On Walkup's class K(d) and a minimal triangulation of (S3 × 90 S1)\#3

Abstract

For d ≥ 2, Walkup's class K(d) consists of the d-dimensional simplicial complexes all whose vertex-links are stacked (d-1)-spheres. Kalai showed that for d ≥ 4, all connected members of 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 satisfies f1 ≥ 5f0 - 15/2 , with equality only for X ∈ K(4). K\"uhnel observed that this implies f0(f0 - 11) ≥ -15, with equality only for 2-neighborly members of K(4). K\"uhnel also asked if there is a triangulated 4-manifold with f0 = 15, = -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 S3 × -2.8mm- S1. Because of K\"uhnel'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 Sd-1 × S1 (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.