Three‐Dimensional Pseudomanifolds on Eight Vertices
Basudeb Datta, Nandini Nilakantan · International Journal of Mathematics and Mathematical Sciences · 2008
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 d ≥ 3. 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.