Articles written in Proceedings – Mathematical Sciences
Volume 104 Issue 2 May 1994 pp 385-388
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.
Volume 112 Issue 2 May 2002 pp 257-281
We explicitly determine all the two-dimensional weak pseudomanifolds on 8 vertices. We prove that there are (up to isomorphism) exactly 95 such weak pseudomanifolds, 44 of which are combinatorial 2-manifolds. These 95 weak pseudomanifolds triangulate 16 topological spaces. As a consequence, we prove that there are exactly three 8-vertex two-dimensional orientable pseudomanifolds which allow degree three maps to the 4-vertex 2-sphere.
Volume 115 Issue 3 August 2005 pp 279-307
A triangulation of a connected closed surface is called weakly regular if the action of its automorphism group on its vertices is transitive. A triangulation of a connected closed surface is called degree-regular if each of its vertices have the same degree. Clearly, a weakly regular triangulation is degree-regular. In , Lutz has classified all the weakly regular triangulations on at most 15 vertices. In , Datta and Nilakantan have classified all the degree-regular triangulations of closed surfaces on at most 11 vertices.
In this article, we have proved that any degree-regular triangulation of the torus is weakly regular. We have shown that there exists an