The non-pure version of the simplex and the boundary of the simplex

TL;DR

This paper introduces minimal non-pure simplicial complexes, called NH-balls and NH-spheres, and characterizes them via their iterated Alexander duals converging to simplexes or their boundaries.

Contribution

It defines and studies minimal non-pure simplicial complexes and links their structure to the behavior of their iterated Alexander duals.

Findings

Minimal NH-balls and NH-spheres are characterized by their Alexander duals.

These complexes have the minimal number of vertices for their type.

The iterated Alexander duals of these complexes converge to simplexes or their boundaries.

Abstract

We introduce the non-pure versions of simplicial balls and spheres with minimum number of vertices. These are a special type of non-homogeneous balls and spheres (NH-balls and NH-spheres) satisfying a minimality condition on the number of maximal simplices. The main result is that minimal NH-balls and NH-spheres are precisely the simplicial complexes whose iterated Alexander duals converge respectively to a simplex or the boundary of a simplex.