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\begin{document}

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


\section*{SO(3)}

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

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

[[!include group theory - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{cohomology}{Cohomology}\dotfill \pageref*{cohomology} \linebreak
\noindent\hyperlink{finite_subgroups}{Finite subgroups}\dotfill \pageref*{finite_subgroups} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The [[special orthogonal group]] in [[dimension]] 3.

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

\hypertarget{cohomology}{}\subsubsection*{{Cohomology}}\label{cohomology}

\begin{prop}
\label{}\hypertarget{}{}
The [[integral cohomology|integral]] [[cohomology ring]] of the [[classifying space]] $B SO(3)$ is

\begin{displaymath}
H^\bullet\big( 
    B SO(3), \mathbb{Z}
  \big) 
  \;\simeq\;
  \mathbb{Z}\big[ p_1, W_3\big] / (2 W_3)
  \,,
\end{displaymath}
where

\begin{itemize}%
\item $p_1 \in H^4\big(B SO(3), \mathbb{Z}\big)$ is the [[universal characteristic class|universal]] [[first Pontryagin class]];


\item $W_3 = \beta(w_2) \in H^3\big( B SO(3), \mathbb{Z}\big)$ is the [[universal characteristic class|universal]] degree-3 [[integral Stiefel-Whitney class]].



\end{itemize}
\end{prop}
This is a special case of \hyperlink{Brown82}{Brown 82, theorem 1.5}, which is also reviewed as \hyperlink{RudolphSchmidt17}{Rudolph-Schmidt 17, Thm. 4.2.23 with Remark 4.2.25}.

$\backslash$linebreak

\hypertarget{finite_subgroups}{}\subsubsection*{{Finite subgroups}}\label{finite_subgroups}

\begin{theorem}
\label{ClassificationOfFiniteSubgroupsOfSO3}\hypertarget{ClassificationOfFiniteSubgroupsOfSO3}{}
\textbf{([[ADE classification]] of [[finite group|finite]] [[subgroups]] of [[SO(3)]] and [[spin group|Spin(3)]]$\simeq$ [[SU(2)]])}

The [[finite group|finite]] [[subgroups]] of the [[special orthogonal group]] $SO(3)$ as well as the [[finite group|finite]] [[subgroups]] of the [[special unitary group]] [[SU(2)]] are, up to [[conjugation]], given by the following classification:

[[!include ADE -- table]]

Here under the [[double cover]] projection (using the \href{spin+group#ExceptionalIsomorphisms}{exceptional isomorphism} $SU(2) \simeq Spin(3)$)

\begin{displaymath}
SU(2) \simeq Spin(3) \overset{\pi}{\longrightarrow} SO(3)
\end{displaymath}
all the finite subgroups of $SU(2)$ except the [[odd number|odd]]-[[order of a group|order]] [[cyclic groups]] are the [[preimages]] of the corresponding finite subgroups of $SO(3)$, in that we have [[pullback]] [[diagrams]]

\begin{displaymath}
\itexarray{
    \left\langle
     \exp
     \left(
       \pi \mathrm{i} \tfrac{1}{n} 
     \right)
    \right\rangle
    &
    =
    &
    \mathbb{Z}/(2n)
    &\overset{\phantom{AA}}{\hookrightarrow}&
    Spin(2)
    &\overset{\phantom{AA}}{\hookrightarrow}&
    Spin(3)
    \\
    &&
    \big\downarrow
    &{}^{(pb)}&
    \big\downarrow
    &{}^{(pb)}&
    \big\downarrow^{ \mathrlap{\pi} }
    \\
    \left\langle
     Ad_{
     \exp
     \left(
       \pi \mathrm{i} \tfrac{1}{n} 
     \right)
     }
    \right\rangle
    &
    =
    &
    \mathbb{Z}/n
    &\overset{\phantom{AA}}{\hookrightarrow}&
    SO(2)
    &\overset{\phantom{AA}}{\hookrightarrow}&
    SO(3)    
  }
\end{displaymath}
exhibiting the [[even number|even]] [[order of a group|order]] [[cyclic groups]] as subgroups of [[Spin(2)]], including the the minimal case of the [[group of order 2]]

\begin{displaymath}
\itexarray{
    \left\langle
     \exp
     \left(
       \pi \mathrm{i}
     \right)
     =
     -1
    \right\rangle
    &
    =
    &
    \mathbb{Z}/2
    &\overset{\phantom{AA}}{\hookrightarrow}&
    Spin(2)
    &\overset{\phantom{AA}}{\hookrightarrow}&
    Spin(3)
    \\
    &&
    \big\downarrow
    &{}^{(pb)}&
    \big\downarrow
    &{}^{(pb)}&
    \big\downarrow^{ \mathrlap{\pi} }
    \\
    \left\langle
     Ad_{
     \exp
     \left(
       \pi \mathrm{i} 
     \right)
     }
     =
     e
    \right\rangle
    &
    =
    &
    1
    &\overset{\phantom{AA}}{\hookrightarrow}&
    SO(2)
    &\overset{\phantom{AA}}{\hookrightarrow}&
    SO(3)    
  }
\end{displaymath}
as well as

\begin{displaymath}
\itexarray{
    \left\langle
      \exp\left( \pi \mathrm{i} \tfrac{1}{n} \right), \, \mathrm{j} 
    \right\rangle
    &=&
    2 D_{2n}
    &\overset{\phantom{AA}}{\hookrightarrow}&
    Pin_-(2)
    &\overset{\phantom{AA}}{\hookrightarrow}&
    Spin(3)
    \\
    &&
    \big\downarrow
    &{}^{(pb)}&
    \big\downarrow
    &{}^{(pb)}&
    \big\downarrow^{ \mathrlap{\pi} }
    \\
    \left\langle
      Ad_{\exp\left( \pi \mathrm{i} \tfrac{1}{n}  \right) }, 
      \, 
      Ad_{\mathrm{j}} 
    \right\rangle
    &&
    D_{2n}
    &\overset{\phantom{AA}}{\hookrightarrow}&
    O(2)
    &\overset{\phantom{AA}}{\hookrightarrow}&
    SO(3)    
  }
\end{displaymath}
exhibiting the [[binary dihedral groups]] as sitting inside the [[Pin(2)]]-[[subgroup]] of [[Spin(3)]],

but only [[commuting diagrams]]

\begin{displaymath}
\itexarray{
    \left\langle
     \exp
     \left(
       2 \pi \mathrm{i} \tfrac{1}{{2n+1}} 
     \right)
    \right\rangle
    &
    =
    &
    \mathbb{Z}/(2n+1)
    &&\overset{\phantom{AA}}{\hookrightarrow}&&
    Spin(3)
    \\
    &&
    \big\downarrow
      &&
      &&
    \big\downarrow^{ \mathrlap{\pi} }
    \\
    \left\langle
     Ad_{
     \exp
     \left(
       2 \pi \mathrm{i} \tfrac{1}{2n+1} 
     \right)
     }
    \right\rangle
    &
    =
    &
    \mathbb{Z}/(2n+1)
    &\overset{\phantom{AA}}{\hookrightarrow}&
    SO(2)
    &\overset{\phantom{AA}}{\hookrightarrow}&
    SO(3)    
  }
\end{displaymath}
for the [[odd number|odd]] [[order of a group|order]] [[cyclic group|cyclic]] [[subgroups]].

\end{theorem}
This goes back to (\href{finite+rotation+group#Klein1884}{Klein 1884, chapter I}). Full proof for $SO(3)$ is spelled out for instance in (\href{finite+rotation+group#Rees05}{Rees 05, theorem 11}, \href{finite+rotation+group#DeVisscher11}{De Visscher 11}). The proof for the case of $SL(2,\mathbb{C})$ is spelled out in (\href{finite+rotation+group#MillerBlichfeldtDickson16}{Miller-Blichfeldt-Dickson 16}) reviewed in (\href{finite+rotation+group#Serrano14}{Serrano 14, section 2}). The proof of the case for $SU(2)$ given the result for $SO(3)$ is spelled out in \href{finite+rotation+group#Keenan03}{Keenan 03, theorem 4}.

$\backslash$linebreak

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

[[!include low dimensional rotation groups -- table]]

\begin{itemize}%
\item [[Euclidean group]]


\item [[rigid body dynamics]]



\end{itemize}
$\backslash$linebreak

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

\begin{itemize}%
\item Jason Hanson, \emph{Rotations in three, four, and five dimensions} (\href{https://arxiv.org/abs/1103.5263}{arXiv:1103.5263})

\end{itemize}
See also

\begin{itemize}%
\item Wikipedia, \emph{\href{https://en.wikipedia.org/wiki/3D_rotation_group}{3D rotation group}}

\end{itemize}
On the [[integral cohomology]] of the [[classifying space]]:

\begin{itemize}%
\item [[Edgar H. Brown]], \emph{The Cohomology of $B SO_n$ and $BO_n$ with Integer Coefficients}, Proceedings of the American Mathematical Society, Vol. 85, No. 2 (Jun., 1982), pp. 283-288 (\href{https://www.jstor.org/stable/2044298}{jstor:2044298})

\end{itemize}
reviewed in

\begin{itemize}%
\item Gerd Rudolph, Matthias Schmidt, around Theorem 4.2.23 of \emph{Differential Geometry and Mathematical Physics: Part II. Fibre Bundles, Topology and Gauge Fields}, Theoretical and Mathematical Physics series, Springer 2017 (\href{https://link.springer.com/book/10.1007/978-94-024-0959-8}{doi:10.1007/978-94-024-0959-8})

\end{itemize}


\end{document}