On $d$-dimensional cycles and the vanishing of simplicial homology

TL;DR

This paper introduces the concept of $d$-dimensional cycles as a homological generalization of graph cycles, exploring their properties and their relation to simplicial homology in higher dimensions.

Contribution

It defines $d$-dimensional cycles, analyzes their properties, and establishes their connection to the vanishing and non-vanishing of simplicial homology over various fields.

Findings

Non-zero $d$-dimensional homology corresponds to the presence of a $d$-dimensional cycle over fields of characteristic 2.

Orientable $d$-dimensional cycles lead to non-zero simplicial homology over any field.

The paper characterizes the relationship between combinatorial structures and homological properties in simplicial complexes.

Abstract

In this paper we introduce the notion of a $d$-dimensional cycle which is a homological generalization of the idea of a graph cycle to higher dimensions. We examine both the combinatorial and homological properties of this structure and use these results to describe the relationship between the combinatorial structure of a simplicial complex and its simplicial homology. In particular, we show that over any field of characteristic 2 the existence of non-zero $d$-dimensional homology corresponds exactly to the presence of a $d$-dimensional cycle in the simplicial complex. We also show that $d$-dimensional cycles which are orientable give rise to non-zero simplicical homology over any field.