The number of small covers over cubes

Choi, Suyoung

doi:10.2140/agt.2008.8.2391

Cited by 22 publications

(42 citation statements)

References 7 publications

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“…Z 2 Df˙1g sending s i to 1 and the others s`(`6 D i ) to 1. This together with (2)(3)(4)(5) shows that the line bundle L j 1 in (2-5) is obtained as the quotient of R j 1 R by the diagonal action of j 1 where the action of j 1 on the second factor R is given through a homomorphism j 1 ! Z 2 sending s i to .…”

Section: Fundamental Groupsmentioning

confidence: 88%

“…Observation (2)(3)(4)(5) implies that the tower (1-1) is completely determined by the matrix A. So we may denote M n by M.A/.…”

Section: Cohomology Ringsmentioning

confidence: 99%

“…This work was motivated by a talk of Suyoung Choi on [4] given at Fudan University in January 2008 and very much stimulated by the discussion with Taras Panov which the second author had during his stay at Fudan University. He would like to thank them and also Zhi Lü for inviting him to Fudan University and organizing fruitful seminars.…”

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confidence: 99%

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Cohomological rigidity of real Bott manifolds

Kamishima

Masuda

2009

Algebr. Geom. Topol.

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A real Bott manifold is the total space of an iterated RP 1 -bundle over a point, whereeach RP 1 -bundle is the projectivization of a Whitney sum of two real line bundles.We prove that two real Bott manifolds are diffeomorphic if their cohomology rings with Z=2-coefficients are isomorphic.A real Bott manifold is a real toric manifold and admits a flat Riemannian metric invariant under the natural action of an elementary abelian 2-group. We also prove that the converse is true, namely a real toric manifold which admits a flat Riemannian metric invariant under the action of an elementary abelian 2-group is a real Bott manifold.57R91; 53C25, 14M25 IntroductionA fundamental result in the theory of toric varieties says that the categories of toric varieties (over the complex numbers C ) and fans are equivalent (see Oda [20] The set X.R/ of real points in a toric manifold X is called a real toric manifold. It appears as the fixed point set of complex conjugation on X . For example, when X is a complex projective space CP n , the subspace X.R/ is a real projective space RP n . It is known thatH .X.R/I Z=2/ Š H 2 .X I Z/˝Z=2 as graded rings;for any toric manifold X where Z denotes the integers and Z=2 D f0; 1g, and one may ask the same question as the rigidity problem above for real toric manifolds with Z=2-coefficients, namely:Cohomological rigidity problem for real toric manifolds Are two real toric manifolds diffeomorphic (or homeomorphic) if their cohomology rings with Z=2-coefficients are isomorphic as graded rings?In this paper we are concerned with a sequence of RP 1 -bundles: : : ; n is the projective bundle of a Whitney sum of two real line bundles over M i 1 , where one of the two line bundles may be assumed to be trivial without loss of generality because projectivizations P .E/ and P .E˝L/ are diffeomorphic for any real vector bundle E and line bundle L over a smooth manifold. Grossberg and Karshon [9] considered the sequence above in the complex case and named it a Bott tower of height n. Following them, we call the sequence above a real Bott tower of height n. Each M i in the tower (1-1) is a real toric manifold. We call M i a real Bott manifold. There are many choices of line bundles in the tower (1-1) so that real Bott towers produce many real Bott manifolds. We note that even if real Bott towers of height n are different, their top manifolds of dimension n might be diffeomorphic.The main purpose of this paper is to prove the following which answers the cohomological rigidity problem affirmatively for real Bott manifolds.Theorem 1.1 Two real Bott manifolds are diffeomorphic if their cohomology rings with Z=2-coefficients are isomorphic as graded rings.Although real toric manifolds have similar properties to toric manifolds, there is one major difference, which is that a real toric manifold is not simply connected while a

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Section: Fundamental Groupsmentioning

confidence: 88%

“…Observation (2)(3)(4)(5) implies that the tower (1-1) is completely determined by the matrix A. So we may denote M n by M.A/.…”

Section: Cohomology Ringsmentioning

confidence: 99%

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Cohomological rigidity of real Bott manifolds

Kamishima

Masuda

2009

Algebr. Geom. Topol.

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“…| we use the correspondence given by Choi [3]. By the nonsingularity condition, for any λ ∈ Λ(P ), the vectors where n(p) is defined as in Corollary 3.3.…”

Section: Theorem 31 (Theorem 28 [3]) the Number Of Dj-equivalence mentioning

confidence: 99%

On small covers over a product of simplices

2018

Turk J Math

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“…For m = 1, Choi studied small covers over cubes [5], which are obtained as the projectivization of a Whitney sum of two real line bundles. In particular he associated an acyclic digraph with labeled n vertices to a small cover over an n -cube and proved that the correspondence is bijective.…”

Section: Introductionmentioning

confidence: 99%

Small covers over products of a simple polytope with a simplex

Dai¹,

Wang²

2018

Turk J Math

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This paper proves that the number of small covers over products of a simple polytope with a n -simplex, up to D-J equivalence, is a polynomial in the variable 2 n . A similar result holds for orientable small covers. We also provide a new way of computation, namely computing the finite number of representatives and interpolating polynomially. The ratio between the number of orientable small covers and the number of small covers is given. As an application, by interpolation, we determine the polynomials related to small covers and orientable small covers over products of a prism with a simplex up to D-J equivalence. A formula for calculating the number of equivariant homeomorphism classes of small covers over the product is also provided.

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The number of small covers over cubes

Cited by 22 publications

References 7 publications

Cohomological rigidity of real Bott manifolds

Cohomological rigidity of real Bott manifolds

On small covers over a product of simplices

Small covers over products of a simple polytope with a simplex

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