The complex property of the boundary operator on 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 $\hat{I}{”}^H$ indeed act as $\hat{I}´\partial +\partial \hat{I}´$ and to decompose $\hat{I}{”}^H$ 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.