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

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.

10 pages, 2 figures. To appear in Topological Methods in Nonlinear Analysis

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