Title: Homology of the boolean complex

Authors: Kári Ragnarsson, Bridget Tenner

Status: Submitted

Abstract: We construct and analyze an explicit basis for the homology of the 
boolean complex of a Coxeter system.  This gives combinatorial meaning to 
the spheres in the wedge sum describing the homotopy type of the complex.  
We assign a set of derangements to any unlabeled Coxeter graph.  For each 
derangement, we construct a corresponding element in the homology of the 
complex, and the collection of these elements forms a basis for the homology 
of the boolean complex.  In this manner, the spheres in the wedge sum 
describing the homotopy type of the complex can be represented by a set of 
derangements.
  We give an explicit, closed-form description of the derangements that can 
be obtained from any graph, and compute this set for several families of graphs.  
In the cases of complete graphs and Ferrers graphs, these calculations give 
bijective proofs of previously obtained enumerative results.