Homology of the Boolean Complex

TL;DR

This paper constructs an explicit basis for the homology of the boolean complex associated with Coxeter systems, linking combinatorial derangements to topological spheres, and provides formulas for specific graph families.

Contribution

It introduces a novel combinatorial basis for the homology of boolean complexes using derangements, connecting graph theory and algebraic topology.

Findings

Basis for homology constructed explicitly using derangements

Provides closed-form descriptions for derangements from graphs

Offers bijective proofs for enumerative results in specific graph families

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 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.