Tight combinatorial manifolds and graded Betti numbers

Abstract

In this paper, we study the conjecture of Kuhnel 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.