arXiv subjects

Naoki Kitazawa

Publications and source records attributed to Naoki Kitazawa.

At least 73 records · Page 4Linked to original sources

arXiv2022

Round fold maps into the plane on some $7$-dimensional closed and simply-connected manifolds

Round fold maps are smooth maps on closed manifolds which are locally represented as the product maps of Morse functions and identity maps on open disks and whose singularity is realized as concentrically embedded spheres. The author previously introduced such maps. Our paper presents round fold maps on some $7$-dimensional simply-connected manifolds whose cohomology rings are isomorphic to that of the product of the $2$-dimensional complex projective space and a $3$-dimensional sphere. Such manifolds have been studied precisely by Wang and round fold maps on spin manifolds in these manifolds have been previously studied by the author. These manifolds form explicit classes of higehr dimensional closed and simply-connected manifolds, which are central objects in classical algeberic topology and differential topology. Understanding these manifolds in geometric and constructive ways is still attractive, which we think as pioneers. Fold maps are defined as smooth maps which are locally represented as the product maps of Morse functions and identity maps on open disks. They are fundamental and strong tools in generalizations of theory of Morse functions and applications to geometry of manifolds. Explicit construction of fold maps are difficult even on elementary or well-known manifolds whereas we can know the (non-)existence from Eliashberg's celebrating theory in the 1970s and related one in considerable cases.

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arXiv2022

Notes on explicit special generic maps into Eulidean spaces whose dimensions are greater than 4

Special generic maps are higher dimensional versions of Morse functions with exactly two singular points, characterizing spheres topologically except 4-dimensional cases and 4-dimensional standard spheres. The class of such maps also contains canonical projections of unit spheres. This class is interesting from the viewpoint of algebraic topology and differential topology of manifolds. These maps have been shown to restrict the topologies and the differentiable structures of the manifolds strongly by Calabi, Saeki and Sakuma before 2010s, and later Nishioka, Wrazidlo and the author. So-called exotic spheres admit no special generic map in considerable cases and homology groups and cohomology rings are shown to be strongly restricted. Moreover, special generic maps into Euclidean spaces whose dimensions are smaller than or equal to 4 have been studied well. The present paper mainly concerns cases where the dimensions of targets are greater than or equal to 5.

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arXiv2022

On the non-existence of special generic maps on complex projective spaces

We prove the non-existence of special generic maps on complex projective space as our extended new result. Simplest special generic maps are Morse functions with exactly two singular points on spheres, or Morse functions in Reeb's theorem, and canonical projections of unit spheres. Manifolds represented as connected sums of products of manifolds diffeomorphic to unit spheres admit such maps in considerable cases. Real and complex projective spaces have been shown to admit no such maps in most cases by the author. This gives a complete answer for complex projective spaces as a corollary to a more general result, which is also our main result.

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arXiv2022

Characterizing certain classes of $6$-dimensional closed and simply-connected manifolds via special generic maps

The present paper finds new necessary and sufficient conditions for $6$-dimensional closed and simply-connected manifolds of certain classes to admit special generic maps into certain Euclidean spaces. The class of special generic maps naturally contains Morse functions with exactly two singular points on spheres in so-called Reeb's theorem, characterizing spheres topologically, and canonical projections of unit spheres. Our paper concerns variants of Reeb's theorem. Several results are known e. g. the cases where the manifolds of the targets are the plane and some cases where the manifolds of the domains are closed and simply-connected. Our paper concerns $6$-dimensional versions of a result of Nishioka, determining $5$-dimensional closed and simply-connected manifolds admitting special generic maps into Euclidean spaces completely. Closed and simply-connected manifolds are central geometric objects in (classical) algebraic topology and differential topology. The $6$-dimensional case is more complicated than the $5$-dimensional one: they are classified via explicit algebraic systems.

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arXiv2022

Restrictions on special generic maps into ${\mathbb{R}}^5$ on $6$-dimensional or higher dimensional closed and simply-connected manifolds

The class of special generic maps is a natural class of smooth maps containing Morse functions on spheres with exactly two singular points and canonical projections of unit spheres. We find new restrictions on such maps on $6$-dimensional or higher dimensional closed and simply-connected manifolds into ${\mathbb{R}}^5$. Spheres which are not diffeomorphic to unit spheres do not admit such maps whose codimensions are negative in considerable cases. They restrict the homeomorphism and the diffeomorphism types of the manifolds in general. On the other hands, some elementary manifolds admit special generic maps into suitable Euclidean spaces: manifolds represented as connected sums of products of unit spheres are of such examples. This motivates us to study the (non-)existence of special generic maps on elementary manifolds such as projective spaces and some closed and simply-connected manifolds. For example, new explicit investigations of cohomology rings are keys in our new study.

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arXiv2022

Closed manifolds admitting no special generic maps whose codimensions are negative and their cohomology rings

Special generic maps are higher dimensional versions of Morse functions with exactly two singular points, characterizing spheres topologically except $4$-dimensional cases: in these cases standard spheres are characterized. Canonical projections of unit spheres are special generic. In suitable cases, it is easy to construct special generic maps on manifolds represented as connected sums of products of spheres for example. It is an interesting fact that these maps restrict the topologies and the differentiable structures admitting them strictly in various cases. For example, exotic spheres, which are not diffeomorphic to standard spheres, admit no special generic map into some Euclidean spaces in considerable cases. In general, it is difficult to find (families of) manifolds admitting no such maps of suitable classes. The present paper concerns a new result on this work where key objects are products of cohomology classes of the manifolds. We can see that manifolds such as closed symplectic manifolds and real projective spaces admit no special generic maps into any connected nin-closed manifold in considerable cases.

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arXiv2022

7-dimensional simply-connected spin manifolds whose integral cohomology rings are isomorphic to that of ${\mathbb{C}P}^2 \times S^3$ admit round fold maps

We have been interested in understanding the class of 7-dimensional closed and simply-connected manifolds in geometric and constructive ways. We have constructed explicit fold maps, which are higher dimensional versions of Morse functions, on some of the manifolds, previously. The studies have been motivated by studies of {\it special generic} maps, higher dimensional versions of Morse functions on homotopy spheres with exactly two singular points, characterizing them topologically except $4$-dimensional cases. The class contains canonical projections of unit spheres for example. This class has been found to be interesting, restricting the topologies and the differentiable structures of the manifolds strictly: Saeki, Sakuma and Wrazidlo found explicit phenomena. The present paper concerns fold maps on $7$-dimensional closed and simply-connected spin manifolds whose integral cohomology rings are isomorphic to that of the product of the $2$-dimensional complex projective space and the $3$-dimensional sphere.

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arXiv2022

On Reeb graphs induced from smooth functions on closed or open manifolds

For a smooth function on a smooth manifold of a suitable class, the space of all connected components of preimages is the graph and called the {\it Reeb graph}. Reeb graphs are fundamental tools in the algebraic and differential topological theory of Morse functions and more general functions which are not so wild. In this paper, we study whether we can construct a smooth function with good geometric properties inducing a given graph as the Reeb graph. This problem has been essentially launched by Sharko in 2000s and various answers have been given by Masumoto, Michalak, Saeki, and so on. Recently the author set a new explicit problem and gave an answer. In the studies before the result of the author, considered functions are smooth functions on closed surfaces or Morse functions such that preimages of regular values are disjoint unions of standard spheres. On the other hand, the author constructed a smooth function on a suitable $3$-dimensional closed, connected and orientable manifold inducing the Reeb graph isomorphic to the given graph such that preimages of regular values are arbitrary closed surfaces. Based on this result and method of the author, with several new ideas, we will consider smooth functions on surfaces and manifolds which may be non-closed and give answers to the problem.

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arXiv2022

Maps on manifolds onto graphs locally regarded as the quotient maps onto Reeb spaces of some differentiable maps and a new construction problem

The Reeb space of a function or a map on a manifold is defined as the space of all connected components of preimages and represents the manifold compactly. In fact, Reeb spaces are fundamental and useful tools in geometric theory of so-called Morse functions and more general maps which are sufficiently tame. Can we construct an explicit good function inducing a given graph as the Reeb space (Reeb graph)? These problems were launched by Sharko in 2000s and have been explicitly solved by several researchers. As related pioneering studies, the author also found and solved problems adding constraints on singularities and preimages for example. The present paper concerns new problems on these works. We define the classes of maps onto graphs locally regarded as ones onto the Reeb spaces induced from smooth functions of suitable classes and consider and challenge the problems for the classes.

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arXiv2022

Smooth functions with simple structures on 3-dimensional closed manifolds with prescribed Reeb graphs and preimages

We give a new answer to so-called realization problems of graphs as Reeb graphs of smooth functions with prescribed preimages of regular values having nice structures. We present a best possible answer for functions on 3-dimensional closed manifolds. The Reeb space of a smooth function is the quotient space of the manifold of the domain induced from the following equivalence relation; two points in the manifold are equivalent if and only if they are points of a same connected component of some preimage. They are in considerable cases graphs (Reeb graphs). Reeb spaces with preimages represent the manifolds well and are important tools in geometry. Recently they play important roles in applications of mathematics such as visualizations. Realization problems ask us whether we can construct smooth functions with prescribed Reeb graphs and preimages. Studies on construction respecting preimages were essentially started by the author.

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arXiv2021

Round fold maps of n--dimensional manifolds into ${\mathbb{R}}^{n-1}$

We determine those smooth $n$--dimensional closed manifolds with $n \geq 4$ which admit round fold maps into ${\mathbb{R}}^{n-1}$, i.e.\ fold maps whose critical value sets consist of disjoint spheres of dimension $n-2$ isotopic to concentric spheres. We also classify such round fold maps up to $C^{\infty}$ $\mathcal{A}$--equivalence.

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arXiv2021

Investigating additional structures on preimages of regular values of smooth maps induced from the manifolds via the Reeb spaces and applications

The Reeb space of a generic map is the space of all connected components of preimages of the map. Reeb spaces are fundamental and useful tools in the theory of Morse functions and higher dimensional variants and their applications to geometry. These spaces are polyhedra compatible with natural simplicial structures of the manifolds in considerable cases. For example, for Morse functions, fold maps, and proper stable maps, which are useful and fundamental tools in these studies, they are polyhedra whose dimensions are equal to those of the manifolds of the targets. In the 2010s, Hiratuka and Saeki found that the top-dimensional homology group of the Reeb space does not vanish for smooth maps such as proper stable maps having some preimages containing a component which is not (oriented) null-cobordant and which has no singular points. Recently the author found some extended versions of this fact formulating and using cobordism-like groups of equivalence classes of smooth and closed manifolds. The present paper concerns cases where manifolds and preimages with no singular points of maps may have additional algebraic topological or differential topological structures other than differentiable structures and orientations. We show a theorem for this case. We also give some explicit applications of the result where the preimages have spin or spin^c structures induced from the manifolds of the domains and structures of framed manifolds for example. As important applications, we present new explicit construction of spin cobordisms and spin^c cobordisms for example.

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arXiv2021

The topologies and the differentiable structures of the images of special generic maps having simple structures

Special generic maps are smooth maps at each singular point of which we can represent as $(x_1, \cdots, x_m) \mapsto (x_1,\cdots,x_{n-1},\sum_{k=n}^{m}{x_k}^2)$ for suitable coordinates. Morse functions with exactly two singular points on homotopy spheres and canonical projections of unit spheres are special generic. They are known to restrict the topologies and the differentiable structures of the manifolds in various situations. On the other hands, various manifolds admit such maps. This article first presents a special generic map on a $7$-dimensional manifold and the image. This results also seems to present a new example of $7$-dimensional closed and simply-connected manifolds having non-vanishing triple Massey products and seems to be a new work related to similar works by Dranishnikov and Rudyak. We also review results on vanishing of products of cohomology classes, previously obtained by the author. The images of special generic maps are smoothly immersed manifolds whose dimensions are equal to the dimensions of the targets. The author studied the topologies of these images previously and studies on homology groups, cohomology rings and structures of them for special generic maps having simple structures are presented as new results.

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arXiv2021

7-dimensional closed simply-connected and spin manifolds having 2nd integral cohomology classes whose squares are not divisible by 2 and stable fold maps on them

This article presents families of 7-dimensional closed and simply-connected manifolds and fold maps on them such that squares of 2nd integral cohomology classes may not be divisible by 2. Fold maps are higher dimensional versions of Morse functions. The author has launched and been challenging the following new area: geometric and constructive studies of higher dimensional, closed and simply-connected manifolds. They are central objects in classical algebraic topology and differential topology. They were classified via algebraic and abstract objects in the last century and their understanding has been studied via concrete algebraic topological theory such as concrete bordism theory since the beginning of this century by Crowley, Kreck, Wang and so on. Fold maps are fundamental objects in the new area and the author has obtained families of these manifolds and fold maps on the manifolds. The present paper presents a new explicit result on the new area.

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arXiv2021

Explicit fold maps on 7-dimensional closed and simply-connected spin manifolds of new classes

Closed (and simply-connected) manifolds whose dimensions are greater than 4 are classified via sophisticated algebraic and abstract theory such as surgery theory and homotopy theory. It is difficult to handle 3 or 4-dimensional closed manifolds in such ways. However, the latter work via geometric and constructive ways is not so difficult. The assumption that the dimensions are not high enables us to handle the manifolds via diagrams for example. It is difficult to study higher dimensional manifolds in these ways, although it is natural and important. In the present paper, we present such studies via {\it fold} maps, which are higher dimensional variants of Morse functions. The author previously constructed fold maps on 7-dimensional closed and simply-connected manifolds satisfying additional conditions on cohomology rings, including so-called {\it exotic} homotopy spheres. This paper concerns fold maps on such manifolds of a wider class.

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arXiv2021

Notes on explicit smooth maps on 7-dimensional manifolds into the 4-dimensional Euclidean space

A fold map is a smooth map at each singular point of which it is represented as the product map of a Morse function and the identity map on an open ball. A special generic map is a fold map such that the Morse function can be taken as a natural height function on an unit disk. The class of special generic maps includes a Morse function with exactly two singular points on a closed manifold, characterizing a sphere topologically (except 4-dimensional cases) as the Reeb's theorem shows, and canonical projections of unit spheres. It has been known that so-called exotic spheres do not admit special generic maps into Euclidean spaces whose dimensions are sufficiently high and smaller than the dimensions of the spheres. Exotic 7-dimensional homotopy spheres do not admit special generic maps into the 4-dimensional Euclidean space for example. We can easily obtain special generic maps on fundamental manifolds such as ones represented as connected sums of products of two standard spheres and in considerable cases, smooth manifolds resembling topologically them and different from them do not admit special generic maps. These interesting results are due to studies of Saeki, Sakuma and Wrazidlo since the 1990s. In the present paper, we present new results on explicit smooth maps including fold maps on 7-dimensional manifolds into the 4-dimensional Euclidean space and meanings in algebraic topology and differential topology of manifolds. Moreover, the author obtained related results before motivated by the studies before and they are reviewed in the presentation of the new results. We also present new discussions and results related to the results for 7-dimensional manifolds and maps on them for fold maps between manifolds of general dimensions.

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arXiv2021

On Reeb graphs induced from smooth functions on 3-dimensional closed manifolds which may not be orientable

The Reeb space of a smooth function is a topological and combinatoric object and fundamental and important in understanding topological and geometric properties of the manifold of the domain. It is the graph and a topological space endowed with a natural topology. This is defined as the quotient space of the manifold of the domain where the equivalence relation is as follows: two points in the manifold are equivalent if and only if they are in a same connected component of a level set or a preimage. In considerable cases they are graphs (Reeb graphs): if the function is a so-called Morse(-Bott) functions for example, then this is the graph such that a point is a vertex if and only if the corresponding connected component of the level set contains some singular points. The author previously constructed explicit smooth functions on suitable 3-dimensional connected, closed and orientable manifolds whose Reeb graphs are isomorphic to prescribed graphs and whose preimages are as prescribed types. This gives a new answer to so-called realization problems of graphs as Reeb graphs of smooth functions of suitable classes. The present paper concerns a variant in the case where the 3-dimensional manifolds may not be non-orientable extending the result before. \end{abstract}

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arXiv2021

On Reeb graphs induced from smooth functions on 3-dimensional closed orientable manifolds with finitely many singular values

The Reeb graph of a function on a smooth manifold is the graph obtained as the space of all connected components of level sets such that the set of all vertices coincides with the set of all connected components of level sets including singular points. Reeb graphs are fundamental and important in the algebraic and differential topological theory of Morse functions and their generalizations. In this paper, as a related fundamental and important study, for given graphs, we construct certain smooth functions inducing the graphs as the Reeb graphs. Such works have been demonstrated by Masumoto, Michalak, Saeki, Sharko etc. and also by the author since 2000s. We present new smooth functions on suitable $3$-dimensional closed orientable manifolds through explicit constructive methods.

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