The simple connectivity of $B\Sol(q)$
by Andrew Chermak, Bob Oliver, and Sergey Shpectorov

Andrew Chermak
Department of Mathematics 
Kansas State University
Manhattan, KS 66502, USA}
chermak@math.ksu.edu

Bob Oliver
LAGA, Institut Galil\'ee 
Av. J-B Cl\'ement 
93430 Villetaneuse, France
bobol@math.univ-paris13.fr

Sergey Shpectorov
School of Mathematics 
University of Birmingham
Edgbaston, Birmingham, B15 2TT, UK
s.shpectorov@bham.ac.uk

Abstract:  A $p$-local finite group is an algebraic structure which 
includes two categories, a fusion system and a linking system, which mimic 
the fusion and linking categories of a finite group over one of its Sylow 
subgroups.  The $p$-completion of the geometric realization of the linking 
system is the classifying space of the finite group.  In this paper, we 
study the geometric realization, \emph{without} completion, of linking 
systems of certain exotic 2-local finite groups whose existence was 
predicted by Solomon and Benson, and prove that they are all simply 
connected.