Harmonic Representatives in Homology over Arbitrary Fields

TL;DR

This paper extends the concept of harmonic chains to chain complexes over fields of positive characteristic, providing conditions for a Hodge decomposition and applications to CW complexes and data spaces.

Contribution

It introduces harmonic chain theory over arbitrary fields, establishes conditions for Hodge decomposition, and applies the theory to finite CW complexes and sampled data spaces.

Findings

Harmonic chains can be constructed over fields of positive characteristic.

Conditions for Hodge decomposition in this setting are identified.

Applications include CW decompositions of surfaces and data-derived spaces.

Abstract

We introduce a notion of harmonic chain for chain complexes over fields of positive characteristic. A list of conditions for when a Hodge decomposition theorem holds in this setting is given and we apply this theory to finite CW complexes. An explicit construction of the harmonic chain within a homology class is described when applicable. We show how the coefficients of usual discrete harmonic chains due to Eckmann can be reduced to localizations of the integers, allowing us to compare classical harmonicity with the notion introduced here. We focus on applications throughout, including CW decompositions of orientable surfaces and examples of spaces arising from sampled data sets.