A characterization of normal 3-pseudomanifolds with at most two singularities

Abstract

Characterizing face-number-related invariants of a given class of simplicial complexes has been a central topic in combinatorial topology. In this regard, one of the well-known invariants is g2. Let K be a normal 3-pseudomanifold such that g2(K)≤g2(lk(v))+9 for some vertex v in K. Suppose either K has only one singularity or K has two singularities (at least) one of which is an RP2-singularity. We prove that K is obtained from some boundary complexes of 4-simplices by a sequence of operations of types connected sums, bistellar 1-moves, edge contractions, edge expansions, vertex foldings, and edge foldings. In case K has one singularity, |K| is a handlebody with its boundary coned off. Further, we prove that the above upper bound is sharp for such normal 3-pseudomanifolds.