Three-dimensional normal pseudomanifolds with relatively few edges
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, wh...
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Published in | arXiv.org |
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Main Authors | , |
Format | Paper Journal Article |
Language | English |
Published |
Ithaca
Cornell University Library, arXiv.org
05.02.2020
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Subjects | |
Online Access | Get full text |
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Summary: | 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. |
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ISSN: | 2331-8422 |
DOI: | 10.48550/arxiv.1803.08942 |