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Proceedings of the London Mathematical Society Advance Access originally published online on March 21, 2008
Proceedings of the London Mathematical Society 2008 97(1):209-238; doi:10.1112/plms/pdn011
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© 2008 London Mathematical Society

The linking systems of the Solomon 2-local finite groups are simply connected

Andrew Chermak

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

Bob Oliver

LAGA, Institut Galilée
Av. J-B Clément
93430 Villetaneuse
France

Sergey Shpectorov

School of Mathematics
University of Birmingham
Edgbaston
Birmingham
B15 2TT
United Kingdom
s.shpectorov@bham.ac.uk

Received 6 October 2006. Revision received 20 December 2007.

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, without completion, of linking systems of certain exotic 2-local finite groups of which the existence was predicted by Solomon and Benson, and prove that they are all simply connected.

2000 Mathematics Subject Classification 55R35 (primary), 20D06, 20D20 (secondary).

The second author is partially supported by UMR 7539 of the CNRS. Part of this work was done during his stay at the Mittag-Leffler institute in Sweden. The third author was partially supported by an NSA grant.


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