arXiv subjects

Naoki Kitazawa

Publications and source records attributed to Naoki Kitazawa.

At least 91 records · Page 5Linked to original sources

arXiv2021

On simple classes of special generic maps and round fold maps and fold maps obtained by composing projections

Fold maps are fundamental tools in the theory of singularities of differentiable maps and its applications to geometry. They are higher dimensional variants of Morse functions. Classes of special generic maps and round fold maps are important classes of fold maps. {\it Special generic} maps are higher dimensional variants of Morse functions on homotopy spheres with exactly two {\it singular points}: canonical projections of unit spheres are special generic. Round fold maps are Morse functions obtained as doubles of Morse functions, or fold maps such that the set of all the singular points are embeddings and that the images are concentric. In the present paper, we discuss compositions of these maps with canonical projections. For example, we observe that these compositions for special generic maps of simple classes are regarded as round fold maps in considerable cases. We also present round fold maps we cannot represent in this way, seeming to be represented so. Note that such compositions are natural operations in related theory of differentiable maps.

math.GN↗

arXiv2021

A differential topological study of compact manifolds having simple structures

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 is also a new study closely related to them. 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.

math.GN↗

arXiv2021

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

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 fold maps with Reeb spaces with non-trivial top homology groups and with good topological properties. This paper presents new families, especially, generalized situations of some known situations.

math.GN↗

arXiv2021

Round fold maps on $3$--manifolds

We show that a closed orientable 3--dimensional manifold admits a round fold map into the plane, i.e. a fold map whose critical value set consists of disjoint simple closed curves isotopic to concentric circles, if and only if it is a graph manifold, generalizing the characterization for simple stable maps into the plane. Furthermore, we also give a characterization of closed orientable graph manifolds that admit directed round fold maps into the plane, i.e.\ round fold maps such that the number of regular fiber components of a regular value increases toward the central region in the plane.

math.GT↗

arXiv2021

New explicit construction of fold maps on general 7-dimensional closed and simply-connected spin manifolds

7-dimensional closed and simply-connected manifolds have been attractive as central and explicit objects in algebraic topology and differential topology of higher dimensional closed and simply-connected manifolds, which were studied actively especially in the 1950s--60s. Attractive studies of the class of these $7$-dimensional manifolds were started by the discovery of so-called exotic spheres by Milnor. It has influenced on the understanding of higher dimensional closed and simply-connected manifolds via algebraic and abstract objects. Recently this class is studied via more concrete notions from algebraic topology such as concrete bordism theory by Crowley, Kreck, and so on. As a new kind of fundamental and important studies, the author has been challenging understanding the class in constructive ways via construction of fold maps, which are higher dimensional versions of Morse functions. The present paper presents a new general method to construct ones on spin manifolds of the class.

math.AT↗

arXiv2020

Surgery operations to fold maps to increase connected components of singular sets by two

In geometry, understanding the topologies and the differentiable structures of manifolds in constructive ways is fundamental and important. It is in general difficult, especially for higher dimensional manifolds. The author is interested in this and trying to understand manifolds via construction of explicit fold maps: differentiable maps locally represented as product maps of Morse functions and identity maps on open balls. Fold maps have been fundamental and useful in investigating the manifolds by observing (the sets of) singular points and values and preimages as Thom and Whitney's pioneering studies and recent studies of Kobayashi, Saeki, Sakuma, and so on, show. Here, construction of explicit fold maps on explicit manifolds is difficult. The author constructed several explicit families of fold maps and investigated the manifolds admitting the maps. Main fundamental methods are surgery operations (bubbling operations), the author recently introduced motivated by Kobayashi and Saeki's studies such as operations to deform generic differentiable maps whose codimensions are negative into the plane preserving the differentiable structure of the manifold in 1996 and so on. We remove a neighborhood of a (an immersed) submanifold consisting of regular values in the target space, attach a new map and obtain a new fold map such that the number of connected components of the set consisting of singular points increases. In this paper, we investigate cases where the numbers increase by two and obtain cases of a new type.

math.KT↗

arXiv2020

Realizing a homology class of a compact manifold by a homology class of an explicit closed submanifold--a new approach to Thom's works on homology classes of submanifolds-

It is a classical important problem of differential topology by Thom; for a homology class of a compact manifold, can we realize this by a closed submanifold with no boundary? This is true if the degree of the class is smaller or equal to the half of the dimension of the outer manifold under the condition that the coefficient ring is Z_2. If the degree of the class is smaller or equal to 6 or equal to k-2 or k-1 under the condition that the coefficient ring is the integer ring where k is the dimension of the manifold, then this is also true. As a specific study, for 4-dimensional closed manifolds, the topologies (genera) of closed and connected surfaces realizing given 2nd homology classes have been actively studied, for example. In the present paper, we consider the following similar problem; can we realize a homology class of a compact manifold by a homology class of an explicit closed manifold embedded in the (interior of the) given compact manifold? This problem is considered as a variant of previous problems. We present an affirmative answer via important theory in the singularity theory of differentiable maps: lifting a given smooth map to an embedding or obtaining an embedding such that the composition of this with the canonical projection is the given map. Presenting this application of lifting smooth maps and related fundamental propositions is also a main purpose of the present paper.

math.AT↗

arXiv2020

Surgery operations to fold maps to construct fold maps whose restrictions to the singular sets may not be embeddings

Constructing Morse functions and their higher dimensional versions or fold maps is fundamental, important and challenging in investigating the topologies and the differentiable structures of differentiable manifolds via Morse functions, fold maps and more general generic maps. It is one of important and interesting branches of the singularity theory of differentiable maps and applications to geometry of manifolds. In this paper we present fold maps with information of cohomology rings of their Reeb spaces. Reeb spaces are defined as the spaces of all connected components of all preimages, and in suitable situations inherit topological information such as homology groups and cohomology rings of the manifolds. Previously, the author demonstrated construction of fold maps in various cases : key methods are surgery operations to manifolds and maps and in this paper, we present more useful surgery operations and by them we construct new fold maps. More precisely, fold maps with singular value sets with crossings: the singular value set of a smooth map is the image of the set of all singular points and note that for fold maps, the set of all singular points are closed submanifolds without boundaries and the restrictions to them are immersions of codimension 1.

math.GT↗

arXiv2020

Special generic maps and fold maps and information on triple Massey products of higher dimensional differentiable manifolds

Closed (and simply-connected) manifolds whose dimensions are larger than 4 are central geometric objects in classical algebraic topology and differential topology. They have been classified via algebraic and abstract objects. On the other hand, It is difficult to understand them in geometric and constructive ways. In the present paper, we show such studies via explicit fold maps, higher dimensional versions of Morse functions. The author captured information of the topologies and the differentiable structures of closed (and simply-connected) manifolds which are not so complicated with respect to homotopy previously and cohomology rings of more general closed (and simply-connected) manifolds via construction of these maps. In the present paper, as a more precise work, we capture so-called (triple) Massey products in this way.

math.AT↗

arXiv2020

Notes on fold maps obtained by surgery operations and algebraic information of their Reeb spaces

The theory of Morse functions and their higher dimensional versions or fold maps on manifolds and its application to geometric theory of manifolds is one of important branches of geometry and mathematics. Studies related to this was started in 1950s by differential topologists such as Thom and Whitney and they have been studied actively. In this paper, we study fold maps obtained by surgery operations to fundamental fold maps, and especially Reeb spaces, defined as the spaces of all connected components of preimages and in suitable situations inheriting fundamental and important algebraic invariants such as (co)homology groups. Reeb spaces are fundamental and important tools in studying manifolds also in general. The author has already studied about homology groups of the Reeb spaces and obtained several results and in this paper, we study about their cohomology rings for several specific cases, as more precise information. These studies are motivated by a problem that construction of explicit fold maps is important in investigating (the worlds of explicit classes of) manifolds in geometric and constructive ways and difficult. It is not so difficult to construct these maps for simplest manifolds such as standard spheres, products of standard spheres and manifolds represented as their connected sums. We see various types of cohomology rings of Reeb spaces via systematic construction of fold maps.

math.GT↗

arXiv2020

New observations on cohomology rings of Reeb spaces of explicit fold maps and manifolds admitting these maps

As a branch of algebraic and differential topology of manifolds, the theory of Morse functions and their higher dimensional versions or fold maps and its application to algebraic and differential topology of manifolds is fundamental, important and interesting. This paper is on explicit construction of fold maps and homology groups and cohomology rings of their Reeb spaces: they are defined as the spaces of all connected components of preimages of the maps, and in suitable situations inherit some topological information such as homology groups and cohomology rings of the manifolds. Explicit construction of the maps is a fundamental and difficult task even on manifolds which are not so complicated. The author has constructed explicit fold maps systematically and performed several calculations of homology groups and cohomology rings of the Reeb spaces. This paper concerns new observations on this task.

math.KT↗

arXiv2019

An elementary study on realizable changes of homology groups of Reeb spaces of fold maps by fundamental surgery operations

In the singularity and differential topological theory of Morse functions and higher dimensional versions or fold maps and application to algebraic and differential topology of manifolds, constructing explicit fold maps and investigating their source manifolds is fundamental, important and difficult. The author has introduced surgery operations (bubbling operations) to fold maps, motivated by studies of Kobayashi, Saeki etc. since 1990 and has explicitly shown that homology groups of Reeb spaces of maps constructed by iterations of these operations are flexible in several cases. Such operations seem to be strong tools in construction of maps and precise studies of manifolds. More precisely, the author has also noticed that the resulting groups are represented as direct sums of the original homology groups and suitable finitely generated commutative groups. The Reeb space of a map is the space of all connected components of inverse images of the maps. Reeb spaces inherit fundamental invariants of the manifolds such as homology groups etc. much in simple cases as polyhedra whose dimensions are equal to those of the target spaces. This paper is on a new explicit study of changes of homology groups of Reeb spaces of fold maps by the surgery operations. We present explicit changes obtained by an approach via elementary theory of sequences of numbers and fundamental continuous or differentiable functions.

math.GT↗

arXiv2019

Explicit remarks on the torsion subgroups of homology groups of Reeb spaces of explicit fold maps

Fold maps are higher dimensional versions of Morse functions and fundamental and important tools in studying algebraic and differential topological properties of manifolds: as the theory established by Morse and the higher dimensional version, started by Thom and Whitney, later actively studied by Eliashberg, Levine etc. and recently studied by Kobayashi, Saeki, Sakuma etc., explicitly show this. One of fundamental, important and difficult studies on this field is, constructing explicit fold maps and investigating their source manifolds. As fundamental and strong tools for systematic construction, the author has introduced surgery operations (bubbling operations) to fold maps, motivated by studies of Kobayashi etc. since 1990. The author has explicitly shown that homology groups of Reeb spaces of maps constructed by iterations of these operations are generally flexible and restricted in several specific cases. The Reeb space of a map is defined as the space of all connected components of inverse images of the maps, inheriting fundamental invariants of manifolds such as homology groups etc. in considerable cases and fundamental and important tools in the field. In this paper, we explicitly remark on the torsion subgroups of the homology groups. More precisely, under explicit algebraic constraints, we see explicit strong restrictions on the torsion subgroups, where the homology groups seem to be very flexible in general. We note that this work is similar to several works by the author before but a work in a new situation and that new technique such as well-known fundamental theory of abstract algebra will be used.

math.GT↗

arXiv2019

Round fold maps and the topologies and the differentiable structures of manifolds admitting explicit ones

Stable fold maps are fundamental tools in a generalization of the theory of Morse functions on smooth manifolds and its application to studies of geometric properties of smooth manifolds. Round fold maps were introduced as stable fold maps such that the sets of all of the singular values of them are concentric spheres by the author in 2013-4. Topological properties of such maps and topological information of their source manifolds such as homology and homotopy groups have been studied under appropriate conditions by the author. In this paper, we redefine round fold maps respecting the definition. As more precise information of manifolds admitting round fold maps, we study the topologies and differentiable structures of manifolds admitting such maps under appropriate differential topological conditions.

math.AT↗

arXiv2019

On defining products of cobordism classes of Morse functions

(Co)bordisms of manifolds and maps are fundamental and important objects in algebraic and differential topology of manifolds and related studies were started by Thom etc.. Cobordisms of Morse functions were introduced and have been studied as a branch of the algebraic and differential topological theory of Morse functions and their higher dimensional versions, or the global singularity theory, by Kalmár, Ikegami, Sadykov, Saeki, Wrazidlo, Yamamoto etc. since 2000s. Cobordism relations are in most cases defined as the following for example; two closed manifolds of a fixed dimension or maps on them into a fixed space are said to be {\it cobordant} if the disjoint union is a boundary of a compact manifold or the restriction of a map satisfying suitable conditions on the manifold into the product of the target space and the closed interval. Such relations induce structures of modules consisting of all obtained equivalence classes such that the sums are defined by procedures of taking disjoint unions and in the cases of manifolds, ring structures such that the products are defined by the procedures of taking products, are also introduced. In this paper, as a new algebraic topological study, we try to define a product of two cobordism classes of Morse functions and show that a natural method fails, by presenting explicit examples which are regarded as an obstruction.

math.AT↗

arXiv2019

Products of elements of cobordism-like modules induced from generic maps

Recently the author has introduced cobordism-like modules induced from generic maps whose codimensions are negative. They are generalizations of cobordism modules of manifolds. They have been introduced in generalizing the following theorem shown by Hiratuka and Saeki in 2013--14; for a generic map whose codimension is negative including a connected component of an inverse image of a regular value being not null-cobordant and for a space defined as all connected components of inverse images, which is a polyhedron of dimension equal to that of the target space, the top-dimensional homology group does not vanish. Note that such spaces are fundamental and important tools in general, in the differential topological theory of Morse functions and their higher dimensional versions and application to algebraic and differential topology of manifolds, or the global singularity theory. In this paper, the author succeeded in defining suitable elements as the products for pairs of elements in cobordism modules which may be distinct, as in the case of the ordinary cobordism modules. This is an extension of the product of two ordinary cobordism classes of manifolds.

math.GN↗

arXiv2019

Lifts of spherical Morse functions

In studies of smooth maps with good differential topological conditions such as immersions, embeddings, Morse functions and their higher dimensional versions including fold maps and application to geometry, especially algebraic and differential topology of manifolds, liftings or desingulizations of maps of appropriate classes to other maps of other appropriate classes are fundamental and important. In this paper, we consider Morse functions such that inverse images of regular values are disjoint unions of spheres, which are extensions of Morse functions with just two singular points on homotopy spheres, and defined and studied by Saeki and Suzuoka in 2000s, and lift them to immersions, embeddings and special generic maps, which are regarded as higher dimensional versions of Morse functions with just 2 singular points before. In lifting smooth maps, we usually lift them to immersions or embeddings and in this paper, as new works, we consider lifts to special generic maps, whose codimensions are not positive. In addition, we construct most of lifts by new methods.

math.GT↗

arXiv2019

Structures of cobordism-like modules induced from generic maps of codimension -2

The Reeb space of a smooth map whose codimension is minus is the space defined as the space of all connected components of inverse images. For generic maps such as Morse functions and their higher dimensional versions, they are polyhedra whose dimensions are equal to those of the target manifolds and which have simplicial structures compatible with (the canonical) simplicial structures of the source and the target manifolds, and in considerable cases they inherit fundamental and important invariants of source manifolds. In fact, Reeb spaces are fundamentall tools in the algebraic and differential topological theory of generic maps or the global singularity theory. As one of studies of global topological properties of Reeb spaces, Hiratuka and Saeki showed in 2013 that for generic maps or more precisely, maps compatible with simplicial structures of the manifolds, inducing simplicial structures on the Reeb spaces and having connected components of inverse images of regular values being not null-cobordant, the top-dimensional homology groups with appropriate coefficient rings of the Reeb spaces do not vanish. Later the author extended this theorem: the author has introduced cobordism-like groups based on adjacent relations of connected components of inverse images of regular values and shown a similar theorem.

math.GT↗