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\section*{Dwyer-Wilkerson H-space}

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

\hypertarget{homotopy_theory}{}\paragraph*{{Homotopy theory}}\label{homotopy_theory}

[[!include homotopy - contents]]

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

[[!include group theory - contents]]

\hypertarget{exceptional_structures}{}\paragraph*{{Exceptional structures}}\label{exceptional_structures}

[[!include exceptional structures -- 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{construction_as_a_homotopy_colimit}{Construction as a homotopy colimit}\dotfill \pageref*{construction_as_a_homotopy_colimit} \linebreak
\noindent\hyperlink{AsA2CompactGroup}{As a 2-compact group}\dotfill \pageref*{AsA2CompactGroup} \linebreak
\noindent\hyperlink{weyl_group}{Weyl group}\dotfill \pageref*{weyl_group} \linebreak
\noindent\hyperlink{homotopy_coset_space_}{Homotopy coset space $G_3/Spin(7)$}\dotfill \pageref*{homotopy_coset_space_} \linebreak
\noindent\hyperlink{RelationtoCo3}{Relation to the Conway group, $Co_3$}\dotfill \pageref*{RelationtoCo3} \linebreak
\noindent\hyperlink{RelationToPctomionic3Times3HermitianMatrices}{Relation to octonionic $3 \times 3$ matrix algebra?}\dotfill \pageref*{RelationToPctomionic3Times3HermitianMatrices} \linebreak
\noindent\hyperlink{homotopy_representation}{Homotopy representation}\dotfill \pageref*{homotopy_representation} \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 \emph{Dwyer-Wilkerson space} $G_3$ (\hyperlink{DwyerWilkerson93}{Dwyer-Wilkerson 93}) is a [[p-adic completion|2-complete]] [[H-space]], in fact a finite [[loop space]]/[[∞-group]], such that the mod 2 [[cohomology ring]] of its [[classifying space]]/[[delooping]] is the mod 2 [[Dickson invariants]] of rank 4. As such, it is the fifth and last space (see \hyperlink{Cohomology}{below}) in a series of [[∞-groups]] that starts with 4 [[compact Lie groups]], namely with the \href{normed+division+algebra#Automorphisms}{automorphism groups of real normed division algebras}:

\begin{tabular}{l|l|l|l|l|l}
$n=$&0&1&2&3&4\\
\hline 
$DI(n)=$&[[trivial group&1]]&[[Z/2]]&[[SO(3)]]&[[G2]]\\
&= Aut(R)&\href{complex+number#AutomorphismsOfComplexNumbersIsZ2}{= Aut(C)}&\href{quaternion#AutomorphismsOfQUatrnionsAlgebraIsSO3}{= Aut(H)}&\href{octonion#AutomorphismsOfOctonionAlgebraIsG2}{= Aut(O)}&\\
\end{tabular}

whence the notation ``$G_3$'' (suggested in \hyperlink{Moller95}{Møller 95, p. 5}).

While $G_3$ is not a [[compact Lie group]], it is a [[p-compact group|2-compact group]], hence a ``[[homotopy Lie group]]'' (see \hyperlink{AsA2CompactGroup}{below}).

The above progression starting with the [[automorphism groups]] of [[real numbers|real]] [[normed division algebras]] suggests that $G_3$ has a geometric or algebraic relevance in a context of [[division algebra and supersymmetry]]. This remains open, but there are speculations, see \hyperlink{RelationToPctomionic3Times3HermitianMatrices}{below}.

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

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

The [[ordinary cohomology]] of the [[classifying space]]/[[delooping]] $B G_3$ with [[coefficients]] in the [[prime field]] $\mathbb{F}_2$ is, as an [[associative algebra]] over the [[Steenrod algebra]], the ring of mod 2 [[Dickson invariants]] of rank 4. This is the ring of invariants of the natural action of $GL(4, \mathbf{F}_2)$ on the rank 4 polynomial algebra $H^{\ast}((B \mathbf{Z}/2)^4, \mathbf{F}_2)$, a polynomial algebra on classes $c_8$, $c_12$, $c_14$, and $c_15$ with $Sq^4 c_8 = c_{12}$, $Sq^2 c_{12} = c_{14}$, and $Sq^1 c_{14} = c_{15}$.

(\hyperlink{DwyerWilkerson93}{Dwyer-Wilkerson 93, Theorem 1.1})

As such, $G_3$ is the last in a series of [[∞-groups]] whose [[classifying spaces]]/[[deloopings]] have as mod 2 [[cohomology ring]] the mod 2 [[Dickson invariants]] for rank $n$, which starts with three ordinary [[compact Lie groups]]:

\begin{tabular}{l|l|l|l|l}
$n=$&1&2&3&4\\
\hline 
$DI(n)=$&[[Z/2]]&[[SO(3)]]&[[G2]]&[[G3]]\\
\end{tabular}

(\hyperlink{DwyerWilkerson93}{Dwyer-Wilkerson 93, top of p. 38 (2 of 28)})

This means in particular that the cohomology is an [[exterior algebra]] on generators of degree 7, 11, 13, 14 so it's (2-locally) a [[Poincaré duality space]] of [[dimension]] 45.

(\ldots{})

\hypertarget{construction_as_a_homotopy_colimit}{}\subsubsection*{{Construction as a homotopy colimit}}\label{construction_as_a_homotopy_colimit}

The space $B G_3$ is the 2-completion of the homotopy colimit of a diagram (\hyperlink{Notbohm}{Notbohm 03, Sec. 2}, \hyperlink{Ziemianski}{Ziemianski, 0.2.3}).

\hypertarget{AsA2CompactGroup}{}\subsubsection*{{As a 2-compact group}}\label{AsA2CompactGroup}

$G_3$ is the only exotic 2-group, or, in other words, the only simple [[p-compact group|2-compact group]] not arising as the 2-completion of a compact connected Lie group (\hyperlink{AndersenGrodal06}{Andersen-Grodal 06}).

\hypertarget{weyl_group}{}\subsubsection*{{Weyl group}}\label{weyl_group}

The analog of the [[Weyl group]] for $G_3$ is $\mathbb{Z}/2 \times GL(3,\mathbb{F}_2)$.

(\hyperlink{DwyerWilkerson93}{Dwyer-Wilkerson 93, middle of p. 38 (2 of 28)})

\hypertarget{homotopy_coset_space_}{}\subsubsection*{{Homotopy coset space $G_3/Spin(7)$}}\label{homotopy_coset_space_}

$G_3$ receives a homomorphism from [[Spin(7)]]. The [[homotopy fiber]] of the corresponding [[delooping]] map is a homotopy-[[coset space]]

\begin{displaymath}
G_3/Spin(7)
\end{displaymath}
The [[ordinary cohomology]] with [[coefficients]] in the [[prime field]] $\mathbb{F}_2$ of this space

\begin{enumerate}%
\item is concentrated in even degrees,


\item has [[Euler characteristic]] 24.



\end{enumerate}
(\hyperlink{DwyerWilkerson93}{Dwyer-Wilkerson 93, Theorem 1.8})

\hypertarget{RelationtoCo3}{}\subsubsection*{{Relation to the Conway group, $Co_3$}}\label{RelationtoCo3}

$B G_3$ receives a map from $B Co_3$, the [[delooping]] of the [[Conway group]], $Co_3$. This map has the property that it injects the mod two cohomology of $B G_3$ as a subring over which the mod two cohomology of $B Co_3$ is finitely generated as a module (see \hyperlink{Benson94}{Benson 94}). This continues a pattern from $B A_5 \to B SO(3)$ and $B M_{12} \to B G_2$, where $M_{12}$ is a [[Mathieu group]]. For further developments see (\href{{#AschbacherChermak10}}{Aschbacher-Chermak 10}).

$G_3$ and $Co_3$ both contain as 2-local subgroups the non-split extension, $(\mathbb{Z}/2)^4.G L(4, \mathbb{F}_2)$.

\hypertarget{RelationToPctomionic3Times3HermitianMatrices}{}\subsubsection*{{Relation to octonionic $3 \times 3$ matrix algebra?}}\label{RelationToPctomionic3Times3HermitianMatrices}

Since, by the \hyperlink{Cohomology}{above}, $G_3$ is (2-locally) a [[Poincaré duality space]] of [[dimension]] 45, there has been speculation that it might be related to the $8 + 2 \cdot 8 + 3 \cdot 7 = 45$-dimensional algebra

\begin{displaymath}
Mat^{skher}_{3 \times 3}(\mathbb{O})
\end{displaymath}
of skew-hermitian [[matrices]] over the [[octonions]] (\hyperlink{SolomonStancu08}{Solomon-Stancu 08, p. 175}, \hyperlink{Wilson09a}{Wilson 09a, slide 94}, \hyperlink{Benson98}{Benson 98, p. 19}). (Wilson's suggestion appears to arise from his construction of a 3-dimensional octonionic [[Leech lattice]], his representation of its automorphism group, the [[Conway group]] $Co_0$, as right multiplications by $3 \times 3$ matrices over the octonions (\hyperlink{Wilson09b}{Wilson 09b}), and the \hyperlink{RelationtoCo3}{relationship} between the latter's subgroup $Co_3$ and $G_3$.)

Incidentally, the algebra of $3\times 3$ [[hermitian matrices]] (as opposed to skew-hermitian) over the octonions

\begin{displaymath}
Mat^{her}_{3 \times 3}(\mathbb{O})
  \;
  \simeq_{\mathbb{R}}
  \;
  \underset{
    dim_{\mathbb{R}} = 26
  }{
  \underbrace{
    \mathbb{R}^{9,1}
      \oplus
    \mathbf{16}
  }}
  \oplus
  \mathbb{R}
  \,.
\end{displaymath}
is the [[exceptional Jordan algebra]] called the \emph{[[Albert algebra]]} (see \href{Albert+algebra#RelationTo10dSuperSpacetime}{there}).

\hypertarget{homotopy_representation}{}\subsubsection*{{Homotopy representation}}\label{homotopy_representation}

The possibility of there being a faithful 15-dimensional real homotopy representation of $G_3$ is raised in (\hyperlink{BakerBauer}{Baker-Bauer 19, p. 8}).

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

[[!include coset space structure on n-spheres -- table]]

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

Due to

\begin{itemize}%
\item [[William Dwyer]], [[Clarence Wilkerson]], \emph{A new finite loop space at the prime two}, J. Amer. Math. Soc. 6 (1993), 37-64 (\href{https://doi.org/10.1090/S0894-0347-1993-1161306-9}{doi:10.1090/S0894-0347-1993-1161306-9})

\end{itemize}
Review:

\begin{itemize}%
\item [[Jesper Møller]], \emph{Homotopy Lie groups}, Bull. Amer. Math. Soc. (N.S.) 32 (1995) 413-428 (\href{https://arxiv.org/abs/math/9510218}{arXiv:math/9510218})


\item [[Jesper Grodal]], \emph{The Classification of $p$–Compact Groups and Homotopical Group Theory}, Proceedings of the International Congress of Mathematicians, Hyderabad 2010 (\href{https://arxiv.org/abs/1003.4010}{arXiv:1003.4010}, \href{http://web.math.ku.dk/~jg/papers/icm.pdf}{pdf}, [[GrodalpCompactGroups2010.pdf:file]])


\item Dietrich Notbohm, \emph{On the compact 2-group $D I(4)$}, Journal für die reine und angewandte Mathematik (Crelles Journal), Volume 2003, Issue 555, Pages 163–185, (\href{https://pdfs.semanticscholar.org/76ee/dfae66955e452fc20e7ab4a19bd1f6284af4.pdf}{pdf})



\end{itemize}
See also

\begin{itemize}%
\item Martin Bendersky, Donald M. Davis, \emph{$v_1$-periodic homotopy groups of the Dwyer-Wilkerson space} (\href{https://arxiv.org/abs/0706.0993}{arXiv:0706.0993})


\item [[Andrew Baker]], [[Tilman Bauer]], \emph{The realizability of some finite-length modules over the Steenrod algebra by spaces} (\href{https://arxiv.org/abs/1903.10288}{arXiv:1903.10288})


\item Kasper Andersen, [[Jesper Grodal]], \emph{The classification of 2-compact groups}, J. Amer. Math. Soc. 22 (2009), 387-436 (\href{https://arxiv.org/abs/math/0611437}{arXiv:math/0611437})


\item Krzysztof Ziemiaski, \emph{A faithful complex representation of the 2-compact group DI(4)}, (\href{https://www.mimuw.edu.pl/~ziemians/pap/Thesis.pdf}{thesis})


\item David Benson, \emph{Conway’s group $Co_3$ and the Dickson invariants}, Manuscripta Math (1994) 85: 177 (\href{https://eudml.org/doc/156016}{dml:156016})


\item Andrew Baker, Tilman Bauer, \emph{The realizability of some finite-length modules over the Steenrod algebra by spaces}, (\href{https://arxiv.org/abs/1903.10288}{arXiv:1903.10288})


\item , Michael Aschbacher, Andrew Chermak, \emph{A group-theoretic approach to a family of 2-local finite groups constructed by Levi and Oliver}, (\href{http://annals.math.princeton.edu/2010/171-2/p06}{paper})



\end{itemize}
Speculation on possible geometric roles of $G_3$:

\begin{itemize}%
\item Eon Solomon, Radu Stancu, p. 175 of: \emph{Conjectures on finite and p-local groups}, L'Enseignement Mathématique (2) 54 (2008) 171-176 ([[SolomonStancuConjectures.pdf:file]], \href{http://doi.org/10.5169/seals-109929}{doi:10.5169/seals-109929})


\item David Benson, \emph{Cohomology of Sporadic Groups, Finite Loop Spaces, and the Dickson Invariants}, in P. Kropholler, G. Niblo, \& R. Stöhr (Eds.), Geometry and Cohomology in Group Theory (London Mathematical Society Lecture Note Series, pp. 10-23), 1998. Cambridge University Press.


\item [[Robert A. Wilson]], Slide 94 of: \emph{A new approach to the Leech lattice}, talk at University of Cambridge, 21st October 2009 (\href{http://www.maths.qmul.ac.uk/~raw/talks_files/Cambridge09.pdf}{slides pdf})

(on an [[octonions|octonionic]] construction of the [[Leech lattice]])


\item [[Robert A. Wilson]], \emph{Conway’s group and octonions}, (\href{http://www.maths.qmul.ac.uk/~raw/pubs_files/octoConway.pdf}{pdf})



\end{itemize}
[[!redirects Dwyer-Wilkerson H-spaces]]

[[!redirects Dwyer-Wilkerson space]] [[!redirects Dwyer-Wilkerson spaces]]

[[!redirects G3]]



\end{document}