2010/05/24 by Kari Ragnarsson, Ragnarsson, Kari, Bridget Eileen Tenner +1
Mathematics · #05E15 #06A07 #Algebraic Topology (math.AT) #Combinatorics (math.CO) #FOS: Mathematics #Primary 20F55 #Secondary 05C99 #math.AT #math.CO #msc:05C99 #msc:05E15 #msc:06A07 #msc:20F55
paper · pdf · doi:10.48550/arxiv.1005.4411
final version, to appear in Journal of Algebraic Combinatorics
arxiv created 2011/03/30 · arxiv updated 2011/04/01
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 finite simple 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.