2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/70837In a previous paper the second author showed that if $M$ is a pseudomanifold with complementarity other than the 6-vertex real projective plane and the 9-vertex complex projective plane, then $M$ must have dimension $\geq 6$, and - in case of equality - $M$ must have exactly 12 vertices. In this paper we prove that such a 6-dimensional pseudomanifold does not exist. On the way to proving our main result we also prove that all combinatorial triangulations of the 4-sphere with at most 10 vertices are combinatorial 4-spheres.11 pages. To appear in Advances in GeometryGeometric TopologyAlgebraic Topology57Q15; 57Q25; 57R05text