Strong relations between discrete surfaces, poset-based connected manifolds, and normal pseudomanifolds

TL;DR

This paper explores the deep connections between discrete surfaces, poset-based connected manifolds, and normal pseudomanifolds, revealing their relationships and conditions for smoothness in discrete topology and geometry.

Contribution

It establishes relationships among these structures and provides conditions under which poset-based connected manifolds are smooth, advancing understanding in discrete topology and geometry.

Findings

Poset-based connected manifolds relate closely to discrete surfaces and pseudomanifolds.

Not all poset-based connected manifolds are smooth, even with additional topological properties.

Sufficient conditions for smoothness of poset-based connected manifolds are identified.

Abstract

In this paper, we study some relationships existing between some particular mathematical structures: discrete surfaces coming from discrete topology and mathematical morphology, poset-based connected manifolds coming from discrete topology, and normal pseudomanifolds which are much used in discrete geometry and topological data analysis. We will also show that, even when poset-based connected manifolds are assumed to be simplicial complexes, and then supplied with many additional topological properties, they are not necessarily smooth. A set of sufficient conditions to ensure that poset-based connected manifolds are smooth will be provided.