On flag spheres with few equators

TL;DR

This paper constructs a minimal 12-vertex flag simplicial 3-sphere with unique equator properties, answering a question about the structure of such spheres and their equators.

Contribution

It provides the first explicit example of a flag 3-sphere with no suspensions, minimal vertices, and only vertex links as equators, addressing a question by Chudnovsky and Nevo.

Findings

Constructed a 12-vertex flag 3-sphere with specific properties

Proved the minimal number of vertices for such complexes

Answered an open question about equator structures in flag spheres

Abstract

In this note we construct a flag simplicial $3$-sphere $\Delta$ with the following properties: - $\Delta$ is not a suspension; - $\Delta$ has no edge that can be contracted to obtain another flag sphere; - The only equators (induced subcomplexes which are spheres of codimension $1$) of $\Delta$ are vertex links. Our construction has $12$ vertices, the minimum number of vertices such a simplicial complex can have. This answers a question posed by Chudnovsky and Nevo.