Minimal flag triangulations of lower-dimensional 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, ${\gamma }_2:=f_1-5f_0+16\ge 16{\beta }_1$, 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.

6 figures, 3 tables, 19 pages. Final version with a few typos corrected