Ear Decomposition and Balanced Neighborly Simplicial Manifolds

TL;DR

This paper constructs new examples of balanced neighborly simplicial manifolds and spheres, revealing limitations of ear decompositions in balanced complexes, and advancing understanding of their combinatorial properties.

Contribution

It presents the first non-octahedral balanced 2-neighborly 3-sphere and a balanced triangulation of a lens space, along with a new non-spherical 5-manifold and insights into ear decompositions.

Findings

First non-octahedral balanced 2-neighborly 3-sphere with 16 vertices

Balanced triangulation of lens space L(3,1) with 16 vertices

Existence of a balanced 3-neighborly non-spherical 5-manifold with 18 vertices

Abstract

We find the first non-octahedral balanced 2-neighborly 3-sphere and the balanced 2-neighborly triangulation of the lens space $L(3,1)$. Each construction has 16 vertices. We show that there exists a balanced 3-neighborly non-spherical 5-manifold with 18 vertices. We also show that the rank-selected subcomplexes of a balanced simplicial sphere do not necessarily have an ear decomposition.