Paper:       A Segal conjecture for p-completed classifying spaces

Author:      Kári Ragnarsson

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

Status:      preprint

Abstract:  We formulate and prove a new variant of the Segal Conjecture describing the group of homotopy classes of stable maps from the $p$-completed classifying 
space of a finite group $G$ to the classifying space of a compact Lie group $K$ as the $p$-adic completion of the Grothendieck group $A_p(G,K)$ of principal 
$(G,K)$-bundles whose isotropy groups are $p$-groups. Collecting the result for different primes $p$, we get a new and simple description of the group of homotopy 
classes of stable maps between (uncompleted) classifying spaces of groups. This description allows us to determine the kernel of the map from the Grothendieck group 
$A(G,K)$ of principal $(G,K)$-bundles to the group of homotopy classes of stable maps from $BG$ to $BK$.