Tight combinatorial manifolds and graded Betti numbers

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Abstract

In this paper, we study the conjecture of Kühnel and Lutz, who state that a combinatorial triangulation of the product of two spheres Si×Sj with j ≥ i is tight if and only if it has exactly i + 2j + 4 vertices. To approach this conjecture, we use graded Betti numbers of Stanley–Reisner rings. By using recent results on graded Betti numbers, we prove that the only if part of the conjecture holds when j > 2i and that the if part of the conjecture holds for triangulations all whose vertex links are simplicial polytopes. We also apply this algebraic approach to obtain lower bounds on the numbers of vertices and edges of triangulations of manifolds and pseudomanifolds.

Original languageEnglish
Pages (from-to)367-386
Number of pages20
JournalCollectanea Mathematica
Volume66
Issue number3
DOIs
Publication statusPublished - 2015 Sep 18
Externally publishedYes

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ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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