arXiv research

Study k-folding map-germs to understand surface geometry.

Let $M$ and $N$ be Nash manifolds, and $f$ and $g$ Nash maps from $M$ to $N$. If $M$ and $N$ are compact and if $f$ and $g$ are analytically R-L equivalent, then they are Nash R-L equivalent. In the local case, $C^infty$ R-L equivalence of two Nash map germs implies Nash R-L equivalence. This shows a difference of Nash…

Generic smooth plane-to-plane map germs are topologically equivalent to cones of mappings of the circle. We carry out a complete topological classification of smooth stable mappings of the circle and show how this classification leads, via the result mentioned above, to a topological classification of finitely determin…

In this paper, two sufficient conditions are provided for given two K-equivalent map-germs to be bi-Lipschitz A-equivalent. These are Lipschitz analogues of the known results on C^r-A-equivalence $(0 \leq r \leq \infty)$ for given two K-equivalent map-germs. As a corollary of one of our results, a Lipschitz version of …

Unimodular classification of symmetric matrix map-germs.

Study on conditions for singular local tube fibrations.

In this paper, we introduce the notions of map-germs of pedal unfolding type and normalized Legendrian map-germs; and then we show that the fundamental theorem of calculus provides a natural one to one correspondence between Whitney umbrellas of pedal unfolding type and normalized swallowtails.

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 p…

We present a complete set of criteria for determining A-types of plane-to-plane map-germs of corank one with A-codimension <7, which provides a new insight into the A-classification theory from the viewpoint of recognition problem. As an application to generic differential geometry, we discuss about projections of smoo…

Study the topological information of map germs using Euler obstruction.

The study investigates deformations of swallowtails in 3D space, preserving curvature signs.

We provide bi-Lipschitz invariants for finitely determined map germs $f: (\mathbb{K}^n,0) \to (\mathbb{K}^p, 0)$, where $\mathbb{K} = \mathbb{R}$ or $\mathbb{C}$. The aim of the paper is to provide partial answers to the following questions: Does the bi-Lipschitz type of a map germ $f: (\mathbb{R}^n, 0) \to (\mathbb{R…

We consider equivalence relations among smooth map germs with respect to geometry of G-structures on the target space germ. These equivalence relations are natural generalization of right-left equivalence (i.e., A-equivalence) in the sense of Thom-Mather depending on geometric structures on the target space germ. Unfor…

The paper improves Zakalyukin's lemma for frontals and applies it to surface singularities.

The paper introduces new invariants to study topological properties of map germs.

In this paper we study the Milnor fibrations associated to real analytic map germs $ψ:(\mathbb{R}^{m},0) \to (\mathbb{R}^2,0)$ with isolated critical point at $0\in \mathbb{R}^{m}$. The main result relates the existence of called Strong Milnor fibrations with a transversality condition of a convenient family of analyti…

We use topology of configuration spaces to give a characterization of Neuwirth--Stallings pairs $(S^5, K)$ with $\dim K = 2$. As a consequence, we construct polynomial map germs $(\mathbb{R}^{6},0)\to (\mathbb{R}^{3},0)$ with an isolated singularity at the origin such that their Milnor fibers are not diffeomorphic to a…

The study examines the topology of map germs and their images.

Study the topology of Milnor boundaries for real analytic map germs.

In this paper we extend the characterization of trivial map-germs for the real Milnor fibrations started by Church and Lamotke. Our main result cover all cases on the three dimensional real Milnor fibers.

We give a new and simple proof for the computation of the oriented and the unoriented fold cobordism groups of Morse functions on surfaces. We also compute similar cobordism groups of Morse functions based on simple stable maps of 3-manifolds into the plane. Furthermore, we show that certain cohomology classes associat…

We study a moduli stratum of A-orbits of plane-to-plane germs of corank 2 with codimension 3. We describe explicitly the bifurcation diagram of its topologically A-versal unfolding. Two geometric applications to parabolic objects are presented.

We introduce several sufficient conditions to guarantee the existence of the Milnor vector field for new classes of singularities of map germs. This special vector field is related with the equivalence problem of the Milnor fibrations for real and complex singularities, if they exit.

Quasi-holomorphic homotopies of immersions of 3-manifolds into 5-manifolds

The paper classifies constraint mappings in optimization problems.

We study singularities of constant positive Gaussian curvature surfaces and determine the way they bifurcate in generic 1-parameter families of such surfaces. We construct the bifurcations explicitly using loop group methods. Constant Gaussian curvature surfaces correspond to harmonic maps, and we examine the relations…

Study describes singularities of height functions on specific singular surfaces.

We generalize a classical result concerning smooth germs of surfaces, by proving that monodromies on links of isolated complex surface singularities associated with reduced holomorphic map germs admit a positive factorization. As a consequence of this and a topological characterization of these monodromies by Anne Pich…

Study describes singularities of distance squared functions on singular surfaces.

In this paper, we study the topology of real analytic map-germs with isolated critical value $f: (\mathbb{R}^m,0) \to (\mathbb{R}^n,0)$, with $1 <n <m$. We compare the topology of $f$ with the topology of the compositions $π_i^* \circ f$, where $π_i^*: \mathbb{R}^n \to \mathbb{R}^{n-1}$ are the projections $(t_1, \dots…

Homotopy equivalence found between Milnor-Lê fibers of specific singularities.

Adapts a short argument to derive a stability theorem for smooth maps.

The paper explores $ρ$-regularity for real analytic maps and its relation to Milnor fibrations.

Introduces quasi-holomorphic maps and their properties.

In this article we study the topology of a family of real analytic germs $F \colon (\mathbb{C}^3,0) \to (\mathbb{C},0)$ with isolated critical point at 0, given by $F(x,y,z)=f(x,y)\bar{g(x,y)}+z^r$, where $f$ and $g$ are holomorphic, $r \in \mathbb{Z}^+$ and $r \geq 2$. We describe the link $L_F$ as a graph manifold us…

The paper explores density of stable mappings and their properties.