A characterization of $g_2$-minimal normal 3-pseudomanifolds with at most four singularities
Biplab Basak, Raju Kumar Gupta, Sourav Sarkar
TL;DR
This paper addresses the problem of characterizing $g_2$-minimal normal $3$-pseudomanifolds with up to four singularities. It extends the Basak–Swartz framework by showing that such complexes can be constructed from a one-vertex suspension of a base link and boundary complexes of $4$-simplices through $\text{connected sums}$, $\text{vertex foldings}$, and $\text{edge foldings}$; the analysis hinges on the structure of links at singular vertices, notably when $\mathbb{RP}^2$-singularities occur, and uses Möbius-strip arguments to locate non-cut edges and missing tetrahedra. The main results describe the precise decomposition: for three singularities (with one $\mathbb{RP}^2$-singularity) or four singularities (with two $\mathbb{RP}^2$-singularities), $\Delta$ arises from a one-vertex suspension plus boundary $4$-simplices via controlled foldings and sums, and corollaries express $\text{lk}(t,\Delta)$ as $\text{lk}(s,\Delta)\#_n (\mathbb{T}^2 \text{ and/or } \mathbb{RP}^2)$ factors. The results yield an explicit, constructive classification for these classes and delineate the limits beyond four singularities.
Abstract
Let $Δ$ be a $g_2$-minimal normal 3-pseudomanifold. A vertex in $Δ$ whose link is not a sphere is called a singular vertex. When $Δ$ contains at most two singular vertices, its combinatorial characterization is known [9]. In this article, we present a combinatorial characterization of such a $Δ$ when it has three singular vertices, including one $\mathbb{RP}^2$-singularity, or four singular vertices, including two $\mathbb{RP}^2$-singularities. In both cases, we prove that $Δ$ is obtained from a one-vertex suspension of a surface, and some boundary complexes of $4$-simplices by applying the combinatorial operations of types connected sums, vertex foldings, and edge foldings.