Homotopy Type of the Boolean Complex of a Coxeter System

TL;DR

This paper proves that the boolean complex of any Coxeter system is homotopy equivalent to a wedge of spheres, introduces the boolean number as a new graph invariant, and computes it for various Coxeter systems.

Contribution

It establishes the homotopy type of the boolean complex for all Coxeter systems and introduces the boolean number as a new invariant derived from the Coxeter graph.

Findings

Boolean complex is homotopy equivalent to a wedge of spheres.

Boolean number can be computed recursively from the Coxeter graph.

Boolean complex is contractible iff a generator is central.

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, the boolean complex is homotopy equivalent to a wedge of (n-1)-dimensional spheres. The number of such spheres can be computed recursively from the unlabeled Coxeter graph, and defines a new graph invariant called the boolean number. Specific calculations of the boolean 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. 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.