On the homology of several number-theoretic set families

TL;DR

This paper investigates the homological properties of simplicial complexes derived from various number-theoretic set families, providing conditions for easier computation and extending techniques to new classes of sets.

Contribution

It introduces a unifying condition for computing homology of complexes from number-theoretic sets and extends methods to coprime-free and generalized primitive sets.

Findings

Identified conditions simplifying homology calculations for certain set families.

Demonstrated that primitive, coprime, and product-free sets satisfy these conditions.

Extended homology computation techniques to coprime-free and generalized primitive sets.

Abstract

This paper describes the homology of various simplicial complexes associated to set families from combinatorial number theory, including primitive sets, pairwise coprime sets, product-free sets, and coprime-free sets. We present a condition on a set family that results in easy computation of the homology groups, and show that the first three examples, among many others, admit such a structure. We then extend our techniques to address the complexes associated to coprime-free sets and a generalization of primitive sets.