 1}). Thus because ${\displaystyle \pi _{p}(S^{2k-1})}$  is trivial and because of the long exact sequence of the fibration, we have

${\displaystyle \pi _{p}(F_{n}(\mathbb {C} ^{k}))=\pi _{p}(F_{n-1}(\mathbb {C} ^{k-1}))}$ 

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

${\displaystyle \pi _{p}(F_{n}(\mathbb {C} ^{k}))=\pi _{p}(F_{n-1}(\mathbb {C} ^{k-1}))=\cdots =\pi _{p}(F_{1}(\mathbb {C} ^{k+1-n}))=\pi _{p}(S^{k-n}).}$ 

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

${\displaystyle EU(n)={\lim _{\rightarrow }}\;_{k\rightarrow \infty }F_{n}(\mathbb {C} ^{k})}$ 

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

${\displaystyle G_{n}(\mathbb {C} ^{\infty })={\lim _{\rightarrow }}\;_{k\rightarrow \infty }G_{n}(\mathbb {C} ^{k})}$ 

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

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

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 ${\displaystyle F_{n}(\mathbb {C} ^{k+1})}$ , resp. ${\displaystyle G_{n}(\mathbb {C} ^{k+1})}$ . Thus EU(n) (and also ${\displaystyle G_{n}(\mathbb {C} ^{\infty })}$ ) is a CW-complex. By Whitehead Theorem and the above Lemma, EU(n) is contractible.

Cohomology of ${\displaystyle BU(n)}$

Proposition: The cohomology of the classifying space ${\displaystyle H^{*}(BU(n))}$  is a ring of polynomials in n variables ${\displaystyle c_{1},\ldots ,c_{n}}$  where ${\displaystyle c_{p}}$  is of degree ${\displaystyle 2p}$ .

Proof: Let us first consider the case n = 1. In this case, U(1) is the circle ${\displaystyle S^{1}}$  and the universal bundle

is ${\displaystyle S^{\infty }\longrightarrow \mathbb {C} P^{\infty }}$ . It is well known^[1] that the cohomology of ${\displaystyle \mathbb {C} P^{k}}$  is isomorphic to ${\displaystyle \mathbb {R} \lbrack c_{1}\rbrack /c_{1}^{k+1}}$ , where ${\displaystyle c_{1}}$  is the Euler class of the U(1)-bundle ${\displaystyle S^{2k+1}\longrightarrow \mathbb {C} P^{k}}$ , and that the injections ${\displaystyle \mathbb {C} P^{k}\longrightarrow \mathbb {C} P^{k+1}}$ , for ${\displaystyle k\in \mathbb {N} ^{*}}$ , 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 ${\displaystyle (\mathbb {C} P^{\infty })^{n}}$  and its cohomology is ${\displaystyle \mathbb {R} \lbrack x_{1},\ldots ,x_{n}\rbrack }$ , where ${\displaystyle x_{i}}$  is the Euler class of the tautological bundle over the i-th ${\displaystyle \mathbb {C} P^{\infty }}$ . The Weyl group acts on T by permuting the diagonal entries, hence it acts on ${\displaystyle (\mathbb {C} P^{\infty })^{n}}$  by permutation of the factors. The induced action on its cohomology is the permutation of the ${\displaystyle x_{i}}$ 's. We deduce
${\displaystyle H^{*}(BU(n))=\mathbb {R} \lbrack c_{1},\ldots ,c_{n}\rbrack ,}$ 
where the ${\displaystyle c_{i}}$ 's are the symmetric polynomials in the ${\displaystyle x_{i}}$ 's.

K-theory of ${\displaystyle BU(n)}$

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

The K-theory reduces to computing ${\displaystyle K_{0}}$ , since K-theory is 2-periodic by the Bott periodicity theorem, and ${\displaystyle BU(n)}$  is a limit of complex manifolds, so it has a CW-structure with only cells in even dimensions, so odd K-theory vanishes.

Thus ${\displaystyle K_{*}(X)=\pi _{*}(K)\otimes K_{0}(X)}$ , where ${\displaystyle \pi _{*}(K)=\mathbf {Z} [t,t^{-1}]}$ , where t is the Bott generator.

${\displaystyle K_{0}(BU(1))}$  is the ring of numerical polynomials in w, regarded as a subring of ${\displaystyle H_{*}(BU(1);\mathbf {Q} )=\mathbf {Q} [w]}$ , where w is element dual to tautological bundle.

For the n-torus, ${\displaystyle K_{0}(BT^{n})}$  is numerical polynomials in n variables. The map ${\displaystyle K_{0}(BT^{n})\to K_{0}(BU(n))}$  is onto, via a splitting principle, as ${\displaystyle T^{n}}$  is the maximal torus of ${\displaystyle U(n)}$ . The map is the symmetrization map

${\displaystyle f(w_{1},\dots ,w_{n})\mapsto {\frac {1}{n!}}\sum _{\sigma \in S_{n}}f(x_{\sigma (1)},\dots ,x_{\sigma (n)})}$ 

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

${\displaystyle {n \choose n_{1},n_{2},\ldots ,n_{r}}f(k_{1},\dots ,k_{n})\in \mathbf {Z} }$ 

where

${\displaystyle {n \choose k_{1},k_{2},\ldots ,k_{m}}={\frac {n!}{k_{1}!\,k_{2}!\cdots k_{m}!}}}$ 

is the multinomial coefficient and ${\displaystyle k_{1},\dots ,k_{n}}$  contains r distinct integers, repeated ${\displaystyle n_{1},\dots ,n_{r}}$  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 ${\displaystyle K_{0}(BG)}$  as a ${\displaystyle K_{0}(K)}$ -comodule for any compact, connected Lie group.
• L. Schwartz (1983), "K-théorie et homotopie stable", Thesis, Université de Paris–VII Explicit description of ${\displaystyle K_{0}(BU(n))}$ 
• A. Baker, F. Clarke, N. Ray, L. Schwartz (1989), "On the Kummer congruences and the stable homotopy of BU", Trans. Amer. Math. Soc., 316 (2), American Mathematical Society: 385–432, doi:10.2307/2001355, JSTOR 2001355