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

Abstract

For $d\ge 2$, Walkup's class $\mathcal{K}(d)$ consists of the $d$-dimensional simplicial complexes all whose vertex-links are stacked $(d-1)$-spheres. Kalai showed that for $d\ge 4$, all connected members of $\mathcal{K}(d)$ 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 $f_1\ge 5f_0-15/2\chi$, with equality only for $X\in \mathcal{K}(4)$. Kühnel observed that this implies $f_0(f_0-11)\ge -15\chi$, with equality only for 2-neighborly members of $\mathcal{K}(4)$. Kühnel 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 Kühnel'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.

To appear in `Discrete Mathematics'

References in corpus (2)

• Triangulated Manifolds with Few Vertices: Combinatorial Manifolds
• Stacked polytopes and tight triangulations of manifolds

Cited by in corpus (1)

• Stacked polytopes and tight triangulations of manifolds