Classifying space for U(n)

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.

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 $e_{i}$ 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.

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.

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

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

is a fibre bundle of fibre $F_{n-1}(\mathbb {C} ^{k-1})$. Thus because $\pi _{p}(S^{2k-1})$ 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 $\pi _{p}(EU(n))$ is trivial for all $p\geq 1$.
Proof Let $\gamma$ be a map from the sphere $S^{p}$ to EU(n). As $S^{p}$ is compact, there exists $k$ such that $\gamma (S^{p})$ is included in $F_{n}(\mathbb {C} ^{k})$. By taking $k$ big enough, we see that $\gamma$ 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.

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 $K_{0}(BT)$ is given by numerical polynomials; more details below.

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 $S^{1}$ and the universal bundle is $S^{\infty }\longrightarrow \mathbb {C} P^{\infty }$. It is well known that the cohomology of $\mathbb {C} P^{k}$ is isomorphic to $\mathbb {R} \lbrack c_{1}\rbrack /c_{1}^{k+1}$, where $c_{1}$ 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 $K_{0}$, 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.

$K_{0}(BU(1))$ 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, $K_{0}(BT^{n})$ is numerical polynomials in n variables. The map $K_{0}(BT^{n})\to K_{0}(BU(n))$ is onto, via a splitting principle, as $T^{n}$ 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)

References

• S. Ochanine, L. Schwartz (1985), "Une remarque sur les générateurs du cobordisme complex", Math. Z., 190: 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., 316 (2): 385–432, doi:10.2307/2001355{{citation}}: CS1 maint: multiple names: authors list (link)