Collectanea Mathematica

, Volume 66, Issue 3, pp 367–386 | Cite as

Tight combinatorial manifolds and graded Betti numbers

  • Satoshi Murai


In this paper, we study the conjecture of Kühnel and Lutz, who state that a combinatorial triangulation of the product of two spheres \(\mathbb S^i \times \mathbb S^j\) with \(j \ge i\) 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.



We would like to thank Isabella Novik for helpful comments on an earlier version of this paper. The author was partially supported by JSPS KAKENHI 25400043.


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© Universitat de Barcelona 2015

Authors and Affiliations

  1. 1.Department of Pure and Applied Mathematics, Graduate School of Information Science and TechnologyOsaka UniversityToyonakaJapan

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