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 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-neighbourly members of K(4). Kühnel also asked if there is a triangulated 4-manifold with f0 = 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 a non-orientable closed 4-manifold with first Betti number β1 = 3. Because of Kühnel’s inequality, the given triangulation of this manifold is a vertex-minimal triangulation. We also present a self-complete proof of Kalai’s result. MSC 2000: 57Q15, 57R05.