The complex property of the boundary operator on simplicial complexes

TL;DR

This paper investigates the boundary operator's property on simplicial complexes, characterizing it in terms of recurrence, and explores its implications for Hodge Laplacians, cohomology, and harmonic forms.

Contribution

It provides a characterization of the boundary operator property in $ ext{ell}^2$ and examines its impact on Hodge theory and transience in simplicial complexes.

Findings

Characterization of $oundaryoundary=0$ in $ ext{ell}^2$ via recurrence of links.

Establishment of Hodge Laplacian decomposition and harmonic eigenforms.

Discussion of transience properties in simplicial complexes.

Abstract

We study the complex property $\partial\partial = 0$ of the boundary operator $\partial$ on a weighted, infinite, and possibly non-locally finite simplicial complex. We give a characterization of this property in $\ell^2$ in terms of the recurrence of the links of simplices. The complex property is essential to ensure that Hodge Laplacians $\Delta^H $ indeed act as $\delta\partial + \partial\delta$ and to decompose $\Delta^H$ into a direct sum of operators acting on $k$-forms. Furthermore, it allows us to define relative cohomology classes, show a respective weak Hodge decomposition, and prove the existence of harmonic Dirichlet eigenforms. We also discuss a transience property for simplicial complexes, that was introduced by Parzanchevski and Rosenthal.