Multibranched Surfaces in 3-Manifolds

Ozawa, Makoto

doi:10.1007/s10958-021-05362-x

Cited by 3 publications

(3 citation statements)

References 24 publications

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“…These polyhedra form a class wider than that of the 2-dimensional branched manifolds here. [44] and [46] are on multibranched surfaces and regarded as slightly generalized versions of these 2-dimensional branched manifolds. Proposition 3.…”

Section: (Stable) Fold Maps and Their Reeb Spacesmentioning

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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Section: (Stable) Fold Maps and Their Reeb Spacesmentioning

confidence: 99%

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

Kitazawa

2021

Preprint

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“…Remark 1. In [22] and [24] for example, multibranched surfaces are defined as 2dimensional polyhedra and these polyhedra embedded in 3-dimensional compact manifolds are studied in the PL category. For example, such polyhedra represented as products of circles and graphs are studied.…”

Section: Special Generic Mapsmentioning

confidence: 99%

A differential topological study of compact manifolds having simple structures

Kitazawa

2021

Preprint

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The present paper mainly presents, for example, explicit classifications of compact smooth manifolds having non-empty boundaries and simple structures where the dimensions are general. Studies of this type is fundamental and important. They also remain to be immature and difficult. This is due to the fact that the dimensions are high and this has prevented us from studying the manifolds in geometric and constructive ways.Moreover, most of the present work is motivated by explicit studies of higher dimensional variants of Morse functions: especially so-called special generic maps. The class of special generic maps is a natural class containing canonical projections of unit spheres and Morse functions on homotopy spheres with exactly two singular points. Their images are in general (compact) manifolds smoothly immersed to the targets and the dimensions of the images and the targets coincide. They know much about the topologies and the differentiable structures of the manifolds of the domains. The author has previously studied related problems and the present study gives new extended results of related previous results. Last, we also present a dream for contribution to studies of special generic maps and higher dimensional variants of Morse functions and manifolds admitting them.

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“…PL embeddings of multibranched surfaces into 3-dimensional closed, connected, and orientable manifolds, are studied in [31], [34], [37], and so on. These studies can be higher dimensional variants of the theory of embeddings of graphs into surfaces.…”

Section: Introductionmentioning

confidence: 99%

Branched surfaces homeomorphic to Reeb spaces of simple fold maps

Kitazawa¹

2021

Preprint

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Classes of branched surfaces extend the classes of surfaces satisfying suitable properties and defined in various manners. Reeb spaces of smooth maps of suitable classes into surfaces whose codimensions are negative are regarded as branched surfaces, the classes of which are defined in suitable manners. They are defined as the spaces of all connected components of preimages in general situations, which are natural quotient spaces of the domains. They are important topological objects in differential topology and also play important roles in applied or applications of mathematics such as projections in data analysis, visualizations, and so on.The present paper concerns global topologies of branched surfaces and explicit construction via fundamental operations. The author has believed that global topologies of Reeb spaces of suitable smooth maps are important to understand and obtained several Reeb spaces with information on global algebraic topological or differential topological properties. The present study concentrates on construction of these objects in 2-dimensional cases for simple fold maps, generalized versions of so-called Morse functions which are general. This is a new work motivated by systematic and various theory of Morse functions and higher dimensional variants. Mainly, we give explicit affirmative answers to the following problem: is a branched surface locally regarded as the Reeb space of a simple fold map of a suitable class regarded as the Reeb space of such a map globally.

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Multibranched Surfaces in 3-Manifolds

Cited by 3 publications

References 24 publications

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

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

A differential topological study of compact manifolds having simple structures

Branched surfaces homeomorphic to Reeb spaces of simple fold maps

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