Abstract
A simplicial complex is said to satisfy complementarity if exactly one of each complementary pair of nonempty vertex-sets constitutes a face of the complex.
We show that if a d-dimensional combinatorial manifold M with n vertices satisfies complementarity then d=0, 2, 4, 8, or 16 with n=3d/2+3 and |M| is a “manifold like a projective plane”. Arnoux and Marin had earlier proved the converse statement.
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Datta, B. Combinatorial manifolds with complementarity. Proc. Indian Acad. Sci. (Math. Sci.) 104, 385–388 (1994). https://doi.org/10.1007/BF02863418
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DOI: https://doi.org/10.1007/BF02863418