Round fold maps on $3$--manifolds

Kitazawa, Naoki; Saeki, Osamu

doi:10.48550/arxiv.2105.00974

Cited by 4 publications

(12 citation statements)

References 11 publications

(23 reference statements)

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“…Recall that for n = 3, if a closed orientable 3-dimensional manifold admits a simple fold map into R 2 , then it also admits a round fold map into R 2 as has been shown in [12]. The above example shows that this is not the case for n ≥ 4 in general.…”

Section: Remarks and Related Resultsmentioning

confidence: 72%

“…Remark 4.5. For n = 3, a characterization of closed orientable 3-dimensional manifolds that admit round fold maps into R 2 has been obtained in [12]. For nonorientable 3-dimensional manifolds, such a characterization has not been known, as far as the authors know.…”

Section: Remarks and Related Resultsmentioning

confidence: 99%

“…Those closed orientable 3-dimensional manifolds which admit directed round fold maps into R 2 have also been characterized in [12].…”

Section: Remarks and Related Resultsmentioning

confidence: 99%

“…It would be a difficult but interesting problem to classify round fold maps on closed orientable 3-dimensional manifolds into R 2 (refer to [12]) up to Aequivalence. Note that for simple stable maps of closed orientable 3-dimensional manifolds into R 2 , a classification result has been given in [21]: however, the classification is up to a certain equivalence strictly weaker than the A-equivalence, and the result is given in terms of links in the 3-dimensional manifolds.…”

Section: Classification Of Round Fold Mapsmentioning

confidence: 99%

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Round fold maps of n--dimensional manifolds into ${\mathbb{R}}^{n-1}$

Kitazawa¹,

Saeki²

2021

Preprint

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Section: Remarks and Related Resultsmentioning

confidence: 72%

Section: Remarks and Related Resultsmentioning

confidence: 99%

“…Those closed orientable 3-dimensional manifolds which admit directed round fold maps into R 2 have also been characterized in [12].…”

Section: Remarks and Related Resultsmentioning

confidence: 99%

Section: Classification Of Round Fold Mapsmentioning

confidence: 99%

See 2 more Smart Citations

Round fold maps of n--dimensional manifolds into ${\mathbb{R}}^{n-1}$

Kitazawa¹,

Saeki²

2021

Preprint

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“…Round fold maps have been introduced by the author in [10,11,12], followed by [13,14,15] for example. Studies such as [20,21] have been presented recently. These maps have been fundamental and strong tools in understanding the topologies and the differentiable structures of the manifolds via geometric and constructive ways or combinatorial ways.…”

Section: Introductionmentioning

confidence: 99%

Round fold maps on $3$-dimensional manifolds and their integral and rational cohomology rings

Kitazawa¹

2023

Preprint

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Fold maps are smooth maps at each singular point of which it is represented as the product map of a Morse function and the identity map. Round fold maps are, in short, such maps the sets of all singular points of which are embedded concentrically. They are, as Morse functions, important in understanding the topologies and the differentiable structures in geometric ways.In the present paper, we study cohomology rings of 3-dimensional manifolds admitting round fold maps into the plane and see that difference of the coefficient rings and topological types of round fold maps are closely related. This is an explicit precise new study on our previous study, showing that a 3-dimensional closed and orientable manifold is a so-called graph manifold, or a manifold obtained by gluing so-called circle bundles over surfaces along tori, if and only if it admits a round fold map into the plane. We also show another exposition on classifications of graph manifolds admitting such maps whose topological types are of a certain simplest class.

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Real algebraic functions on closed manifolds whose Reeb graphs are given graphs

Kitazawa

2022

Methods of Functional Analysis and Topology

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In this paper, we construct a real-algebraic function on some closed manifold whose Reeb (Kronrod-Reeb) graph is a graph respecting some algebraic domain: a graph for this is called a Poincaré-Reeb graph.The Reeb graph of a smooth function is defined as a natural graph which is the quotient space of the manifold of the domain under a natural equivalence relation for some wide and nice class of smooth functions. The vertex set is defined as the set of all connected components containing some singular points of the function: a singular point of a smooth function is a point where the differential vanishes. Morse-Bott functions give very specific cases. The relation is to contract each connected component of each preimage to a point.Sharko has posed a natural and important problem: can we construct a nice smooth function whose Reeb graph is a given graph? Explicit answers have been given first by Masumoto-Saeki in a generalized manner for closed surfaces. After that various answers have been presented by various researchers and most of them are essentially for functions on closed surfaces and Morse functions such that connected components of preimages that contain no singular points are spheres. Recently the author has also considered questions and answered them in the cases where the preimages are general manifolds.У статтi побудовано дiйсну алгебраїчну функцiю на деякому замкнутому многовидi, графом Реба (Кронрода-Реба) для якого є граф, який зберiгає деяку алгебраїчну область: його графiк називається графiком Пуанкаре-Реба.Граф Реба гладкої функцiї визначається як природний граф, який є факторпростором многовида, що вiдповiдає областi, вiдносно природньому вiдношенню еквiвалентностi для деякого широкого класу гладких функцiй. Множина вершин визначається як множина всiх зв'язаних компонентiв, що мiстять деякi особливi точки функцiї: особливою точкою гладкою функцiї є точка, в якiй диференцiал дорiнює нулю. Функцiї Морсе-Ботта є конкретними випадками таких функцiй. Вiдношення еквiвалентностi полягає в тому, щоб звести кожен зв'язаний компонент кожного прообразу до точки.Шарко поставив природну i важливу проблему: чи можемо ми побудувати хорошу гладку функцiю, граф Реба якої є заданим графом? Чiткi вiдповiдi були данi спочатку Масумото-Саекi в узагальненому виглядi для замкнутих поверхонь. Пiсля цього були данi вiдповiдi рiзними дослiдниками, i бiльшiсть з них були для функцiй на замкнутих поверхнях i функцiй Морса для випадку, коли зв'язанi компоненти прообразiв, що не мiстять особливих точок, є сферами. Нещодавно автор також розглянув i вiдповiв на цi питання в випадках, де прообрази є загальними многовидами.

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Round fold maps on $3$--manifolds

Cited by 4 publications

References 11 publications

Round fold maps of n--dimensional manifolds into ${\mathbb{R}}^{n-1}$

Round fold maps of n--dimensional manifolds into ${\mathbb{R}}^{n-1}$

Round fold maps on $3$-dimensional manifolds and their integral and rational cohomology rings

Real algebraic functions on closed manifolds whose Reeb graphs are given graphs

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