On Walkup's class ${\cal K}(d)$ and a minimal triangulation of $(S^3 \times \rotatebox{90}{\ltimes} S^1)^{\#3}$

TL;DR

This paper constructs a minimal and tight triangulation of a specific 4-manifold, expanding the known examples of such manifolds and confirming theoretical bounds related to face vectors and Euler characteristics.

Contribution

It provides the first known minimal triangulation of the connected sum of three twisted sphere products, demonstrating a new member of the rare class of tight triangulated manifolds.

Findings

Constructed a vertex-minimal triangulation of a specific 4-manifold.

Proved the triangulation is tight, adding to the small set of known tight manifolds.

Confirmed the theoretical bounds relating face vectors and Euler characteristic.

Abstract

For $d \geq 2$, Walkup's class ${\cal K}(d)$ consists of the $d$-dimensional simplicial complexes all whose vertex-links are stacked $(d-1)$-spheres. Kalai showed that for $d \geq 4$, all connected members of ${\cal K}(d)$ are obtained from stacked $d$-spheres by finitely many elementary handle additions. According to a result of Walkup, the face vector of any triangulated 4-manifold $X$ with Euler characteristic $\chi$ satisfies $f_1 \geq 5f_0 - {15/2} \chi$, with equality only for $X \in {\cal K}(4)$. K\"{u}hnel observed that this implies $f_0(f_0 - 11) \geq -15\chi$, with equality only for 2-neighborly members of ${\cal K}(4)$. K\"{u}hnel also asked if there is a triangulated 4-manifold with $f_0 = 15$, $\chi = -4$ (attaining equality in his lower bound). In this paper, guided by Kalai's theorem, we show that indeed there is such a triangulation. It triangulates the connected sum of three copies of the twisted sphere product $S^3 \times {-2.8mm}_{-} S^1$. Because of K\"{u}hnel's inequality, the given triangulation of this manifold is a vertex-minimal triangulation. By a recent result of Effenberger, the triangulation constructed here is tight. Apart from the neighborly 2-manifolds and the infinite family of $(2d+ 3)$-vertex sphere products $S^{d-1} \times S^1$ (twisted for $d$ odd), only fourteen tight triangulated manifolds were known so far. The present construction yields a new member of this sporadic family. We also present a self-contained proof of Kalai's result.