Minimal flag triangulations of lower-dimensional manifolds

TL;DR

This paper investigates minimal flag triangulations of 2- and 3-manifolds, providing vertex bounds, an algorithm for constructing such triangulations, and computational evidence supporting a generalized Charney-Davis conjecture.

Contribution

It establishes minimal vertex counts for certain 2-manifolds, introduces an algorithm for flag 3-manifold triangulations, and offers evidence for a generalized conjecture relating face numbers and Betti numbers.

Findings

Vertex-minimal flag triangulation of $ ext{RP}^2$ has 11 vertices.

Vertex-minimal flag triangulation of $ ext{S}^1 imes ext{S}^1$ has 12 vertices.

Supported a generalized Charney-Davis conjecture for flag 3-manifolds.

Abstract

We prove the following results on flag triangulations of 2- and 3-manifolds. In dimension 2, we prove that the vertex-minimal flag triangulations of $\mathbb{R} P^2$ and $\mathbb{S}^1\times \mathbb{S}^1$ have 11 and 12 vertices, respectively. In general, we show that $8+3k$ (resp. $8+4k$) vertices suffice to obtain a flag triangulation of the connected sum of $k$ copies of $\mathbb{R} P^2$ (resp. $\mathbb{S}^1\times \mathbb{S}^1$). In dimension 3, we describe an algorithm based on the Lutz-Nevo theorem which provides supporting computational evidence for the following generalization of the Charney-Davis conjecture: for any flag 3-manifold, $\gamma_2:=f_1-5f_0+16\geq 16 \beta_1$, where $f_i$ is the number of $i$-dimensional faces and $\beta_1$ is the first Betti number over a field. The conjecture is tight in the sense that for any value of $\beta_1$, there exists a flag 3-manifold for which the equality holds.