Lab Notes on the exceptional Lie group $E_8$ at the prime $2$
\author[C. W. Wilkerson]{Clarence W. Wilkerson, Jr.}
\dedicatory{Dedicated to Morton L. Curtis (1921-1989).}
\address{Department of Mathematics, Purdue University, West Lafayette, 
Indiana 47907-1395}
\thanks{Thanks to the National Science Foundation, Purdue University, 
Johns Hopkins University, and Fukuoka University for financial support 
during this research and the 2000 sabbatical of the author. Thanks to 
the Clay Foundation for travel support during this research.}
\email{wilker@math.purdue.edu}

 This is an account of the author's use 
of computer algebra tools to explore the structure 
of the maximal elementary abelian $2$-subgroups of the exceptional 
Lie group $E_8$. The principal result obtained thus far by these
methods is that any rank $8$ connected $2$-compact group $(BX,X)$ with
Weyl group isomorphic to that of the exceptional Lie group $E_8$  has
its normalizer of the maximal torus isomorphic to that of
$E_8$ at the prime $2$. Similar results hold for the comparison of 
possible exotic forms of $G_2$, $DI(4)$, $F_4$,
and $E_7/\Center(E_7)$ to the standard forms.\\

Corollaries of this result include that the Krull dimension of the
mod $2$ cohomology of such $BX$ is $9$ and that the cohomology
ring is not Cohen-Macaulay. \\