Toric cohomological rigidity of simple convex polytopes

Abstract

A simple convex polytope P is cohomologically rigid if its combinatorial structure is determined by the cohomology ring of a quasitoric manifold over P. Not every P has this property, but some important polytopes such as simplices or cubes are known to be cohomologically rigid. In this paper we investigate the cohomological rigidity of polytopes and establish it for several new classes of polytopes, including products of simplices. The cohomological rigidity of P is related to the bigraded Betti numbers of its Stanley-Reisner ring, another important invariant coming from combinatorial commutative algebra.