Example of C-rigid polytopes which are not B-rigid

Choi, Suyoung; Park, Kyoungsuk

doi:10.1515/ms-2017-0236

Cited by 5 publications

(7 citation statements)

References 18 publications

(15 reference statements)

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“…In [5, 10, 14], the notion of B -rigidity is defined for simple polytopes rather than simplicial complexes. A simple polytope P is called B-rigid over k if for every simple polytope , a graded ring isomorphism implies a combinatorial equivalence .…”

Section: B-rigiditymentioning

confidence: 99%

Product decompositions of moment-angle manifolds and B-rigidity

Amelotte

Briggs²

2023

Can. Math. Bull.

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A simple polytope P is called B-rigid if its combinatorial type is determined by the cohomology ring of the moment-angle manifold $\mathcal {Z}_P$ over P. We show that any tensor product decomposition of this cohomology ring is geometrically realized by a product decomposition of the moment-angle manifold up to equivariant diffeomorphism. As an application, we find that B-rigid polytopes are closed under products, generalizing some recent results in the toric topology literature. Algebraically, our proof establishes that the Koszul homology of a Gorenstein Stanley–Reisner ring admits a nontrivial tensor product decomposition if and only if the underlying simplicial complex decomposes as a join of full subcomplexes.

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Section: B-rigiditymentioning

confidence: 99%

Product decompositions of moment-angle manifolds and B-rigidity

Amelotte

Briggs²

2023

Can. Math. Bull.

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“…B-rigidity was introduced and defined as above in [9] to address Question 1.1 (cf. [4, Lecture IV, Problem 7.6]), and has since been studied in [5,10,13,14,15], where some variations on the definition above have appeared. For example, in [15], bigraded isomorphisms of cohomology rings are used to define B-rigidity while complexes satisfying the definition above are called strongly Brigid.…”

Section: B-rigiditymentioning

confidence: 99%

“…In [5,10,13], the notion of B-rigidity is defined for simple polytopes rather than simplicial complexes. A simple polytope P is called B-rigid over k if for every simple polytope P ′ , a graded ring isomorphism H * (Z P ; k) ∼ = H * (Z P ′ ; k) implies a combinatorial equivalence P ≃ P ′ .…”

Section: B-rigiditymentioning

confidence: 99%

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Product decompositions of moment-angle manifolds and $B$-rigidity

Amelotte¹,

Briggs²

2022

Preprint

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A simple polytope P is called B-rigid if its combinatorial type is determined by the cohomology ring of the moment-angle manifold Z P over P . We show that any tensor product decomposition of this cohomology ring is geometrically realized by a product decomposition of the moment-angle manifold up to equivariant diffeomorphism. As an application, we find that B-rigid polytopes are closed under products, generalizing some recent results in the toric topology literature. Algebraically, our proof establishes that the Koszul homology of a Gorenstein Stanley-Reisner ring admits a nontrivial tensor product decomposition if and only if the underlying simplicial complex decomposes as a join of full subcomplexes.

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“…It is known that B-rigid polytope is C-rigid [CPS10] (see also [BEMPP17]). The converse is not true [CP19].…”

Section: Introductionmentioning

confidence: 97%

$B$-rigidity of ideal almost Pogorelov polytopes

Erokhovets

2020

Preprint

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Toric topology assigns to each n-dimensional combinatorial simple convex polytope P with m facets an (m+n)-dimensional moment-angle manifold Z P with an action of a compact torus T m such that Z P /T m is a convex polytope of combinatorial type P . A simple n-polytope is called B-rigid, if any isomorphism of graded rings H * (Z P , Z) = H * (Z Q , Z) for a simple n-polytope Q implies that P and Q are combinatorially equivalent. An ideal almost Pogorelov polytope is a combinatorial 3-polytope obtained by cutting off all the ideal vertices of an ideal right-angled polytope in the Lobachevsky (hyperbolic) space L 3 . These polytopes are exactly the polytopes obtained from any, not necessarily simple, convex 3-polytopes by cutting off all the vertices followed by cutting off all the "old" edges. The boundary of the dual polytope is the barycentric subdivision of the boundary of the old polytope (and also of its dual polytope). We prove that any ideal almost Pogorelov polytope is B-rigid. This produces three cohomologically rigid families of manifolds over ideal almost Pogorelov manifolds: moment-angle manifolds, canonical 6-dimensional quasitoric manifolds and canonical 3-dimensional small covers, which are "pullbacks from the linear model".

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Example of C-rigid polytopes which are not B-rigid

Cited by 5 publications

References 18 publications

Product decompositions of moment-angle manifolds and B-rigidity

Product decompositions of moment-angle manifolds and B-rigidity

Product decompositions of moment-angle manifolds and $B$-rigidity

$B$-rigidity of ideal almost Pogorelov polytopes

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