Paper: Homotopy type of the boolean complex of a Coxeter system

Authors:     Kari Ragnarsson and Bridget Eileen Tenner

Institution: The Mathematical Sciences Research Institute
             Berkeley, California
             USA

             DePaul University
             Chicago, Illinois
             USA

Status:      preprint

Abstract: In any Coxeter group, the set of elements whose principal order
ideals are boolean forms a simplicial poset under the Bruhat order.  This
simplicial poset defines a cell complex, called the boolean complex.  In
this paper it is shown that, for any Coxeter system of rank n, this
boolean complex is homotopy equivalent to a wedge of (n-1)-dimensional
spheres, and the number of such spheres can be computed recursively.
Specific calculations of this number are given for all finite and affine
irreducible Coxeter systems, as well as for systems with graphs that are
disconnected, complete, or stars.  One implication of these results is
that the boolean complex is contractible if and only if a generator of the
Coxeter system is in the center of the group.