Sphere Triangulations and their Double Homology

TL;DR

This paper investigates the double homology of triangulated spheres, providing explicit computations for minimal cases, analyzing the effects of simplex removal, and exploring the relationship with neighborliness, supported by computational methods.

Contribution

It offers explicit calculations of double homology for sphere triangulations and links double homology to neighborliness, advancing understanding of topological properties.

Findings

Computed double homology for minimal sphere triangulations.

Analyzed the impact of removing maximal simplices using spectral sequences.

Generated complexes with exotic double homology ranks and related homology to neighborliness.

Abstract

We study the double homology associated to triangulated spheres and present two results. First, we explicitly compute the double homology for minimum degree sphere triangulations. Using a spectral sequence argument, we compute the effect of removing a maximal simplex of a non-neighborly sphere triangulation. Using these results and computational aid we generate complexes with exotic double homology rank. We also relate the double homology of a complex with how neighborly it is.