Classification problems of toric manifolds via topology

Masuda, Mikiya; Suh, Dong Youp

doi:10.1090/conm/460/09024

Cited by 49 publications

(60 citation statements)

References 40 publications

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“…Recently the second author has shown in [7] that toric manifolds as varieties can be distinguished by their equivariant cohomology. So we are led to ask how much information ordinary cohomology contains for toric manifolds and we posed the following problem in [9]. Throughout this paper, an isomorphism of cohomology rings is as graded rings unless otherwise stated.…”

Section: Introductionmentioning

confidence: 99%

Topological classification of generalized Bott towers

Choi

Masuda

Suh

2009

Trans. Amer. Math. Soc.

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114

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Abstract. If B is a toric manifold and E is a Whitney sum of complex line bundles over B, then the projectivization P (E) of E is again a toric manifold. Starting with B as a point and repeating this construction, we obtain a sequence of complex projective bundles which we call a generalized Bott tower. We prove that if the top manifold in the tower has the same cohomology ring as a product of complex projective spaces, then every fibration in the tower is trivial so that the top manifold is diffeomorphic to the product of complex projective spaces. This gives supporting evidence to what we call the cohomological rigidity problem for toric manifolds, "Are toric manifolds diffeomorphic (or homeomorphic) if their cohomology rings are isomorphic?" We provide two more results which support the cohomological rigidity problem.

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Section: Introductionmentioning

confidence: 99%

Topological classification of generalized Bott towers

Choi

Masuda

Suh

2009

Trans. Amer. Math. Soc.

Self Cite

114

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“…It is well-known from a result of Hirzebruch in [10] Related to these observations, the following conjecture has been posed (refer to [14] and [15]). …”

Section: Introduction and Main Resultsmentioning

confidence: 97%

On a Generalization of Hirzebruch's Theorem to Bott Towers

Kim¹

2016

Journal of the Korean Mathematical Society

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Abstract. The primary aim of this paper is to generalize a theorem of Hirzebruch for the complex 2-dimensional Bott manifolds, usually called Hirzebruch surfaces, to more general Bott towers of height n. To do so, we first show that all complex vector bundles of rank 2 over a Bott manifold are classified by their total Chern classes. As a consequence, in this paper we show that two Bott manifolds Bn(α 1 , . . . , α n−1 , αn) and Bn(α 1 , . . . , α n−1 , α ′ n ) are isomorphic to each other, as Bott towers if and only if both αn ≡ α ′ n mod 2 and α 2 n = (α ′ n ) 2 hold in the cohomology ring of B n−1 (α 1 , . . . , α n−1 ) over integer coefficients. This result will complete a circle of ideas initiated in [11] by Ishida. We also give some partial affirmative remarks toward the assertion that under certain condition our main result still holds to be true for two Bott manifolds just diffeomorphic, but not necessarily isomorphic, to each other.

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“…It was proved in [3] that any real Bott manifold admits a flat Riemannian metric and two real Bott manifolds are homeomorphic or diffeomorphic if and only if their cohomology rings are isomorphic. This is called cohomological rigidity of real Bott manifolds (see [6]). …”

Section: Theorem 12 For Any a ∈ A(n) If The Space M (A) Is A Closementioning

confidence: 99%

Discrete Group Actions and Generalized Real Bott Manifolds

2011

Mathematical Research Letters

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Abstract. We study a class of discrete group actions on the Euclidean space. In particular, we will investigate when the orbit spaces of such group actions are closed manifolds. The answer turns out to be a class of real toric manifolds called generalized real Bott manifolds which are the total spaces of some kind of iterated real projective space bundles. This relation provides a new viewpoint on generalized real Bott manifolds which might be useful for the future study.

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Classification problems of toric manifolds via topology

Abstract: Abstract. We propose some problems on the classification of toric manifolds from the viewpoint of topology and survey related results.

Cited by 49 publications

References 40 publications

Topological classification of generalized Bott towers

Topological classification of generalized Bott towers

On a Generalization of Hirzebruch's Theorem to Bott Towers

Discrete Group Actions and Generalized Real Bott Manifolds

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