arXiv subjects

Naoki Kitazawa

Publications and source records attributed to Naoki Kitazawa.

At least 55 records · Page 3Linked to original sources

arXiv2023

Notes on Reeb graphs of real algebraic functions which may not be planar

The Reeb graph of a smooth function is a graph being a natural quotient space of the manifold of the domain and the space of all connected components of preimages. Such a combinatorial and topological object roughly and compactly represents the manifold. Since the proposal by Sharko in 2006, reconstructing nice smooth functions and the manifolds from finite graphs in such a way that the Reeb graphs are the graphs has been important. The author has launched new studies on this, discussing construction of real algebraic functions. We concentrate on Reeb graphs we cannot realize as (natural) planar graphs here. Previously the graphs were planar and embedded in the plane naturally.

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arXiv2023

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

The present paper is on a new result on so-called realization problems of graphs as Reeb graphs of Morse functions with prescribed preimages. The Reeb graph of a smooth function is a graph obtained by identifying two points in the manifold of the domain if and only if they are in a same connected component of some preimage. In considerable cases such as cases where functions are Morse-Bott functions on closed manifolds we obtain such graphs. Reeb graphs represent the manifolds of the domains compactly. They are not only fundamental tools in geometry using differentiable functions and maps as fundamental tools, but also in applications of mathematics such as visualizations. Realization problems are also fundamental and important. Construction of functions with prescribed preimages was considered first by the author essentially and before the present study, the author has obtained a result.

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arXiv2023

Explicit smooth real algebraic functions which may have both compact and non-compact preimages on smooth real algebraic manifolds

In our previous work, we have constructed explicit smooth real algebraic functions which may have both compact and non-compact preimages on smooth real algebraic manifolds. This paper presents its variant. Our result is new in obtaining non-proper smooth real algebraic functions on smooth real algebraic manifolds satisfying explicit conditions on (non-)compactness of preimages whereas previously the manifolds are only semi-algebraic. Explicitly, this mainly contributes to two different regions of mathematics. One is singularity theory of differentiable maps and applications to differential topology. More precisely, construction of nice smooth maps with desired preimages. The other is real algebraic geometry. More precisely, explicit construction of smooth real algebraic functions and maps whereas we can know the existence and consider approximations of smooth maps by maps of such classes in considerable cases.

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arXiv2023

On real algebraic maps whose images are domains surrounded by the products of hyperbolas and real affine spaces

Previously, we have systematically constructed explicit real algebraic functions which are represented as the compositions of smooth real algebraic maps whose images are domains surrounded by hypersurfaces of degree 1 or 2 with canonical projections. Here we give new examples with the hypersurfaces each of which is the product of a connected component of a hyperbola and a copy of the $1$-dimensional affine space explicitly. As a related future work we also discuss problems to obtain the zero sets of some real polynomials explicitly from increasing sequences of real numbers. This is motivated by a problem in theory of smooth functions proposed first by Sharko: can we construct nice smooth functions whose Reeb graphs are as prescribed? The Reeb space of a smooth function is the naturally obtained graph whose underlying space is the quotient space of the manifold consisting of connected components of preimages. The author first considered variants respecting the topologies of the preimages and obtained several results before. Our work is also motivated by real algebraic geometry, pioneered by Nash. We can know existence of real algebraic structures of smooth manifolds and some general sets and we already know several approximations of smooth maps by real algebraic maps. Our interest lies in explicit construction, which is difficult.

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arXiv2023

Explicit real algebraic functions which may have both compact and non-compact preimages

As a pioneering work we construct explicit real algebraic functions which may have both compact and non-compact preimages. The author has obtained explicit real algebraic functions with preimages satisfying some nice conditions. More precisely, we have given answers to a considerably revised version of Sharko's question. Sharko originally asked whether we can have nice smooth functions whose Reeb graphs are as desired. The Reeb graph of a smooth function of a certain nice class is a natural graph whose underlying space is the space of all connected components. Such graphs can have some important topological information of the manifolds. Our answers are new in having real algebraic functions on non-compact manifolds with no boundaries. We also avoid using so-called existence theory and approximation theory whereas we also avoid in our previous studies.

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arXiv2023

Restrictions on manifolds admitting certain explicit special-generic-like maps and construction of maps with the manifolds

Special-generic-like maps or SGL maps are introduced by the author motivated by observing and investigating algebraic topological or differential topological properties of manifolds via nice smooth maps whose codimensions are negative. The present paper says that manifolds admitting certain very explicit SGL maps are topologically restricted strongly and this also constructs these maps with the manifolds explicitly. Morse functions with exactly two singular points on spheres or functions in Reeb's theorem and canonical projections of naturally embedded spheres in Euclidean spaces are generali5zed as special generic maps. Their nice global structures motivate us to study such maps and the manifolds. Manifolds represented as connected sums of the product of spheres and similar ones admit such maps in considerable cases. The topologies and the differentiable structures of the manifolds of the domains of such maps are also strongly restricted: due to Saeki and Sakuma, followed by Nishioka, Wrazidlo and the author. Respecting their nice global structures, these maps are generalized and this yields SGL maps.

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arXiv2023

A class of naturally generalized special generic maps

Special generic maps are generalizations of Morse functions with exactly two singular points on spheres and canonical projections of unit spheres. They restrict the manifolds of the domains strongly in considerable cases and are important in algebraic topology and differential topology of manifolds of specific classes and manifolds regarded as elementary in some senses admit such maps in considerable cases. We propose a class of generalized special generic maps in our present paper and extend a fundamental result on structures and some algebraic topological properties of special generic maps by the author. Our present study will be a pioneering study on nice classes of generalized special generic maps. Studies of algebraic topological properties and differential topological ones of special generic maps have developed due to their nice structures for example.

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arXiv2023

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

In our paper, we construct a real-algebraic function whose Reeb (Kronrod-Reeb) graph is a graph respecting some algebraic domain: a graph for this is called 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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arXiv2023

Smooth maps like special generic maps

In our paper, we introduce special-generic-like maps or SGL maps as smooth maps and study their several algebraic topological and differential topological properties. The new class generalize the class of so-called special generic maps. Special generic maps are smooth maps which are locally projections or the product maps of Morse functions and the identity maps on disks. Morse functions with exactly two singular points on spheres or Morse functions in Reeb's theorem are simplest examples. Special generic maps and the manifolds of their domains have been studied well. Their structures are simple and this help us to study explicitly. As important properties, they have been shown to restrict the topologies and the differentiable structures of the manifolds strongly by Saeki and Sakuma, followed by Nishioka, Wrazidlo and the author. To cover wider classes of manifolds as the domains, the author previously introduced a class generalizing the class of special generic maps and smaller than our class: simply generalized special generic maps.

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arXiv2023

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

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 {\it 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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arXiv2022

Decompositions of manifolds into submanifolds compatible with specific fold maps

We present new explicit decompositions of manifolds via so-called fold maps into lower dimensional spaces. Fold maps form a nice class of so-called generic maps, generalizing Morse functions naturally. To understand the topologies and the differentibale structures of manifolds globally, decomposing manifolds are important and this presents interesting topics and problems on geometry of manifolds. The notion of a Heegaard splitting of a $3$-dimensional closed and connected manifold presents a pioneering study. A $3$-dimensional closed and connected manifold is always decomposed into two copies of a so-called $3$-dimensional handlebody via a so-called Heegaard surface, which is a closed and connected surface. Heegaard splitiings are generalized as multisections of smooth or PL manifolds in the 2010s. As a way of understanding, these decompositions are understood via Morse functions and general generic smooth maps whose codimensions are negative.

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arXiv2022

Manifolds admitting special generic maps and their nice generalized multisections

We show that manifolds admitting special generic maps also admit nice generalized multisections. Special generic maps are natural generalized versions of Morse functions with exactly two singular points on closed manifolds, characterizing spheres whose dimensions are not $4$ topologically and the $4$-dimensional unit sphere, and canonical projections of unit spheres. They are shown to restrict the differentiable structures of spheres etc. and topologies of more general manifolds strongly by Saeki, Sakuma etc., followed by Nishioka, Wrazidlo etc. and followed by the author. Some elementary or important manifolds also admit such maps. (Generalized) multisections of manifolds are nice decompositions of (compact) manifolds, generalizing so-called Heegaard splittings of $3$-dimensional manifolds. PL manifolds have been shown to have (generalized) multisections enjoying certain properties by Rubinstein and Tillmann.

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arXiv2022

New families of manifolds with similar cohomology rings admitting special generic maps

As Reeb's theorem shows, Morse functions with exactly two singular points on closed manifolds are very simple and important. They characterize spheres whose dimensions are not $4$ topologically and the $4$-dimensional unit sphere. Special generic maps are generalized versions of these maps. Canonical projections of unit spheres are special generic. Studies of Saeki and Sakuma since the 1990s, followed by Nishioka and Wrazidlo, show that the differentiable structures of the spheres and the homology groups of the manifolds (in several classes) are restricted. We see special generic maps are attractive. Our paper studies the cohomology rings of manifolds admitting such maps. As our new result, we find a new family of manifolds whose cohomology rings are similar and find that the (non-)existence of special generic maps are closely related to the topologies. More explicitly, we have previously found related families and our new manifolds add to these discoveries.

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arXiv2022

The images of special generic maps of an elementary class

The class of special generic maps contains Morse functions with exactly two singular points, characterizing spheres topologically which are not $4$-dimensional and the $4$-dimensional unit sphere. This class is for higher dimensional versions of such functions. Canonical projections of unit spheres are simplest examples and suitable manifolds diffeomorphic to ones represented as connected sums of products of spheres admit such maps. They have been found to restrict the topologies and the differentiable structures of the manifolds strongly. The present paper focuses on images of special generic maps on closed manifolds. They are smoothly immersed compact manifolds whose dimensions are same as those of the targets. Some studies imply that they have much information on homology groups and cohomology rings. We present new construction and explicit examples of special generic maps by investigating the images and the present paper is essentially on construction and explicit algebraic topological and differential topological studies of immersed compact manifolds.

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arXiv2022

Special generic maps into ${\mathbb{R}}^5$ on closed and simply-connected manifolds and information on the cohomology of the manifolds

Morse functions with exactly two singular points on spheres and canonical projections of spheres belong to the class of a certain good class of smooth maps: special generic maps. We mainly investigate information on cohomology of closed and simply-connected manifolds admitting such maps into the $5$-dimensional Euclidean spaces by investigating the embedded curves and submanifolds and their preimages. Studies on homology groups for ones into the Euclidean spaces (whose dimensions are lower than $5$ in most cases) have been pioneered by Saeki and Sakuma since 1990s and later by Nishioka and Wrazidlo since 2010s. Recently the author has started pioneering studies on the cohomology for cases where the dimensions of the Euclidean spaces may not be lower than $5$. Our new cases are difficult due to the situation that the dimensions of manifolds we consider are higher. Previously, we have found several restrictions on the cohomology rings. We present new restrictions by new investigations.

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arXiv2022

Simple polyhedra homeomorphic to Reeb spaces of stable fold maps

Simple polyhedra are $2$-dimensional polyhedra and important objects in low-dimensional geometry and in the applications of {\it fold} maps, defined as smooth maps regarded as higher dimensional variants of Morse functions. For example, they are locally so-called {\it Reeb spaces} of (so-called stable) fold maps into the plane and represent the manifolds compactly. The Reeb space of a fold map is the space of all connected components of preimages of it and is a polyhedron whose dimension is same as that of the manifold of the target. Is a given simple polyhedron homeomorphic to the Reeb space of a suitable stable fold map? What are their global topologies like? Previously the author has challenged this for a specific case and presented fundamental construction and topological properties of the polyhedra as new results. The present paper extends some of these works and results and present results of new types.

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arXiv2022

Characterizing families of graph manifolds via suitable classes of simple fold maps into the plane and embeddability of the Reeb spaces in some 3-dimensional manifolds

Graph manifolds form important classes of $3$-dimensional closed and orientable manifolds. For example, {\it Seifert} manifolds are graph manifolds where hyperbolic manifolds are not. In applying singularity theory of differentiable maps to understanding global topologies of manifolds, graph manifolds have been shown to be characterized as ones admitting so-called simple fold maps into the plane of explicit classes by Saeki and the author. The present paper presents several related new results. Fold maps are higher dimensional variants of Morse functions and simple ones form simple classes, generalizing the class of general Morse functions. Such maps into the plane on $3$-dimensional closed and orientable manifolds induce quotient maps to so-called simple polyhedra with no vertices, which are $2$-dimensional. This is also closely related to the theory of {\it shadows} of $3$-dimensional manifolds. We also discuss invariants for graph manifolds via embeddability of these polyhedra in some $3$-dimensional manifolds.

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arXiv2022

Branched surfaces homeomorphic to Reeb spaces of simple fold maps

Classes of branched surfaces extend the classes of surfaces or 2-dimensional manifolds 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. They are the spaces of all connected components of preimages and natural quotient spaces of the manifolds of the domains. They are defined for general smooth maps and important topological objects in differential topology. They also play important roles in applied or applications of mathematics such as projections in data analysis and visualizations. The present paper concerns global topologies of branched surfaces and explicit construction of canonically obtained maps from the branched surfaces into surfaces of the targets via fundamental operations. The class of these induced maps extends the class of smooth immersions of compact surfaces into surfaces with no boundaries. It is also regarded as a variant of the class of so-called generic smooth maps between these surfaces. We study so-called "geography" of such maps as a natural, important and new study and also study global topological properties of the branched surfaces such as embeddability into $3$-dimensional closed and connected manifolds.

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