nLab
stable unitary group

Definition

Definition

For $n \in \mathbb{N}$ write $U(n)$ for the unitary group in dimension $n$ and $O(n)$ for the orthogonal group in dimension $n$, both regarded as topological groups in the standard way. Write $B U(n) , B O(n) \in$ Top for the corresponding classifying space.

Write

$$[X, B O(n)] := \pi_0 Top(X, B O(n))$$

and

$$[X, B U(n)] := \pi_0 Top(X, B U(n))$$

for the set of homotopy-classes of continuous functions $X \to B U(n)$.

Proposition

This is equivalently the set of isomorphism classes of $O(n)$- or $U(n)$-principal bundles on $X$ as well as of rank-$n$ real or complex vector bundles on $X$, respectively:

$$[X, B O(n)] \simeq O(n) Bund(X) \simeq Vect_{\mathbb{R}}(X,n) \,,$$

$$[X, B U(n)] \simeq U(n) Bund(X) \simeq Vect_{\mathbb{C}}(X,n) \,.$$

Definition

For each $n$ let

$$U(n) \to U(n+1)$$

be the inclusion of topological groups given by inclusion of $n \times n$ matrices into $(n+1) \times (n+1)$-matrices given by the block-diagonal form

$$\left[g\right] \mapsto \left[ \array{ 1 & [0] \\ [0] & [g] } \right] \,.$$

This induces a corresponding sequence of morphisms of classifying spaces, def. , in Top

$$B U(0) \hookrightarrow B U(1) \hookrightarrow B U(2) \hookrightarrow \cdots \,.$$

Write

$$B U := {\lim_{\to}}_{n \in \mathbb{N}} B U(n)$$

for the homotopy colimit (the “homotopy direct limit”) over this diagram (see at homotopy colimit the section Sequential homotopy colimits).

Remark

The topological space $B U$ is not equivalent to $B U(\mathcal{H})$, where $U(\mathcal{H})$ is the unitary group on a separable infinite-dimensional Hilbert space $\mathcal{H}$. In fact the latter is contractible, hence has a weak homotopy equivalence to the point

$$B U(\mathcal{H}) \simeq *$$

while $B U$ has nontrivial homotopy groups in arbitrary higher degree (by Kuiper's theorem).

But there is the group $U(\mathcal{H})_{\mathcal{K}} \subset U(\mathcal{H})$ of unitary operators that differ from the identity by a compact operator. This is essentially $U = \Omega B U$. See below.

Properties

Classifying space for topological K-theory

Proposition

Write $\mathbb{Z}$ for the set of integers regarded as a discrete topological space.

The product spaces

$$B O \times \mathbb{Z}\,,\;\;\;\;\;B U \times \mathbb{Z}$$

are classifying spaces for real and complex topological K-theory, respectively: for every compact Hausdorff topological space $X$, we have an isomorphism of groups

$$\tilde K(X) \simeq [X, B U ] \,.$$

$$K(X) \simeq [X, B U \times \mathbb{Z}] \,.$$

See for instance (Friedlander, prop. 3.2) or (Karoubi, prop. 1.32, theorem 1.33).

Proof

First consider the statement for reduced cohomology $\tilde K(X)$:

Since a compact topological space is a compact object in Top (and using that the classifying spaces $B U(n)$ are (see there) paracompact topological spaces, hence normal, and since the inclusion morphisms are closed inclusions (…)) the hom-functor out of it commutes with the filtered colimit

$$\begin{aligned} Top(X, B U) &= Top(X, {\lim_\to}_n B U(n)) \\ & \simeq {\lim_\to}_n Top(X, B U (n)) \end{aligned} \,.$$

Since $[X, B U(n)] \simeq U(n) Bund(X)$, in the last line the colimit is over vector bundles of all ranks and identifies two if they become isomorphic after adding a trivial bundle of some finite rank.

For the full statement use that by prop. we have

$$K(X) \simeq H^0(X, \mathbb{Z}) \oplus \tilde K(X) \,.$$

Because $H^0(X,\mathbb{Z}) \simeq [X, \mathbb{Z}]$ it follows that

$$H^0(X, \mathbb{Z}) \oplus \tilde K(X) \simeq [X, \mathbb{Z}] \times [X, B U] \simeq [X, B U \times \mathbb{Z}] \,.$$

There is another variant on the classifying space

Definition

Let

$$U_{\mathcal{K}} = \left\{ g \in U(\mathcal{H}) | g - id \in \mathcal{K} \right\}$$

be the group of unitary operators on a separable Hilbert space $\mathcal{H}$ which differ from the identity by a compact operator.

Palais showed that

Proposition

$U_\mathcal{K}$ is a homotopy equivalent model for $B U$. It is in fact the norm closure of the evident model of $B U$ in $U(\mathcal{H})$.

Moreover $U_{\mathcal{K}} \subset U(\mathcal{H})$ is a Banach Lie normal subgroup.

Since $U(\mathcal{H})$ is contractible, it follows that

$$B U_{\mathcal{K}} \coloneqq U(\mathcal{H})/U_{\mathcal{K}}$$

is a model for the classifying space of reduced K-theory.

Related concepts

• stable orthogonal group

References

Introductions are in

• Allen Hatcher, Vector bundles and K-theory (web)

• Eric Friedlander, An introduction to K-theory (pdf)

• Max Karoubi, K-theory: an introduction

• H. Blaine Lawson, Marie-Louise Michelsohn, Spin geometry, Princeton University Press (1989)

The H-space structure on $B U \times \mathbb{Z}$ is discussed in

• Peter May, p. 205 (213 of 251 in A Concise Course in Algebraic Topology.