arXiv subjects

Naoki Kitazawa

Publications and source records attributed to Naoki Kitazawa.

99 records · Page 6Linked to original sources

arXiv2019

Constructing fold maps by surgery operations and their Reeb spaces

In this paper, as a fundamental study on the theory of Morse functions and their higher dimensional versions or fold maps and applications to geometric theory of manifolds, which were started in 1950s by differential topologists such as Thom and Whitney and have been studied actively, we study algebraic and differential topological properties of certain fold maps and their source manifolds. More precisely, we investigate fold maps obtained by surgery operations to fundamental fold maps and especially, homology groups of Reeb spaces, which are defined as the space of all connected components of inverse images, often inheriting important invariants of manifolds such as homology groups and fundamental and important tools in studying manifolds. Studies of this paper are especially motivated by the stream of studies of fold maps satisfying good (differential) topological properties such as special generic maps, which were defined in 1970s and studied since 1990s by Saeki and Sakuma, and round fold maps, which were introduced by the author in 2012--2014, and their source manifolds. Moreover, constructions of generic maps by fundamental surgeries to investigate manifolds by using generic maps, studied by Kobayashi and Saeki etc., also have motivated the present study.

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arXiv2018

Smooth maps compatible with simplicial structures and inverse images

As a higher dimensional version of the theory of Morse functions, there have been various studies of smooth manifolds using generic smooth maps. As fundamental results, in these studies, they have found that inverse images of such maps often restrict the types of the source manifolds. For example, if a generic map such that the inverse image of a regular value is not null-cobordant, then the homology group of its {\it Reeb} space, which is defined as the space of all the connected components of inverse images of the map and a fundamental tool in the theory of generic smooth maps, is known to be non-trivial. In this paper, we show similar results in new appropriate situations. These works are regarded as extensions of works by Hiratuka and Saeki in 2013--4.

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arXiv2018

A new explicit way of obtaining special generic maps into the 3-dimensional Euclidean space

A special generic map is a smooth map regarded as a natural generalization of Morse functions with just 2 singular points on homotopy spheres. Canonical projections of unit spheres are simplest examples of such maps and manifolds admitting special generic maps into the plane are completely determined by Saeki in 1993 and ones admitting such maps into general Euclidean spaces are determined under appropriate conditions. Moreover, if the difference of dimensions of source and target manifolds are not so large, then the diffeomorphism types of source manifolds are often limited. These explicit facts make special generic maps attractive objects in the theory of Morse functions and higher dimensional versions and application to algebraic and differentiable topology of manifolds, which is an important study in both singuarity theory of maps and algebraic and differential topology of manifolds. In this paper, we demonstrate a way of construction of special generic maps into the 3-dimensional Euclidean space. For this, first we prepare maps onto 2-dimensional polyhedra regarded as simplicial maps naturally called pseudo quotient maps, which are generalizations of the quotient maps to the spaces of all the connected components of inverse images, so-called Reeb spaces of original smooth maps, being fundamental and important tools in the studies. The success of the construction explicitly shows that a class of maps which seems to cover a larger class of source manifolds may not be not so large and that the diffeomorphism types of source manifolds may be restricted as strongly as special generic maps. We also explain differential topological facts and problems related to this.

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arXiv2018

Generalizations of Reeb spaces of special generic maps and applications to a problem of lifts of smooth maps

A Reeb space is defined as the space of all the connected components of inverse images of a smooth map, which is a fundamental tool in studying smooth manifolds using generic smooth maps whose codimensions are not positive such as Morse functions, their higher dimensional versions including fold maps and general stable maps. A special generic map is a fold map and a generalization of Morse functions with just 2 singular points on homotopy spheres and the Reeb space is a compact manifold whose dimension is equal to that of the target manifold and which can be immersed into the target manifold. In this paper, we generalize a quotient map onto a Reeb space of a special generic map. We define a map onto a polyhedron locally a quotient map induced from a special generic map. Moreover, we take advantage of the generalized maps to construct lifts of Morse functions of a certain class; the composition of the lift and the canonical projection is the original funciton. It is an answer of an explicit problem in the studies of lifts of smooth maps, or maps such that the compositions of the found maps and the canonical projections are original maps, which are fundamental and important in the studies of smooth maps and applications to algebraic and differential topology of manifolds.

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arXiv2015

Simple fold maps and manifolds bounded by the source manifolds

Fold maps are higher dimensional versions of Morse functions, which play important roles in the studies of smooth manifolds, and such general maps also have been fundamental tools in the studies of smooth manifolds by using generic maps. In this paper, we study {\it simple} fold maps, which are fold maps such that any connected component of the inverse image of each singular value includes at most one singular point. More precisely, we consider simple fold maps having simple structures locally or globally and show that the source manifolds are bounded by (PL) manifolds obtained by considering the structures of maps under appropriate conditions. Such studies are regarded as extensions of results obtained by Saeki, Suzuoka etc. by 2005, which state that closed manifolds admitting simple fold maps and more generally stable maps into manifolds of lower dimensions without boundaries inverse images of whose regular values are always disjoint unions of spheres are bounded by compact manifolds obtained by observing the given maps.

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arXiv2015

Round fold maps on manifolds regarded as the total spaces of linear and more general bundles

Stable fold maps are fundamental tools in studying a generalized theory of the theory of Morse functions on smooth manifolds and its application to geometry of the manifolds. It is important to construct explicit fold maps systematically to study smooth manifolds by the theory of fold maps easy to handle. However, such constructions have been difficult in general. Round fold maps are defined as stable fold maps such that the sets of all the singular values are concentric spheres and it was first introduced in 2012--2014. The author studied algebraic and differential topological properties of such maps and their manifolds and constructed explicit round fold maps. For example, the author succeeded in constructing such maps on manifolds regarded as the total spaces of bundles over smooth homotopy spheres by noticing that smooth homotopy spheres admit round fold maps whose singular sets are connected and more generally, new such maps on manifolds regerded as the total space of circle bundles over another manifold admitting a round fold map. In this paper, as an advanced work, we construct new explicit round fold maps on manifolds regarded as the total spaces of bundles such that the fibers are closed smooth manifolds and that the structure groups are linear and more general bundles over a manifold admitting a round fold map.

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arXiv2014

Explicit round fold maps on some fundamental manifolds

Stable fold maps are fundamental tools in a generalization of the theory of Morse functions on smooth manifolds and its application to studies of topological properties of smooth manifolds. Round fold maps were introduced as stable fold maps with singular value sets, defined as the set consisting of all the singular values, of concentric spheres by the author in 2013; for example, some special generic maps on spheres are regarded as round fold maps whose singular value sets are connected. Algebraic invariants such as homology and homotopy groups of manifolds admitting round fold maps and more precisely, the homeomorphism and diffeomorphism types of manifolds admitting such maps having appropriate differential topological structures were studied. Moreover, explicit round fold maps into the Eucidean space of dimension larger than $1$ are constructed on some fundamental manifolds such as manifolds having the structures of bundles over the standard sphere of dimension equal to the Euclidean space whose fibers are closed smooth manifolds and manifolds of dimension not smaller than twice the dimension of the Euclidean space represented as the connected sum of manifolds having the structures of bundles over the standard sphere of dimension equal to the Euclidean space whose fibers are diffeomorphic to standard spheres. In this paper, we construct new explicit fold maps on some fundamental manifolds including the manifolds before by using extended methods of ones used in the constructions before.

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arXiv2014

Constructions of round fold maps on circle bundles

{\it Fold maps} are fundamental tools in generalizing the theory of Morse functions and its application to studies of geometric properties of manifolds. One of the fundamental and important problems in the theory of fold maps is to construct explicit fold maps, which are often difficult. In this paper, we construct new examples of {\it round fold maps}, which are defined as {\it stable fold maps} with singular value sets of concentric spheres introduced by the author on manifolds having the structures of circle bundles. The class of round fold maps includes some {\it special generic} maps on homotopy spheres and such maps have been constructed on manifolds having the structures of smooth bundles over standard spheres and manifolds represented as connected sums of manifolds admitting bundle structures over a standard sphere with fibers diffeomorphic to a standard sphere, for example, in previous studies by the author in the 2010s. Furthermore, such maps on manifolds admitting the structures of smooth bundles over spheres or more general manifolds including families of circle bundles over given manifolds were constructed by applying operations derived from the theory of bundles ({\it P-operations}), and in this paper, we use the operations to obtain new round fold maps.

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arXiv2013

Constructions of round fold maps on $C^{\infty}$ bundles

In this paper, we construct round fold maps or stable fold maps with concentric singular value sets introduced by the author on smooth bundles over spheres or bundles over more general manifolds. The class of round fold maps includes special generic maps on spheres and such maps have been constructed on smooth bundles over standard spheres of dimensions larger than 1 and connected sums of smooth bundles over standard spheres of dimensions larger than 1 whose fibers are standard spheres, for example, in previous studies by the author. In this paper, we obtain round fold maps and the diffeomorphism types of their source manifolds which do not appear in these studies in new manners.

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