The boolean complex of a Coxeter system, part 1

In any Coxeter group, the set of elements whose principal order ideals are boolean under the Bruhat order defines a cell complex, called the boolean complex. We show that for any Coxeter system of rank n, this boolean complex is homotopy equivalent to a edge of (n-1)-dimensional spheres, and the number of such spheres can be computed recursively from the Coxeter graph. Specific calculations of this number are given for certain families of Coxeter systems and graphs. The proof of this homotopy equivalence uses discrete Morse theory, and the proof is phrased in terms of finite simple graphs, thought of as unlabeled Coxeter graphs. We hint at the homological structure of this complex as well, which yields combinatorial proofs of many of our identities. This is joint work with Kari Ragnarsson, and with Anders Claesson and Sergey Kitaev.