Toric cohomological rigidity of simple convex polytopes

Abstract

A simple convex polytope $P$ is \emph{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 article we investigate the cohomological rigidity of polytopes and establish it for several new classes of polytopes including products of simplices. Cohomological rigidity of $P$ is related to the \emph{bigraded Betti numbers} of its \emph{Stanley--Reisner ring}, another important invariants coming from combinatorial commutative algebra.

18 pages, 1 figure, 2 tables; revised version

Cited by in corpus (8)

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• Torus manifolds and non-negative curvature
• Properties of Bott manifolds and cohomological rigidity
• Invariance of Pontrjagin classes for Bott manifolds
• Remarks on the classification of quasitoric manifolds up to equivariant homeomorphism
• On descriptions of products of simplices
• Different moment-angle manifolds arising from two polytopes having the same bigraded Betti numbers
• Classification of toric manifolds over an $n$-cube with one vertex cut