Tight triangulations of some 4-manifolds

Walkup’s class K(d) consists of the d-dimensional simplicial complexes all whose vertex links are stacked (d − 1)-spheres. According to a result of Walkup, the face vector of any triangulated 4-manifold X with Euler characteristic χ satisfies f1 ≥ 5f0 − 15 2 χ, with equality only for X ∈ K(4). Kühnel observed that this implies f0(f0− 11) ≥ −15χ, with equality only for 2-neighborly members of K(4). For n = 6, 11 and 15, there are triangulated 4-manifolds with f0 = n and f0(f0 − 11) = −15χ. In this article, we present triangulated 4-manifolds with f0 = 21, 26 and 41 which satisfy f0(f0 − 11) = −15χ. All these triangulated manifolds are tight and strongly minimal. MSC 2000 : 57Q15, 57R05.