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Tight combinatorial manifolds and graded Betti numbers. (English) Zbl 1409.13041
Summary: 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 \(j \geq i\) 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.

MSC:
13F55 Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes
05E45 Combinatorial aspects of simplicial complexes
13D02 Syzygies, resolutions, complexes and commutative rings
57Q15 Triangulating manifolds
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