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Toric cohomological rigidity of simple convex polytopes. (English) Zbl 1229.52008

We assume \(P\) to be a simple convex polytope of dimension \(n\). A quasitoric manifold over \(P\) is a closed \(2n\)-manifold \(M\) with a locally standard action of an \(n\)-torus \(G\) and a surjective map \(M\to P\) whose fibres are the \(G\)-orbits. The polytope \(P\) is cohomologically rigid if its combinatorial structure is determined by the cohomology ring of a quasitoric manifold over \(P\). Certain types of polytopes, such as simplices and cubes, are known to have this strong property. This paper investigates the (cohomological) rigidity of polytopes and extends it for some new classes of polytopes.
Two key generalizations are proven: First, finite products of simplices are proven to be rigid. And when \(P\) is such a product, the polytope obtained by truncating a vertex of \(P\) (a “vertex cut”) is also shown to be rigid. Second, \(n\)-dimensional polytopes which are triangle-free having less than \(2n+3\) facets are established to be rigid. Combining both of these results, the rigidity classification of \(3\)-dimensional polytopes having up to nine faces is provided in a tabular format.
The paper closes with other notions of rigidity, and considers some open problems.

MSC:

52B05 Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.)
52B10 Three-dimensional polytopes
52B11 \(n\)-dimensional polytopes
13F55 Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes
55N91 Equivariant homology and cohomology in algebraic topology
57S15 Compact Lie groups of differentiable transformations
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
52C25 Rigidity and flexibility of structures (aspects of discrete geometry)
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