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arXiv · 2605.21069

The complex property of the boundary operator on simplicial complexes

Philipp Bartmann·Matthias Keller
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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.

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Philipp Bartmann, Matthias Keller. 2026-05-20. The complex property of the boundary operator on simplicial complexes. https://arxiv.org/abs/2605.21069

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