A characterization of normal $3$-pseudomanifolds with $g_2\leq4$

TL;DR

This paper characterizes normal 3-pseudomanifolds with g_2 ≤ 4, showing they have at most two singular vertices and are constructed from boundary of 4-simplices via specific operations, extending previous results.

Contribution

It provides a detailed characterization of normal 3-pseudomanifolds with g_2 ≤ 4, including their singular vertices and construction methods, extending known classifications.

Findings

Normal 3-pseudomanifolds with g_2 ≤ 4 have at most two singular vertices.

Such pseudomanifolds are obtained from boundary of 4-simplices through connected sum, edge expansion, and edge folding.

The paper extends the classification to cases with no singular vertices and g_2 ≤ 9.

Abstract

We characterize normal $3$-pseudomanifolds with $g_2\leq4$. We know that if a $3$-pseudomanifold with $g_2\leq4$ does not have any singular vertices then it is a $3$-sphere. We first prove that a normal $3$-pseudomanifold with $g_2\leq4$ has at most two singular vertices. Then we prove that a normal $3$-pseudomanifold with $g_2 \leq 4$, which is not a $3$-sphere is obtained from some boundary of $4$-simplices by a sequence of operations connected sum, edge expansion and an edge folding. In addition, by using [17], we re-framed the characterization of normal $3$-pseudomanifolds with $g_2\leq 9$, when it has no singular vertices.