On Walkup’s class K(d)K(d) and a minimal triangulation of (S3S1)#3

For d≥2d≥2, Walkup’s class K(d)K(d) consists of the dd-dimensional simplicial complexes all whose vertex-links are stacked (d−1)(d−1)-spheres. Kalai showed that for d≥4d≥4, all connected members of K(d)K(d) are obtained from stacked dd-spheres by finitely many elementary handle additions. According to a result of Walkup, the face vector of any triangulated 4-manifold XX with Euler characteristic χχ satisfies f1≥5f0−152χ, with equality only for X∈K(4)X∈K(4). Kühnel observed that this implies f0(f0−11)≥−15χf0(f0−11)≥−15χ, with equality only for 2-neighborly members of K(4)K(4). Kühnel also asked if there is a triangulated 4-manifold with f0=15f0=15, χ=−4χ=−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 S3S1. 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)(2d+3)-vertex sphere products Sd−1×S1 (twisted for dd 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.