Tight triangulations of closed 3-manifolds (Q5964259)

The authors prove numerous interesting results regarding closed triangulated, \(\mathbb{F}\)-tight, stacked 3-manifolds. For instance the authors prove: (i) a triangulated closed 3-manifold is tight with respect to a field of odd characteristic if and only if it is neighborly, orientable and stacked; (ii) a \(\mathbb{Z}_2\)-tight, closed, triangulated 3-manifold is stacked if one of the following three conditions holds (a) it has less than 72 vertices (b) a torsion subgroup of its first homology group is of odd order (c) the rank of its first homology group is less than 189; moreover in the last case the 3-manifold can only be one of the following 3-manifolds: \(S^3\), \((S^2 \times S^1)^k\), \((S^2 \ltimes S^1)^k\); (iii) some inequalities relating the number of vertices with the rank of first homology group of a triangulated tight closed 3-manifold are given. The paper is very well written and very well organized; the authors have put tremendous efforts to make the paper easily understandable.