Three-dimensional normal pseudomanifolds with relatively few edges

• Published in: arXiv.org
• Main Authors: Basak, Biplab, Swartz, Ed
• Format: Paper Journal Article
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 05.02.2020
• Subjects: Combinatorial analysis; Lower bounds; Mathematics - Combinatorics; Mathematics - Geometric Topology; Singularities; Topology
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.1803.08942

Let \(\Delta\) be a \(d\)-dimensional normal pseudomanifold, \(d \ge 3.\) A relative lower bound for the number of edges in \(\Delta\) is that \(g_2\) of \(\Delta\) is at least \(g_2\) of the link of any vertex. When this inequality is sharp \(\Delta\) has relatively minimal \(g_2\). For example, whenever the one-skeleton of \(\Delta\) equals the one-skeleton of the star of a vertex, then \(\Delta\) has relatively minimal \(g_2.\) Subdividing a facet in such an example also gives a complex with relatively minimal \(g_2.\) We prove that in dimension three these are the only examples. As an application we determine the combinatorial and topological type of \(3\)-dimensional \(\Delta\) with relatively minimal \(g_2\) whenever \(\Delta\) has two or fewer singularities. The topological type of any such complex is a pseudocompression body, a pseudomanifold version of a compression body. Complete combinatorial descriptions of \(\Delta\) with \(g_2(\Delta) \le 2\) are due to Kalai [12] \((g_2=0)\), Nevo and Novinsky [13] \((g_2=1)\) and Zheng [21] \((g_2=2).\) In all three cases \(\Delta\) is the boundary of a simplicial polytope. Zheng observed that for all \(d \ge 0\) there are triangulations of \(S^d \ast \mathbb{RP}^2\) with \(g_2=3.\) She asked if this is the only nonspherical topology possible for \(g_2(\Delta)=3.\) As another application of relatively minimal \(g_2\) we give an affirmative answer when \(\Delta\) is \(3\)-dimensional.