THE HOMOTOPY THEORY OF FUSION SYSTEMS
             by Carles Broto, Ran Levi, and Bob Oliver

We define and characterize a class of $p$-complete spaces $X$ which have 
many of the same properties as the $p$-completions of classifying spaces 
of finite groups.  For example, each such $X$ has a Sylow subgroup 
$BS--->X$, maps $BQ--->X$ for a $p$-group $Q$ are described via 
homomorphisms $Q--->S$, and $H^*(X;F_p)$ is isomorphic to a certain 
ring of ``stable elements'' in $H^*(BS;F_p)$.  These spaces arise as the 
``classifying spaces'' of certain algebraic objects which we call 
``$p$-local finite groups''.  Such an object consists of a system of 
fusion data in $S$, as formalized by Ll. Puig, extended by some extra 
information carried in a category which allows rigidification of the 
fusion data.

Since we distributed preprints of this paper, we have made several major
changes to its terminology; as well as some changes to the statement
and proof of Proposition 9.1.  Any suggestions or comments people might
want to make on the new terminology will be welcome.