Different moment-angle manifolds arising from two polytopes having the same bigraded Betti numbers

TL;DR

This paper presents two 3-dimensional simple polytopes with identical bigraded Betti numbers but non-isomorphic Tor-algebras, leading to homotopically different moment-angle manifolds, illustrating cohomological but not combinatorial rigidity.

Contribution

It provides the first examples of polytopes that are cohomologically rigid but not combinatorially rigid, highlighting distinctions in polytope rigidity properties.

Findings

Two polytopes with same bigraded Betti numbers but different Tor-algebras.

Homotopically different moment-angle manifolds from these polytopes.

First examples of cohomologically rigid but not combinatorially rigid polytopes.

Abstract

Two simple polytopes of dimension 3 having the identical bigraded Betti numbers but non-isomorphic Tor-algebras are presented. These polytopes provide two homotopically different moment-angle manifolds having the same bigraded Betti numbers. These two simple polytopes are the first examples of polytopes that are (toric) cohomologically rigid but not combinatorially rigid.