# 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
* the Grassmannian of n-planes in an infinite-dimensional complex Hilbert space; or,
* 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
    $EU(n)=\left\{e_{1},\ldots ,e_{n}\ :\ (e_{i},e_{j})=\delta _{ij},e_{i}\in H\right\}.$
Here, H denotes 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,
    $BU(n)=\{V\subset H\ :\ \dim V=n\}$
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
    $BU(1)=PU(H),$
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 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 → ∞, 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
    ${\begin{aligned}F_{n}(\mathbf {C} ^{k})&\longrightarrow \mathbf {S} ^{2k-1}\\(e_{1},\ldots ,e_{n})&\longmapsto e_{n}\end{aligned}}$
is a fibre bundle of fibre F$_{n−1}$(C$^{k−1}$). Thus because $\pi _{p}(\mathbf {S} ^{2k-1})$ 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 ring of $\operatorname {BU} (n)$ with coefficients in the ring $\mathbb {Z}$ of integers is generated by the Chern classes:
    $H^{*}(\operatorname {BU} (n);\mathbb {Z} )=\mathbb {Z} [c_{1},\ldots ,c_{n}].$
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$^∞$ → CP$^∞$. It is well known that the cohomology of CP$^k$ is isomorphic to $\mathbb {Z} \lbrack c_{1}\rbrack /c_{1}^{k+1}$, where c$_1$ 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.
There are homotopy fiber sequences
    $\mathbb {S} ^{2n-1}\to BU(n-1)\to BU(n)$
Concretely, a point of the total space $BU(n-1)$ is given by a point of the base space $BU(n)$ classifying a complex vector space $V$, together with a unit vector $u$ in $V$; together they classify $u^{\perp }<V$ while the splitting $V=(\mathbb {C} u)\oplus u^{\perp }$, trivialized by $u$, realizes the map $BU(n-1)\to BU(n)$ representing direct sum with $\mathbb {C} .$
Applying the Gysin sequence, one has a long exact sequence
    $H^{p}(BU(n)){\overset {\smile d_{2n}\eta }{\longrightarrow }}H^{p+2n}(BU(n)){\overset {j^{*}}{\longrightarrow }}H^{p+2n}(BU(n-1)){\overset {\partial }{\longrightarrow }}H^{p+1}(BU(n))\longrightarrow \cdots$
where $\eta$ is the fundamental class of the fiber $\mathbb {S} ^{2n-1}$. By properties of the Gysin Sequence, $j^{*}$ is a multiplicative homomorphism; by induction, $H^{*}BU(n-1)$ is generated by elements with $p<-1$, where $\partial$ must be zero, and hence where $j^{*}$ must be surjective. It follows that $j^{*}$ must always be surjective: by the universal property of polynomial rings, a choice of preimage for each generator induces a multiplicative splitting. Hence, by exactness, $\smile d_{2n}\eta$ must always be injective. We therefore have short exact sequences split by a ring homomorphism
    $0\to H^{p}(BU(n)){\overset {\smile d_{2n}\eta }{\longrightarrow }}H^{p+2n}(BU(n)){\overset {j^{*}}{\longrightarrow }}H^{p+2n}(BU(n-1))\to 0$
Thus we conclude $H^{*}(BU(n))=H^{*}(BU(n-1))[c_{2n}]$ where $c_{2n}=d_{2n}\eta$. This completes the induction.

## K-theory of BU(n)

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 K$_0$, 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.
K$_0$(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, K$_0$(BT$^n$) is numerical polynomials in n variables. The map K$_0$(BT$^n$) → 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.

## Infinite classifying space

The canonical inclusions $\operatorname {U} (n)\hookrightarrow \operatorname {U} (n+1)$ induce canonical inclusions $\operatorname {BU} (n)\hookrightarrow \operatorname {BU} (n+1)$ on their respective classifying spaces. Their respective colimits are denoted as:
    $\operatorname {U} :=\lim _{n\rightarrow \infty }\operatorname {U} (n);$
    $\operatorname {BU} :=\lim _{n\rightarrow \infty }\operatorname {BU} (n).$
$\operatorname {BU}$ is indeed the classifying space of $\operatorname {U}$.