\title{Classifying spaces of sporadic groups}


\author{David J. Benson}
\address{Department of Mathematical Sciences, University of Aberdeen,
         Aberdeen AB24 3UE, Scotland, UK}


\author{Stephen D. Smith}
\address{Department of Mathematics (m/c 249),
         University of Illinois at Chicago,
         851 S. Morgan, Chicago IL 60607-7045, USA}



\begin{document} 

\begin{abstract}\label{abstract}
For each of the 26 sporadic finite simple groups,
we construct a $2$-completed classifying space,
via a homotopy decomposition
in terms of classifying spaces of suitable $2$-local subgroups;
this leads to an additive decomposition of the mod $2$ group cohomology.
We also summarize the current status of knowledge in the literature
about the ring structure of the mod $2$ cohomology of those groups.

Our decompositions arise
via recent homotopy colimit theorems of various authors: in those results,
the colimit is indexed by a collection of $2$-subgroups,
which is ``ample'' in the sense of affording
the desired cohomology decomposition.
Furthermore our decompositions are ``sharp'' in the sense
of being simplified by the collapse of an underlying spectral sequence.

Among the various standard ample collections
available in the topological literature,
we make a suitably minimal choice for each group---and 
we further interpret that collection in terms of
an equivalent ``2-local geometry'' from the group theory literature:
namely as a simplicial complex determined by certain 2-local subgroups.
In particular,
we complete the verification that for each sporadic group,
an appropriate 2-local geometry affords a small ample collection.
One feature which emerges in each case
is that the geometry has ``flag-transitive'' action by the group,
so that the orbit complex (the quotient space modulo that action) is a simplex:
and then the diagram of classifying spaces of subgroups
which indexes the homotopy decomposition
is a pushout $n$-cube of the relevant dimension $n$
(though sometimes that orbit simplex diagram
is further simplified via cancellations).

The work begins with a fairly extensive initial exposition,
intended for non-experts, of background material
on the relevant constructions from algebraic topology,
and on local geometries from group theory.
The subsequent chapters then use those structures
to develop the main results on individual sporadic groups.
\end{abstract}