Ear Decomposition and Balanced Neighborly Simplicial Manifolds

Hailun Zheng

doi:10.37236/8341

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.

Highlights

A simplicial complex is called k-neighborly if every subset of vertices of size at most k is the set of vertices of one of its faces
The notion of neighborliness was extended to other classes of objects: for instance, neighborly cubical polytopes were defined and studied in [8], [9], and [17], and neighborly centrally symmetric polytopes and spheres were studied in [1], [3], [7] and [14]
It is interesting to ask whether every rank-selected subcomplex of a balanced simplicial polytope or sphere has a convex ear decomposition

Summary

Introduction

A simplicial complex is called k-neighborly if every subset of vertices of size at most k is the set of vertices of one of its faces. It is interesting to ask whether every rank-selected subcomplex of a balanced simplicial polytope or sphere has a convex ear decomposition. This statement, if true, would imply that rank-selected subcomplexes of balanced simplicial polytopes possess certain weak Lefschetz properties, see Theorem 3.9 in [22]. As a consequence, it would provide an alternative proof of the balanced Generalized Lower Bound Theorem, see Theorem 3.3 and Remark 3.4 in [13].

Basic properties of balanced neighborly spheres

Balanced 2-neighborly 3-sphere with 16 vertices

Balanced spheres and ear decomposition