On the cusped fan in a planar portrait of a manifold

Kobayashi, Mahito

doi:10.1007/s10711-012-9715-3

Cited by 4 publications

(3 citation statements)

References 17 publications

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“…(1) For example, in [10], [12] and [11], stable maps into the plane from fundamental 4-dimensional closed manifolds with good differential topological structures such as S 4 , the total space of a S 2 -bundle over S 2 , complex projective plane and manifolds represented as their connected sums whose Reeb spaces are 2-dimensional compact manifolds are presented. Moreover, the total space of a S 2 -bundle over S 2 , complex projective plane and manifolds represented as connected sums admit symplectic structures and in such cases, each inverse image of each regular value is a Lagrangian submanifold.…”

Section: Examplementioning

confidence: 99%

Investigating additional structures on preimages of regular values of smooth maps induced from the manifolds via the Reeb spaces and applications

Kitazawa¹

2018

Preprint

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Morse functions and higher dimensional versions and their singularity theory are useful in studying geometric properties of smooth manifolds. In these studies, the Reeb space of a generic smooth map, defined as the space of all the connected components of inverse images of the map, is a fundamental and useful tool. Reeb spaces are often polyhedra compatible with natural simplicial structures of source manifolds and target ones. For example, for Morse functions, fold maps, and proper stable maps, which are useful and fundamental tools in these studies, the Reeb spaces are polyhedra.In 2010s, Hiratuka and Saeki found that the top-dimensional homology group of the Reeb space does not vanish for a smooth map such as a proper stable map having an inverse image of a regular value containing a component which is not (oriented) null-cobordant, and recently the author has found some extended versions of this fact considering cobordism-like groups of equivalence classes of smooth and closed manifolds.In this paper, we show similar facts in the cases where source manifolds and inverse images of maps of regular values may have additional algebraic and differential topological structures other than differentiable structures and orientations. We introduce one result as a statement not only for Reeb spaces but also for generalized spaces called pseudo quotient spaces, which are first introduced by Kobayashi and Saeki in 1996 to study algebraic and differential topological properties of stable maps into the plane on closed manifolds of dimension larger than 2 such as the types of the source manifolds and invariance of source manifolds by certain deformations. We also give some explicit applications of the result and another result.

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Section: Examplementioning

confidence: 99%

Investigating additional structures on preimages of regular values of smooth maps induced from the manifolds via the Reeb spaces and applications

Kitazawa¹

2018

Preprint

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“…Each piece is diffeomorphic to D 2 x D 2 and the momentum map on a piece is given by (|z| 2 , |u>| 2 ), where z, w are complex numbers in D 2 identified with the unit disc in C. We replace the momentum map on Mi with the cusped fan projection of r = 1. Since the last map is a perturbation of (\z\ 2 , |u>| 2 ), one can thus obtain a global perturbation of the momentum map ( [Kl,example 7.3 (c) t)^2 x 5 2 ttmCP 2 «"CP2.…”

Section: Toric 4-manifoldsmentioning

confidence: 99%

A Survey on Polygonal Portraits of Manifolds

Kobayashi¹

2010

Demonstratio Mathematica

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Abstract. Planar portraits are geometric representations of smooth manifolds defined by their generic maps into the plane. A simple subclass called the polygonal portraits is introduced, their realisations, and relations of their shapes to the topology of source manifolds are discussed. Generalisations and analogies of the results to other planar portraits are also mentioned. A list of manifolds which possibly admit polygonal portraits is given, up to diffeomorphism and up to homotopy spheres. This article is intended to give a summary on our research on the topic, and hence precise proofs will be given in other papers.

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“…In the study of topology of mappings, Kobayashi studied a stable map whose singular value set consists of two concentric circles, the outer one is the image of definite folds and the inner one is the image of indefinite folds and cusps, where cusps are outward [8]. In his further study [7], he constructed an infinite number of stable maps on each of S 2 × S 2 and CP 2 #CP 2 that have the same singular value set but that are not right-left equivalent, using what he called four cusped fans. This theorem is proved by observing the 3-manifolds obtained as the preimage of an arc in the target space.…”

Section: Introductionmentioning

confidence: 99%

Vertical 3-manifolds in simplified (2, 0)-trisections of 4-manifolds

Asano¹

2020

Preprint

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We classify the 3-manifolds obtained as the preimages of arcs on the plane for simplified (2, 0)-trisection maps, which we call vertical 3-manifolds. Such a 3-manifold is a connected sum of a 6-tuple of vertical 3-manifolds over specific 6 arcs. Consequently, we show that each of the 6-tuples determines the source 4-manifold uniquely up to orientation reversing diffeomorphisms. We also show that, in contrast to the fact that summands of vertical 3-manifolds of simplified (2, 0)-trisection maps are lens spaces, there exist infinitely many simplified (2, 0)-4-section maps that admit hyperbolic vertical 3-manifolds.

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On the cusped fan in a planar portrait of a manifold

Cited by 4 publications

References 17 publications

Investigating additional structures on preimages of regular values of smooth maps induced from the manifolds via the Reeb spaces and applications

Investigating additional structures on preimages of regular values of smooth maps induced from the manifolds via the Reeb spaces and applications

A Survey on Polygonal Portraits of Manifolds

Vertical 3-manifolds in simplified (2, 0)-trisections of 4-manifolds

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