Discrete Differential Calculus on Simplicial Complexes and Constrained Homology

TL;DR

This paper develops a discrete differential calculus framework on finite sets and simplicial complexes, enabling the construction of constrained homology and cohomology groups, with applications to hypergraph theory.

Contribution

It introduces a novel differential calculus approach on finite sets and simplicial complexes, leading to new methods for constructing constrained homology and cohomology groups.

Findings

Defined constrained homology groups using differential calculus.

Constructed constrained cohomology groups for independent hypergraphs.

Established connections between simplicial complexes and hypergraph complements.

Abstract

Let $V$ be a finite set. Let $\mathcal{K}$ be a simplicial complex with its vertices in $V$. In this paper, we discuss some differential calculus on $V$. We construct some constrained homology groups of $\mathcal{K}$ by using the differential calculus on $V$. Moreover, we define an independent hypergraph to be the complement of a simplicial complex in the complete hypergraph on $V$. Let $\mathcal{L}$ be an independent hypergraph with its vertices in $V$. We construct some constrained cohomology groups of $\mathcal{L}$ by using the differential calculus on $V$.