#### Vol. 13, No. 4, 2020

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Minimal flag triangulations of lower-dimensional manifolds

### Christin Bibby, Andrew Odesky, Mengmeng Wang, Shuyang Wang, Ziyi Zhang and Hailun Zheng

Vol. 13 (2020), No. 4, 683–703
##### 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 $ℝ{P}^{2}$ and ${\mathbb{𝕊}}^{1}×{\mathbb{𝕊}}^{1}$ 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 $ℝ{P}^{2}$ (resp. ${\mathbb{𝕊}}^{1}×{\mathbb{𝕊}}^{1}$). 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, ${\gamma }_{2}:={f}_{1}-5{f}_{0}+16\ge 16{\beta }_{1}$, where ${f}_{i}$ is the number of $i$-dimensional faces and ${\beta }_{1}$ is the first Betti number over a field $k$. 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.

##### Keywords
flag complexes, triangulations of manifolds, Charney–Davis conjecture
##### Mathematical Subject Classification
Primary: 05E45, 57Q15