Three-Dimensional Pseudomanifolds on Eight Vertices

Abstract EN

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.