Some rigidity problems in toric topology: I

Abstract

We study the cohomological rigidity problem of two families of manifolds with torus actions: the so-called moment-angle manifolds, whose study is linked with combinatorial geometry and combinatorial commutative algebra; and topological toric manifolds, which can be seen as topological generalizations of toric varieties. These two families are related by the fact that a topological toric manifold is the quotient of a moment-angle manifold by a subtorus action.In this paper, we prove that when a simplicial sphere satisfies some combinatorial condition, the corresponding moment-angle manifold and topological toric manifolds are cohomological rigid, i.e. their homeomorphism classes in their own families are determined by their cohomology rings. Our main strategy is to show that the combinatorial types of these simplicial spheres (or more generally, the Gorenstein * complexes in this class) are determined by the Tor-algebras of their face rings. This is a solution to a classical problem (sometimes know as the B-rigidity problem) in combinatorial commutative algebra for a class of Gorenstein * complexes in all dimensions 2.

$B$-rigidity of ideal almost Pogorelov polytopes

“…A very important tools for study of families of 3-dimensional polytopes are given by so-called separable circuit conditions (SCC for short) [FMW20].…”

“…Remark 4. In [FMW20] SCC for Pogorelov polytopes was generalized to higher dimensions (we call it SCC' for short). In particular, the product of two flag polytopes with SCC' is also a flag polytope with SCC'.…”

“…Remark 5. In [FMW20,Proposition F.1] this result was first proved in the dual setting for simplicial polytopes in "only if" direction. In [E20] this result was proved independently for simple polytopes in both directions.…”

“…The triangulation ∂P * is the barycentric subdivision of ∂Q * . Such triangulations appeared also in [FMW20,Example 6.3(2)] as examples of flag triangulations such that any 4-circuit full subcomplex is simple. Also they appeared in [CK11] in study of combinatorial rigidity (a simple n-polytope P is called combinatorially rigid, if there are no other simple n-polytopes with the same bigraded Betti numbers β −i,2j (Z P )).…”

“…Proposition 2.3 (SCC for almost Pogorelov polytopes[FMW20,E20]). A simple 3-polytope P is an almost Pogorelov polytope or the polytope P 8 if and only if for any three pairwise different faces…”