Title: Control of fixed points and existence and uniqueness of centric
linking systems
Authors: George Glauberman and Justin Lynd

Abstract: A. Chermak has recently proved that to each saturated fusion
system over a finite $p$-group, there is a unique associated centric
linking system.  B.  Oliver extended Chermak's proof by showing that
all the higher cohomological obstruction groups relevant to unique
existence of centric linking systems vanish. Both proofs indirectly
assume the classification of finite simple groups. We show how to
remove this assumption, thereby giving a classification-free proof of
the Martino-Priddy conjecture concerning the $p$-completed classifying
spaces of finite groups.  Our main tool is a 1971 result of the first
author on control of fixed points by $p$-local subgroups.  This result
is directly applicable for odd primes, and we show how a slight
variation of it allows applications for $p=2$ in the presence of
offenders.