International Journal of Mathematics and Mathematical Sciences
Volume 2008 (2008), Article ID 254637, 21 pages
Research Article

Three-Dimensional Pseudomanifolds on Eight Vertices

Basudeb Datta1 and Nandini Nilakantan2

1Department of Mathematics, Indian Institute of Science, Bangalore 560 012, India
2Department of Mathematics & Statistics, Indian Institute of Technology, Kanpur 208 016, India

Received 9 April 2008; Revised 11 June 2008; Accepted 25 June 2008

Academic Editor: Pentti Haukkanen

Copyright © 2008 Basudeb Datta and Nandini Nilakantan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


A normal pseudomanifold is a pseudomanifold in which the links of simplices are also pseudomanifolds. So, a normal 2-pseudomanifold triangulates a connected closed 2-manifold. But, normal d-pseudomanifolds form a broader class than triangulations of connected closed d-manifolds for d3. Here, we classify all the 8-vertex neighbourly normal 3-pseudomanifolds. This gives a classification of all the 8-vertex normal 3-pseudomanifolds. There are 74 such 3-pseudomanifolds, 39 of which triangulate the 3-sphere and other 35 are not combinatorial 3-manifolds. These 35 triangulate six distinct topological spaces. As a preliminary result, we show that any 8-vertex 3-pseudomanifold is equivalent by proper bistellar moves to an 8-vertex neighbourly 3-pseudomanifold. This result is the best possible since there exists a 9-vertex nonneighbourly 3-pseudomanifold which does not allow any proper bistellar moves.