Contents

group theory

Contents

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

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

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.

Introductions are in

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