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

TL;DR

This paper classifies minimal triangulations of specific 4-manifolds, revealing exactly 12 tight neighborly triangulations with 15 vertices and automorphism group , including 10 orientable cases.

Contribution

It provides a complete classification of tight neighborly triangulations of certain 4-manifolds with 15 vertices, expanding understanding of minimal triangulations in topology.

Findings

Exactly 12 such triangulations up to isomorphism

10 of these triangulations are orientable

Constructed examples include one previously known by Bagchi and Datta

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.