Three dimensional pseudomanifolds on eight vertices

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 $d \geq 3$. 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 73 such 3-pseudomanifolds, 38 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 non-neighbourly 3-pseudomanifold ($B^3_9$ in Example 7 below) which does not allow any proper bistellar moves.