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\begin{document}

%-------------------------------------------------------------------


\section*{stable unitary group}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{lie_theory}{}\paragraph*{{Lie theory}}\label{lie_theory}

[[!include infinity-Lie theory - contents]]

\hypertarget{group_theory}{}\paragraph*{{Group Theory}}\label{group_theory}

[[!include group theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{classifying_space_for_topological_ktheory}{Classifying space for topological K-theory}\dotfill \pageref*{classifying_space_for_topological_ktheory} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\begin{defn}
\label{BUn}\hypertarget{BUn}{}
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 group]]s in the standard way. Write $B U(n) , B O(n) \in$ [[Top]] for the corresponding [[classifying space]].

Write

\begin{displaymath}
[X, B O(n)] := \pi_0 Top(X, B O(n))
\end{displaymath}
and

\begin{displaymath}
[X, B U(n)] := \pi_0 Top(X, B U(n))
\end{displaymath}
for the set of [[homotopy]]-classes of [[continuous function]]s $X \to B U(n)$.

\end{defn}
\begin{prop}
\label{BUnClassifyingSpace}\hypertarget{BUnClassifyingSpace}{}
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:

\begin{displaymath}
[X, B O(n)] \simeq O(n) Bund(X) \simeq Vect_{\mathbb{R}}(X,n)
  \,,
\end{displaymath}
\begin{displaymath}
[X, B U(n)] \simeq U(n) Bund(X) \simeq Vect_{\mathbb{C}}(X,n)
  \,.
\end{displaymath}
\end{prop}
\begin{defn}
\label{InclusionOfUns}\hypertarget{InclusionOfUns}{}
For each $n$ let

\begin{displaymath}
U(n) \to U(n+1)
\end{displaymath}
be the inclusion of [[topological group]]s given by inclusion of $n \times n$ [[matrices]] into $(n+1) \times (n+1)$-matrices given by the block-diagonal form

\begin{displaymath}
\left[g\right] \mapsto 
  \left[
    \itexarray{
      1 & [0]
      \\ 
      [0] & [g]
    }
  \right]
  \,.
\end{displaymath}
This induces a corresponding sequence of morphisms of classifying spaces, def. \ref{BUn}, in [[Top]]

\begin{displaymath}
B U(0) \hookrightarrow B U(1) \hookrightarrow B U(2) \hookrightarrow \cdots
  \,.
\end{displaymath}
Write

\begin{displaymath}
B U := {\lim_{\to}}_{n \in \mathbb{N}} B U(n)
\end{displaymath}
for the [[homotopy colimit]] (the ``homotopy [[direct limit]]'') over this diagram (see at \emph{[[homotopy colimit]]} the section \emph{\href{homotopy+limit#SequentialHocolims}{Sequential homotopy colimits}}).

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
The [[topological space]] $B U$ is \textbf{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

\begin{displaymath}
B U(\mathcal{H}) \simeq *
\end{displaymath}
while $B U$ has nontrivial [[homotopy group]]s 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 \hyperlink{Uk}{below}.

\end{remark}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{classifying_space_for_topological_ktheory}{}\subsubsection*{{Classifying space for topological K-theory}}\label{classifying_space_for_topological_ktheory}

\begin{prop}
\label{}\hypertarget{}{}
Write $\mathbb{Z}$ for the set of [[integers]] regarded as a [[discrete topological space]].

The product spaces

\begin{displaymath}
B O \times \mathbb{Z}\,,\;\;\;\;\;B U \times \mathbb{Z}
\end{displaymath}
are [[classifying spaces]] for real and complex [[topological K-theory]], respectively: for every compact Hausdorff topological space $X$, we have an isomorphism of groups

\begin{displaymath}
\tilde K(X)
  \simeq
  [X, B U ]
  \,.
\end{displaymath}
\begin{displaymath}
K(X)
  \simeq
  [X, B U \times \mathbb{Z}]
  \,.
\end{displaymath}
\end{prop}
See for instance (\hyperlink{Friedlander}{Friedlander, prop. 3.2}) or (\hyperlink{Karoubi}{Karoubi, prop. 1.32, theorem 1.33}).

\begin{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 space]]s $B U(n)$ are (see there) [[paracompact topological space]]s, hence normal, and since the inclusion morphisms are closed inclusions (\ldots{})) the [[hom-functor]] out of it commutes with the [[filtered colimit]]

\begin{displaymath}
\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}
  \,.
\end{displaymath}
Since $[X, B U(n)] \simeq U(n) Bund(X)$, in the last line the colimit is over [[vector bundle]]s 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. \ref{missing} we have

\begin{displaymath}
K(X) \simeq H^0(X, \mathbb{Z}) \oplus \tilde K(X)
  \,.
\end{displaymath}
Because $H^0(X,\mathbb{Z}) \simeq [X, \mathbb{Z}]$ it follows that

\begin{displaymath}
H^0(X, \mathbb{Z}) \oplus \tilde K(X)
  \simeq
  [X, \mathbb{Z}] \times [X, B U]
  \simeq
  [X, B U \times \mathbb{Z}]
  \,.
\end{displaymath}
\end{proof}
There is another variant on the classifying space

\begin{defn}
\label{Uk}\hypertarget{Uk}{}
Let

\begin{displaymath}
U_{\mathcal{K}}
  = 
  \left\{
    g \in U(\mathcal{H}) | g - id \in \mathcal{K}
  \right\}
\end{displaymath}
be the group of unitary operators on a [[separable Hilbert space]] $\mathcal{H}$ which differ from the identity by a [[compact operator]].

\end{defn}
Palais showed that

\begin{prop}
\label{}\hypertarget{}{}
$U_\mathcal{K}$ is a [[homotopy equivalence|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 group|Banach Lie]] [[normal subgroup]].

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

\begin{displaymath}
B U_{\mathcal{K}} \coloneqq U(\mathcal{H})/U_{\mathcal{K}}
\end{displaymath}
is a model for the [[classifying space]] of reduced K-theory.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[stable orthogonal group]]

\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

Introductions are in

\begin{itemize}%
\item [[Allen Hatcher]], \emph{Vector bundles and K-theory} (\href{http://www.math.cornell.edu/~hatcher/VBKT/VBpage.html}{web})


\item [[Eric Friedlander]], \emph{An introduction to K-theory} (\href{http://users.ictp.it/~pub_off/lectures/lns023/Friedlander/Friedlander.pdf}{pdf})


\item [[Max Karoubi]], \emph{K-theory: an introduction}


\item [[H. Blaine Lawson]], [[Marie-Louise Michelsohn]], \emph{[[Spin geometry]]}, Princeton University Press (1989)



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

\begin{itemize}%
\item [[Peter May]], p. 205 (213 of 251 in \emph{[[A Concise Course in Algebraic Topology]]}.

\end{itemize}
[[!redirects stable unitary groups]]



\end{document}