Operations on 3-dimensional small covers

Kuroki, Shintaro

doi:10.1007/s11401-008-0417-y

Cited by 5 publications

(14 citation statements)

References 8 publications

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“…In addition, Kuroki [6] studied the relations among six operations v , e , eve , , , c on P and found that e = • ( v P 3 (3)) and eve = 2 • ( v P 3 − (3)). Furthermore, Nishimura [11] discovered more relations among the operations in Theorem 5.5 and obtained another algebraic system by the following theroem, which improved Theorem 5.5.…”

Section: 2mentioning

confidence: 99%

Lickorish-type Construction of Manifolds Over Simple Polytopes

Lü

Wang

2019

Trends in Mathematics

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This paper is a survey on the Lickorish type construction of some kind of closed manifolds over simple convex polytopes. Inspired by Lickorish's theorem, we propose a method to describe certain families of manifolds over simple convex polytopes with torus action. Under this construction, many classical classification results of these families of manifolds could be interpreted by this construction and some further problems will be discussed. What is Lickorish type construction?This paper is a survey on the Lickorish type construction of some kind of closed manifolds over simple convex polytopes. We first explain what is "Lickorish type construction".In algebra, it is natural to describe algebraic systems, such as rings and algebras, by generators and relations. In geometry and topology, it is often convenient to construct spaces from some very special examples by certain type of operations. We write this construction in terms of algebraic system by AS generators | some operations , where AS is the abbreviation for "algebraic system". One typical example is the following theorem obtained by Lickorish in 1962.Theorem 1.1 (Lickorish [8]). Any orientable closed connected 3-manifold can be obtained from S 3 by a finite number of Dehn surgeries on knots.This theorem provides a global viewpoint of the construction of orientable closed connected 3-manifolds under algebraic system with generators and operations. We call this kind of construction or description Lickorish type construction. Under this point of view, we can rewrite the above theorem as:All orientable closed connected 3-manifolds = AS S 3 | Dehn surgeries on knots .

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Section: 2mentioning

confidence: 99%

Lickorish-type Construction of Manifolds Over Simple Polytopes

Lü

Wang

2019

Trends in Mathematics

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“…If k = 1, then the projective characteristic function is the ordinary characteristic function, i.e., the fibre dimension is 0. Therefore, we can regard the 0-dimensional projective fibre sum ♯ ∆ 0 as the ordinary (equivariant) connected sum ♯ appeared in [12,13,17,24].…”

Section: Combinatorial Interpretation Of the Fibre Summentioning

confidence: 99%

Projective bundles over small covers and the bundle triviality problem

Kuroki

Lü

2015

Forum Mathematicum

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Abstract. This paper investigates the projectivization of real vector bundles over small covers. We first give a necessary and sufficient condition for such a projectivization to be a small cover. Then associated with moment-angle manifolds, we further study the structure of such a projectivization as a small cover. As an application, we characterize the real projective bundles over 2-dimensional small covers by interpreting the fibre sum operation to some combinatorial operation. Finally, we study when the projectivization of Whitney sum of the tautological line bundle and the tangent bundle over real projective space is diffeomorphic to the product of two real projective spaces.

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“…Then M is generated by the classes of S 1 ×RP 2 over the 3-sided prism and RP 3 over the 3-simplex ( [12,14]). Much further research on small covers over the 3-sided prism has been carried on ( [11,16]). An example of a cohomologically non-rigid polytope was obtained from a 3-sided prism by iterating the operation of vertex cut twice ( [8]).…”

Section: Introductionmentioning

confidence: 99%