On the structure of cellular pseudomanifolds

TL;DR

This paper investigates the structure of cellular pseudomanifolds, a generalization of polytopal spheres, establishing their closure properties under duality and product, and classifying those with low excess.

Contribution

It demonstrates that cellular pseudomanifolds are closed under duality and product operations, and provides a complete classification for excess less than 2.

Findings

Cellular pseudomanifolds are closed under duality and product.

The geometric carriers of duals are homeomorphic.

Complete classification of pseudomanifolds with excess < 2.

Abstract

In this paper we study the structure of cellular pseudomanifolds (aka abstract polytopes). These are natural combinatorial generalisations of polytopal spheres (i.e., boundary complexes of convex polytopes). This class is closed under natural notions of duality and product. We show that they are also closed under an operation of direct product. Any cellular pseudomanifold and it's dual have homeomorphic geometric carriers, while the geometric carrier of the product of two of them is homeomorphic to the product of the carriers of the factors. The excess of a cellular pseudomanifold is defined as the non-negative integer $n - d - 2$ where $d$ is the dimension and $n$ is the number of vertices. We completely classify the cellular pseudo manifolds of excess $< 2$, and make some progress towards classifying those of excess 2.