On Walkup’s class [formula omitted] and a minimal triangulation of [formula omitted

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 χ satisfies f 1 ≥ 5 f 0 − 15 2 χ , with equality only for X ∈ K ( 4 ) . Kühnel observed that this implies f 0 ( f 0 − 11 ) ≥ − 15 χ , with equality only for 2-neighborly members of K ( 4 ) . Kühnel also asked if there is a triangulated 4-manifold with f 0 = 15 , χ = − 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 ▪ 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 ( 2 d + 3 ) -vertex sphere products S d − 1 × 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.