Rigidity problems in toric topology: A survey

Choi, Suyoung; Masuda, Mikiya; Suh, Dong Youp

doi:10.1134/s0081543811080128

Cited by 36 publications

(34 citation statements)

References 33 publications

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“…So far, many results on the smooth classification of generalized Bott manifolds have been established. We refer the reader to a survey paper [7] on this topic. Remarkably, the results lead us to conjecture that all non-singular complete toric varieties are smoothly classified by their cohomology rings.…”

Section: Introductionmentioning

confidence: 99%

Projective bundles over toric surfaces

Choi

Park

2016

Int. J. Math.

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Let E be the Whitney sum of complex line bundles over a topological space X. Then, the projectivization P (E) of E is called a projective bundle over X. If X is a non-singular complete toric variety, so is P (E). In this paper, we show that the cohomology ring of a non-singular projective toric variety M determines whether it admits a projective bundle structure over a non-singular complete toric surface. In addition, we show that two 6-dimensional projective bundles over 4-dimensional quasitoric manifolds are diffeomorphic if their cohomology rings are isomorphic as graded rings. Furthermore, we study the smooth classification of higher dimensional projective bundles over 4-dimensional quasitoric manifolds.

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Section: Introductionmentioning

confidence: 99%

Projective bundles over toric surfaces

Choi

Park

2016

Int. J. Math.

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“…In toric topology, cohomological rigidity is studied for (quasi)toric manifolds and moment-angle manifolds. We refer to [47], [23] and [14, §7.8] for a more detailed survey of related results and problems. The main question here is as follows.…”

Section: Cohomological Rigiditymentioning

confidence: 99%

Cohomological rigidity of manifolds defined by 3-dimensional polytopes

Buchstaber¹,

Erokhovets²,

Masuda³

et al. 2017

Russ. Math. Surv.

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Abstract. A family of closed manifolds is called cohomologically rigid if a cohomology ring isomorphism implies a diffeomorphism for any two manifolds in the family. We establish cohomological rigidity for large families of 3-dimensional and 6-dimensional manifolds defined by 3-dimensional polytopes.We consider the class P of 3-dimensional combinatorial simple polytopes P , different from a tetrahedron, whose facets do not form 3-and 4-belts. This class includes mathematical fullerenes, i. e. simple 3-polytopes with only 5-gonal and 6-gonal facets. By a theorem of Pogorelov, any polytope from P admits a right-angled realisation in Lobachevsky 3-space, which is unique up to isometry.Our families of smooth manifolds are associated with polytopes from the class P. The first family consists of 3-dimensional small covers of polytopes from P, or hyperbolic 3-manifolds of Löbell type. The second family consists of 6-dimensional quasitoric manifolds over polytopes from P. Our main result is that both families are cohomologically rigid, i. e. two manifolds M and M from either of the families are diffeomorphic if and only if their cohomology rings are isomorphic. We also prove that if M and M are diffeomorphic, then their corresponding polytopes P and P are combinatorially equivalent. These results are intertwined with the classical subjects of geometry and topology, such as combinatorics of 3-polytopes, the Four Colour Theorem, aspherical manifolds, diffeomorphism classification of 6-manifolds and invariance of Pontryagin classes. The proofs use techniques of toric topology.

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“…The cohomological rigidity problem for Bott towers and the cohomological rigidity problem for Bott manifolds (see Choi et al [6]) are fundamental problems in algebraic and geometric topology. The former concerns whether the iterated fibre bundle structures of Bott towers can be distinguished by their filtered cohomologies.…”

Section: Introductionmentioning

confidence: 99%

Classification of Bott towers by matrix

Bai

Fang

2015

Front. Math. China

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A criterion for the classification of Bott towers is presented, i.e., two Bott towers B * (A) and B * (A ) are isomorphic if and only if the matrices A and A are equivalent. The equivalence relation is defined by two operations on matrices. And it is based on the observation that any Bott tower B * (A) is uniquely determined by its structure matrix A, which is a strictly upper triangular integer matrix. The classification of Bott towers is closely related to the cohomological rigidity problem for both Bott towers and Bott manifolds.

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Rigidity problems in toric topology: A survey

Abstract: Abstract. Several rigidity problems in toric topology are addressed in [35]. In this paper, we survey results on those problems including recent development.

Cited by 36 publications

References 33 publications

Projective bundles over toric surfaces

Projective bundles over toric surfaces

Cohomological rigidity of manifolds defined by 3-dimensional polytopes

Classification of Bott towers by matrix

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