Toric cohomological rigidity of simple convex polytopes

TL;DR

This paper investigates the cohomological rigidity of simple convex polytopes, establishing it for new classes such as products of simplices, and explores its relation to algebraic invariants like bigraded Betti numbers.

Contribution

It extends the class of polytopes known to be cohomologically rigid, including products of simplices, and links this property to Stanley--Reisner ring invariants.

Findings

Cohomological rigidity established for new classes of polytopes.

Relation between rigidity and bigraded Betti numbers clarified.

Examples include simplices, cubes, and products of simplices.

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.