A characterization of $g_2$-minimal normal 3-pseudomanifolds with at most four singularities

TL;DR

This paper provides a detailed combinatorial classification of $g_2$-minimal normal 3-pseudomanifolds with up to four singularities, including $ ext{RP}^2$-singularities, using specific topological operations.

Contribution

It extends the characterization of such pseudomanifolds to cases with three or four singularities, detailing their construction from simpler complexes via specific operations.

Findings

Characterization of 3-singular and 4-singular $g_2$-minimal pseudomanifolds.

Identification of their construction from suspensions and boundary complexes.

Use of connected sums, vertex foldings, and edge foldings in their formation.

Abstract

Let $\Delta$ be a $g_2$-minimal normal 3-pseudomanifold. A vertex in $\Delta$ whose link is not a sphere is called a singular vertex. When $\Delta$ contains at most two singular vertices, its combinatorial characterization is known [9]. In this article, we present a combinatorial characterization of such a $\Delta$ 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 $\Delta$ 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.