In 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 ≥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.Comment: 11 pages. To appear in Advances in Geometr