The complex property of the boundary operator on simplicial complexes

We study the complex property $\mathrm{\partial }\mathrm{\partial }=0$ of the boundary operator $\mathrm{\partial }$ on a weighted, infinite, and possibly non-locally finite simplicial complex. We give a characterization of this property in ${\mathrm{\ell }}^2$ in terms of the recurrence of the links of simplices. The complex property is essential to ensure that Hodge Laplacians ${\mathrm{\Delta }}^H$ indeed act as $\delta \mathrm{\partial }+\mathrm{\partial }\delta$ and to decompose ${\mathrm{\Delta }}^H$ 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.