Paper:       Alternative stable homotopy classification of BG^_p

Author:      Kári Ragnarsson

Institution: Department of Mathematical Sciences,
             University of Aberdeen
             Aberdeen AB243UE
             United Kingdom

Status:      Published in  Topology  45 (2006) 601--609

Abstract:  We give an alternative to the stable classification of $p$-completed homotopy types of classifying spaces of finite groups offered by Martino-Priddy in 
\cite{MP3}. For a finite group $G$ with Sylow subgroup $S$, we regard the stable $p$-completed classifying space $\Stable{\pComp{BG}}$ as an object under 
$\Stable{BS}$ via the canonical inclusion map. Thus we get a classification in terms of induced fusion systems. Applying Oliver's solution \cite{Ol1,Ol2} to the 
Martino-Priddy conjecture \cite{MP}, we obtain the surprising result that the unstable homotopy type of $\pComp{BG}$ is determined by the map \mbox{$ \Stable{BS} \to 
\Stable{\pComp{BG}}$}, but not by the homotopy type of $\Stable{\pComp{BG}}$.