Search

In this paper, the concordance of Morse functions is defined, and a necessary and sufficient condition for given two Morse functions to be concordant is presented and is compared with the cobordism criterion. Cobordism of Morse functions on smooth closed manifolds is an equivalence relation defined by using cobordisms of manifolds and fold maps. Given two Morse functions, it is important to decide whether they are cobordant or not, and this problem was first solved for surfaces and then for manifolds of general dimensions by Ikegami-Saeki, Kalm\'ar, and Ikegami. On the other hand, for Morse functions on the same manifold, we can consider a stronger equivalence relation called concordance., Comment: 9 pages, 5 figures

Sufficiency of jets is a very important notion introduced by Rene Thom in order to establish the structural stability theory. The criteria for some sufficiency of jets are known as the Kuo condition and Thom type inequality, which are defined using the Kuo quantity and Thom quantity. Therefore these quantities are meaningful. In this paper we show the equivalence of Kuo and Thom quantities. Then we apply this result to the relative conditions to a given closed set., Comment: 10 pages, no figure. arXiv admin note: substantial text overlap with arXiv:1703.07069

Given a null-cobordant oriented framed link $L$ in a closed oriented $3$--manifold $M$, we determine those links in $M \setminus L$ which can be realized as the singular point set of a generic map $M \to \mathbb{R}^2$ that has $L$ as an oriented framed regular fiber. Then, we study the linking behavior between the singular point set and regular fibers for generic maps of $M$ into $\mathbb{R}^2$., Comment: 13 pages, 8 figures

We consider the problems of sufficiency of jets relative to a given closed set. In the non-relative case, criteria for r-jets to be V-sufficient and C^0-sufficient in C^r mappings or C^{r+1} mappings have been obtained. In particular, it is shown that V-sufficiency and C^0-sufficiency in C^r functions or C^{r+1} functions are equivalent. In this paper we discuss characterisations of V-sufficiency and C^0-sufficiency in the relative case, corresponding to the above non-relative results. Applying the results obtained in the relative case, we construct examples of polynomial functions whose relative r-jets are V-sufficient in C^r functions and C^{r+1} functions but not C^0-sufficient in C^r functions and C^{r+1} functions, respectively. In addition, we give characterisations of relative finite V-determinacy and also relative finite C^r contact determinacy., Comment: 34 pages, no figure

We define an equivalence relation called A-isotopy between finitely determined map-germs, which is a strengthened version of A-equivalence. We consider the number of A-isotopy classes of equidimensional Morin singularities, and some other well-known low-dimensional singularities. We also give an application to stable perturbations of simple equi-dimensional map-germs., Comment: 17pages

Lips and swallow-tails are generic local moves of singularities of a smooth map to a 2-manifold. We prove that these moves of singularities of the product map of two functions on a 3-manifold can be realized by isotopies of the functions., Comment: 11 pages, 2 figures, v4: a missed case in Figure 2 has been added, and some minor corrections

The aim of this paper is to give a new result of the differential Galois theory of linear ordinary differential equations. In particular, we compute differential Galois group for special type non-resonant Fuchsian system., Comment: Reports of Enlarged Sessions of the Seminar of I. Vekua Institute of Applied Mathematics, Vol.25, 2011

A singular point of a smooth map F: M -> N of manifolds is a point in M at which the rank of the differential dF is less than the minimum of dimensions of M and N. The classical invariant of the set S of singular points of F of a given type is defined by taking the fundamental class [\bar{S}]\in H_*(M) of the closure of S. We introduce and study new invariants of singular sets for which the classical invariants may not be defined, i.e., for which \bar{S} may not possess the fundamental class. The simplest new invariant is defined by carefully choosing the fundamental class of the intersection of \bar{S} and its slight perturbation in M. Surprisingly, for certain singularity types such an invariant is well-define (and not trivial) despite the fact that \bar{S} does not possess the fundamental class. We determine new invariants for maps with Morin singularities---i.e., singularities of types A_k for k>0 in the ADE-classification of simple singularities by Dynkin diagrams---and, as an application, show that these invariants together with generalized Miller-Morita-Mumford classes form a commutative graded algebra of characteristic classes that completely determine the cobordism groups of maps with at most A_k-singularities for each k>0., Comment: 64 pages

We correct the proof in the unoriented case of Theorem 1.2 in the paper "Cobordisms of maps with singularities of given class"

Given a closed manifold N and a self-indexing Morse function f: N --> R with up to four distinct Morse indices, we construct a symplectic Lefschetz fibration pi: E --> C which models the complexification of f on the disk cotangent bundle, f_C : D(T*N) --> C, when f is real analytic. By construction, pi: E --> C comes with an explicit regular fiber M and vanishing spheres V_1,...,V_m in M, one for each critical point of f. Our main result is that (E,pi) is a good model for the complexification (D(T*N),f_C) in the sense that N embeds in E as an exact Lagrangian submanifold, and in addition, pi|N = f and E is homotopy equivalent to N. There are several potential applications in symplectic topology, which we discuss in the introduction., Comment: 54 pages, 11 figures

We shall give useful criteria of lips, beaks and swallowtail singularities of smooth map from the plane into the plane. As an application of criteria, we will discuss the singularities of Cauchy problem of single conservation law., Comment: 9 pages, 4 figures

Let W be a finite reflection group acting orthogonally on R^n, P be the Chevalley polynomial mapping determined by an integrity basis of the algebra of W-invariant polynomials, and h be the highest degree of the coordinate polynomials in P.There exists a linear mapping from (C^r(R^n))^W to C^[r/h](R^n), f\mapsto F such that f=F \circ P, continuous for the natural Fr\'echet topologies. A general counterexample shows that this result is the best possible. The proof uses techniques of division by linear forms and a study of compensation phenomenons. An extension to P^{-1}(R^n) of invariant formally holomorphic regular fields is needed., Comment: 13 pages

In this paper, we discuss the recognition problem for A_k-type singularities on wave fronts. We give computable and simple criteria of these singularities, which will play a fundamental role in generalizing the authors' previous work "the geometry of fronts" for surfaces. The crucial point to prove our criteria for A_k-singularities is to introduce a suitable parametrization of the singularities called the "k-th KRSUY-coordinates". Using them, we can directly construct a versal unfolding for a given singularity. As an application, we prove that a given nondegenerate singular point p on a real (resp. complex) hypersurface (as a wave front) in R^{n+1} (resp. C^{n+1}) is differentiably (resp. holomorphically) right-left equivalent to the A_{k+1}-type singular point if and only if the linear projection of the singular set around p into a generic hyperplane R^n (resp. C^n) is right-left equivalent to the A_k-type singular point in R^n (resp. C^{n}). Moreover, we show that the restriction of a C-infinity-map f:R^n --> R^n to its Morin singular set gives a wave front consisting of only A_k-type singularities. Furthermore, we shall give a relationship between the normal curvature map and the zig-zag numbers (the Maslov indices) of wave fronts., Comment: 15 pages, 2 figures

Sz\H ucs proved in 2000 that the $r$-tuple-point manifold of a generic immersion is cobordant to the $\Sigma^{1_{r-1}}$-point manifold of its generic projection. Here we slightly extend this by showing that the natural mappings of these manifolds are bordant to each other. The main novelty of our approach is that we construct the bordism explicitly., Comment: 8 pages

We compute the oriented cobordism group of fold maps of 4-manifolds into the space with all the possible restrictions (and also with no restriction) to the singular fibers. We also give geometric invariants which describe completely the cobordism group of fold maps., Comment: 11 pages, revised version

Maps between manifolds $M^m\to N^{m+\ell}$ ($\ell>0$) have multiple points, and more generally, multisingularities. The closure of the set of points where the map has a particular multisingularity is called the multisingularity locus. There are universal relations among the cohomology classes represented by multisingularity loci, and the characteristic classes of the manifolds. These relations include the celebrated Thom polynomials of monosingularities. For multisingularities, however, only the form of these relations is clear in general (due to Kazarian), the concrete polynomials occurring in the relations are much less known. In the present paper we prove the first general such relation outside the region of Morin-maps: the general quadruple point formula. We apply this formula in enumerative geometry by computing the number of 4-secant linear spaces to smooth projective varieties. Some other multisingularity formulas are also studied, namely 5, 6, 7 tuple point formulas, and one corresponding to $\Sigma^2\Sigma^0$ multisingularities., Comment: 23 pages

In the first part of the paper we construct a ring structure on the rational cobordism classes of Morin maps (i. e. smooth generic maps of corank 1). We show that associating to a Morin map its singular strata defines a ring homomorphism to $\Omega_* \otimes \Q$, the rational oriented cobordism ring. This is proved by analyzing multiple-point sets of product immersion. Using these homomorphisms we are able to identify the ring of Morin maps. In the second part of the paper we compute the oriented Thom polynomial of the $\Sigma^2$ singularity type with $\Q$ coefficients. Then we provide a product formula for the $\Sigma^2$ and the $\Sigma^{1,1}$ singularities., Comment: Corrected some small misprints and made lot of minor (mainly grammatical) alterations. 10 pages

This is a survey paper of author's results on cobordism groups and semigroups of fold maps and simple fold maps. The results include: establishing a relation between fold maps and immersions through geometrical invariants of cobordism classes of fold maps and simple fold maps in terms of immersions with prescribed normal bundles, detecting stable homotopy groups of spheres as direct summands of the cobordism semigroups of fold maps, Pontryagin-Thom type construction for -1 codimensional fold maps and estimations about the cobordism classes of manifolds which have fold maps into stably parallelizable manifolds. In the last section some of these results are extended and we show that our invariants also detect stable homotopy groups of the classifying spaces $BO(k)$ as direct summands of the cobordism semigroups of fold maps., Comment: 13 pages

We obtain complete geometric invariants of cobordism classes of oriented simple fold maps of (n+1)-dimensional manifolds into an n-dimensional manifold N in terms of immersions with prescribed normal bundles. We compute that this cobordism group of simple fold maps is isomorphic to the direct sum of the (n-1)th stable homotopy group of spheres and the (n-1)th stable homotopy group of the infinite dimensional projective space. By using geometric invariants defined in the author's earlier works, we also describe the natural map of the simple fold cobordism group to the fold cobordism group by natural homomorphisms between cobordism groups of immersions. We also compute the ranks of the oriented (right-left) bordism groups of simple fold maps., Comment: revised, 16 pages, 1 figure

We show that the cobordism groups of negative codimensional folds maps contain direct sums of stable homotopy groups of Thom spaces of vector bundles like the circle and the infinite dimensional projective space. We give geometrical invariants which detect these direct summands., Comment: 10 pages

Let $M^2$ be an oriented 2-manifold and $f:M^2\to R^3$ a $C^\infty$-map. A point $p\in M^2$ is called a singular point if $f$ is not an immersion at $p$. The map $f$ is called a front (or wave front), if there exists a unit $C^\infty$-vector field $\nu$ such that the image of each tangent vector $df(X)$ $(X\in TM^2)$ is perpendicular to $\nu$, and the pair $(f,\nu)$ gives an immersion into $R^3\times S^2$. In our previous paper, we gave an intrinsic formulation of wave fronts in $R^3$. In this paper, we shall investigate the behavior of cuspidal edges near corank one singular points and establish Gauss-Bonnet-type formulas under the intrinsic formulation., Comment: 20 pages, 12 figures