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

Biplab Basak, Raju Kumar Gupta, Sourav Sarkar

Published 2021-04-08Version 1

Characterizing face-number related invariants of a given class of simplicial complexes has been a central topic in combinatorial topology. In this regards, one of the most well-known invariant is $g_2$. Kalai's relative lower bound [9] for $g_2$ says that if $K$ is a normal $d$-pseudomanifold with $d \geq 3,$ then $g_2(K) \geq g_2(lk (v))$ for any vertex $v$ of $K.$ In [6], two combinatorial tools - `vertex folding' and `edge folding' were defined. Let $K$ be a normal 3-pseudomanifold with at most two singularities and $t$ be a vertex of $K$ such that $g_2(lk (t)) \geq g_2(lk(v))$ for any other vertex $v$. They proved that if $g_2(K) = g_2(lk (t))$ then $K$ is obtained from a triangulation of $3$-sphere by a sequence of vertex folding and edge folding. This leads to a natural question - what will be the maximum value of $n\in\mathbb{N}$, for which $g_2{(K)} \leq g_2(lk (t)) + n$ implies $K$ is such combinatorial normal $3$-pseudomanifold? In this article we give the complete answer of this question. Let $K$ be a normal $3$-pseudomanifold with at most two singularities (in case of two singularities, we take one singularity is $\mathbb{RP}^2$). We prove that if $g_2{(K)} \leq g_2(lk (t)) + 9$ then $K$ is obtained from a triangulation of $3$-sphere by a sequence of vertex folding and edge folding. Further, we prove that the upper bound is sharp for such combinatorial normal $3$-pseudomanifolds.