Minimal triangulations of (S^3\times S^1)^{#3} and (S^3 \(twisted product) S^1)^{#3}

Abstract

A triangulated $d$-manifold $K$, satisfies the inequality $\binom{f_0(K)-d-1}{2}\geq \binom{d+2}{2}\beta_1(K;\mathbb{Z}_2)$ for $d\geq 3$. The triangulated $d$-manifolds that meet the bound with equality are called {\em tight neighborly}. In this paper, we present tight neighborly triangulations of 4-manifolds on 15 vertices with $\mathbb{Z}_3$ as automorphism group. One such example wasconstructed by Bagchi and Datta in 2011. We show that there are exactly 12 such triangulations up to isomorphism, 10 of which are orientable.