Classifying space for U(n)

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In mathematics, the classifying space for U(n) may be constructed as the Grassmannian of n-planes in an infinite-dimensional complex Hilbert space. This construction is detailed here.

1 Construction
2 Validity of the construction
3 Case of n=1
4 Cohomology of the classifying space BU(n)
5 References

[edit] Construction

The total space $E U (n)$ 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 $δ i j$ 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

B U (n) = E U (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.

[edit] Validity of the construction

In this section, we will define the topology on EU(n) and prove that EU(n) is indeed contractible.

Let $F_n(\mathbb{C}^k)$ 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
$\begin{align} F_n(\mathbb{C}^k) & \longrightarrow & S^{2k-1} \\ (e_1,\ldots,e_n) & \longmapsto & e_n \end{align}$
is a fibre bundle of fibre $F_{n-1}(\mathbb{C}^{k-1})$ . Thus because $π p (S 2 k - 1)$ 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
$\pi_p(F_n(\mathbb{C}^k))=\pi_p(F_{n-1}(\mathbb{C}^{k-1}))=\ldots=\pi_p(F_1(\mathbb{C}^{k+1-n}))=\pi_p(S^{k-n})$ .
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 $π p (E U (n))$ is trivial for all $p\ge 1$ .
Proof Let $γ$ be a map from the sphere $S p$ to EU(n). As $S p$ is compact, there exists $k$ such that $γ(S p)$ is included in $F_n(\mathbb{C}^k)$ . 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 $F_n(\mathbb{C}^k)$ 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-complexe. By Whitehead Theorem and the above Lemma, EU(n) is contractible.

[edit] Case of n=1

In the case of n=1, one has

$EU(1)= \mathbb {C}^\infty$

Taking the quotient of $\mathbb{C}^\infty$ by an action of $\mathbb R^+$ , the group of positive numbers by multiplication (this does not change the homotopy type of the space, $\mathbb R^+$ being isomorphic to $R$ ), one sees that the space is essentially a unit ball in a complex countable-dimension vector 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.

[edit] Cohomology of the classifying space BU(n)

Proposition
The cohomology of the classifying space $H * (B U (n))$ 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 $S^\infty\longrightarrow CP^\infty$ . It is well known^[1] that the cohomology of $C P k$ 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 $S^{2k+1}\longrightarrow CP^k$ , and that the injections $CP^k\longrightarrow CP^{k+1}$ , 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 $(CP^\infty)^n$ 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 $CP^\infty$ . The Weyl group acts on $T$ by permuting the diagonal entries, hence it acts on $(CP^\infty)^n$ by permutation of the factors. The induce 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. $\Box$