Realization problems of graphs as Reeb graphs of Morse functions with prescribed preimages

Kitazawa, Naoki

doi:10.48550/arxiv.2108.06913

Cited by 3 publications

(3 citation statements)

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“…We answer this question in Section 5 for some cases (see Theorems 5.3, 5.4 and 5.5). Recently, many authors have dedicated themselves to the realization problem of R-graphs associated with Morse functions or Morse-Bott functions (see, for instance, the works of L.P. Michalak [12], I. Gelbukh [6], N. Kitazawa [9], among others). The second and third named authors of this paper, have also a recent work about the realization problem of the calls MB-Reeb graphs [2].…”

Section: (A) (B) (C)mentioning

confidence: 99%

Stable Bi-Maps on Surfaces and Their Graphs

Jesus

Batista

Costa

2023

TCAM

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In this paper we study stable bi-maps F = (f1, f2): M →R×R^2 from a global viewpoint,where M is a smooth closed orientable surface and f1: M→R, f2: M→R^2 are stable maps.We associate a graph to F, so-called RM-graph and study its properties. The RM-graph captures more information about the topological structure of M than other graphs that appear in literature. Moreover, some graph realization theorems are obtained.

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Section: (A) (B) (C)mentioning

confidence: 99%

Stable Bi-Maps on Surfaces and Their Graphs

Jesus

Batista

Costa

2023

TCAM

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“…These study cases for smooth functions on closed surfaces and Morse functions such that each connected component of each preimage containing no singular points is always a sphere for example. In [5,6,9,10], the author has studied cases where such connected components are general manifolds with mild conditions on singularities of the functions for example. [21] appears as a paper motivated by [5] and through related informal discussions by us.…”

Section: Introductionmentioning

confidence: 99%

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

Kitazawa¹

2023

Preprint

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In our 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 asked 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 containing no singular points are spheres. Recently the author has also considered questions and solved in cases the preimages are general manifolds.

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“…These works study cases of smooth functions on closed surfaces and Morse functions for which each connected component of each preimage containing no singular points is always a sphere, for example. In [10,11,3,4], the author has studied cases where such connected components are general manifolds satisfying mild conditions on singularities of the functions, for example. Paper [21] appears as a paper motivated by [10] and through related informal discussions by us.…”

Section: Introductionmentioning

confidence: 99%

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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Realization problems of graphs as Reeb graphs of Morse functions with prescribed preimages

Cited by 3 publications

References 19 publications

Stable Bi-Maps on Surfaces and Their Graphs

Stable Bi-Maps on Surfaces and Their Graphs

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

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

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