Non-existence of 6-dimensional pseudomanifolds with complementarity

In a previous paper ([10]) 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 ≥ 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. 2000 Mathematics Subject Classification. 57Q15, 57Q25, 57R05.