Reeb spaces of smooth functions on manifolds

Saeki, Osamu

doi:10.48550/arxiv.2006.01689

Cited by 8 publications

(5 citation statements)

References 21 publications

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“…For a function f : X → R on a topological space X, the Reeb graph of a function f : X → R on X is a quotient space X/ ∼ Reeb defined by x ∼ Reeb y if there are a number c ∈ R and a connected component of f −1 (c) which contains x and y. Notice that the Reeb graph of a Morse function (or, more generally, a function with finitely many critical points) on a closed manifold is a finite graph (see [34,Theorem 3.1] for details). The inverse image of a value of R is called the level set.…”

Section: Preliminaries For the Description Of Topological And Combina...mentioning

confidence: 99%

Morse hyper-graphs and abstract weak orbit spaces of semi-decompositions

Yokoyama¹

2022

Preprint

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We introduce topological invariants of semi-decompositions (e.g. filtrations, semi-group actions, multi-valued dynamical systems, combinatorial dynamical systems) on a topological space to analyze semi-decompositions from a dynamical systems point of view. In fact, we construct Morse hypergraphs and abstract weak elements of semi-decompositions. Moreover, the Morse hyper-graphs of the set of sublevel sets of a Morse function of a compact manifold is the Reeb graph of such function as abstract multi-graphs.

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Section: Preliminaries For the Description Of Topological And Combina...mentioning

confidence: 99%

Morse hyper-graphs and abstract weak orbit spaces of semi-decompositions

Yokoyama¹

2022

Preprint

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“…Then the inverse image of a value of R is called the level set. Notice that the Reeb graph of a Morse function (or more generally a function with finitely many critical points) on a closed manifold is a finite graph (see [37,Theorem 3.1] for details). Moreover, by the proof of Proposition 7.6, notice that any edge of the Reeb graph of a Morse function on a closed manifold is the leaf space of a codimension one product compact foliation, because the leaf space of a continuous codimension two (and so one) compact foliation of a compact manifold is Hausdorff [18][19][20]39].…”

Section: Preliminaries For the Description Of Topological And Combina...mentioning

confidence: 99%

Morse hyper-graphs of topological spaces and decompositions

Yokoyama¹

2021

Preprint

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We introduce topological invariants for topological spaces and decompositions, which are analogous to abstract (weak) orbit spaces and Morse graphs for flows. To achieve these, we define analogous concepts of recurrence and "chain-recurrence" for topological spaces and decompositions such that the original and new recurrences correspond to each other for a flow orbits on a locally compact Hausdorff space and the orbit space. We show that the Morse hyper-graphs for any topological spaces and any invariant decompositions (e.g. compact foliated spaces) exist. Moreover, the abstract weak element spaces generalize the Reeb graphs of Morse functions and the Morse (hyper-)graphs and refine abstract cell complexes.

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“…For Morse functions and smooth functions which are not so wild, the Reeb spaces are graphs where the vertex sets are the sets of all (elements representing) connected components of preimages containing at least one singular point: see [50] for example. Reeb spaces are in such cases Reeb graphs.…”

Section: Introductionmentioning

confidence: 99%

“…[54] is a pioneering study. [40], [43], [50], and so on, are important works and there exist other closely related works. [24], [26] and [27] are related works by the author.…”

Section: Introductionmentioning

confidence: 99%

Global topologies of Reeb spaces of stable fold maps with non-trivial top homology groups

Kitazawa

2021

Preprint

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The Reeb space of a continuous map is the space of all (elements representing) connected components of preimages endowed with the quotient topology induced from the natural equivalence relation on the domain. These objects are strong tools in (differential) topological theory of Morse functions, fold maps, which are their higher dimensional variants, and so on: they are in general polyhedra whose dimensions are same as those of the targets. In suitable cases Reeb spaces inherit topological information such as homology groups, cohomology rings, and so on, of the manifolds.This presents the following problem: what are global topologies of Reeb spaces of these smooth maps of suitable classes like? The present paper presents families of stable fold maps having Reeb spaces with non-trivial top homology groups with their (co)homology groups (and rings). Related studies on the global topologies from the viewpoints of the singularity theory of differentiable maps and differential topology have been presented by various researchers including the author. The author previously constructed families of stable fold maps with Reeb spaces with non-trivial top homology groups and with good topological properties and this paper presents new families.

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Reeb spaces of smooth functions on manifolds

Cited by 8 publications

References 21 publications

Morse hyper-graphs and abstract weak orbit spaces of semi-decompositions

Morse hyper-graphs and abstract weak orbit spaces of semi-decompositions

Morse hyper-graphs of topological spaces and decompositions

Global topologies of Reeb spaces of stable fold maps with non-trivial top homology groups

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