A characterization of normal $3$-pseudomanifolds with $g_2\le 4$

Abstract

We characterize normal $3$-pseudomanifolds with $g_2\le 4$. We know that if a $3$-pseudomanifold with $g_2\le 4$ does not have any singular vertices then it is a $3$-sphere. We first prove that a normal $3$-pseudomanifold with $g_2\le 4$ has at most two singular vertices. Then we prove that a normal $3$-pseudomanifold with $g_2\le 4$, 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 $g_2\le 9$, when it has no singular vertices.

13 pages, 2 figures. arXiv admin note: text overlap with arXiv:2104.03751