Singularities of Mappings (original) (raw)

Critical points of higher order for the normal map of immersions in

Topology and its Applications, 2012

We study the critical points of the normal map ν : N M → R k+n , where M is an immersed k-dimensional submanifold of R k+n , N M is the normal bundle of M and ν(m, u) = m + u if u ∈ N m M. Usually, the image of these critical points is called the focal set. However, in that set there is a subset where the focusing is highest, as happens in the case of curves in R 3 with the curve of the centers of spheres with contact of third order with the curve. We give a definition of r-critical points of a smooth map between manifolds, and apply it to study the 2 and 3-critical points of the normal map in general and the 2-critical points for the case k = n = 2 in detail. In the later case we analyze the relation with the strong principal directions of Montaldi [2].

Differentiable maps with 0-dimensional critical set. I

Pacific Journal of Mathematics, 1972

Let /: M n-> N p be C n with n-p = 0 or 1, let p ^ 2, and let R P-i(f) be the critical set of /. If dim (J? P-i(/)) ^ 0 and dim (/CR P-i(/)) ^ p-2, then (1.1) at each α? e M n , f is locally topologically equivalent to one of the following maps: (a) the projection map p: R n-> R p , (b) σ:C^C defined by σ(z) = z d (d = 2, 3,. . .), where C is the complex plane, or (c) τ: CxC-> Cxi? defined by T(S,W) = (2S M;, |W| 2-|Z| 2),

Variation mappings on singularities with a 1-dimensional critical locus

Topology, 1991

DtIfinifion. /is called a k-isolated singularity if dim C = k. The case k = 0 is already known as an isolated singularity. 1.2. In this paper we consider singularities with a l-dimensional critical locus and study the vanishing homology in a full neighborhood of the origin. In this case the vanishing homology is concentrated on the l-dimensional set E. We can write where each Xi is an irreducible curve. At the origin there is the Milnor fibre F of/and on each Xi-(0) there is a local system of transversal singularities: Take at any XEZ~-(0) th e g erm of a generic transversal section. This gives an isolated singularity whose p-class is well-defined. We denote a typical Milnor fibre of this transversal singularity by Fi. On the level of homology we get in this way a local system with fibre A,_, (F,). More precisely we consider in the l-isolated case the following data: a. The Milnor fibre F. The vanishing homology is concentrated in dimensions n-I and n: H,(F) = Z'm, which is free. H,_ , (F), which can have torsion, On the fibre F there is acting the Milnor monodromy (or horizontal monodromy):

Global Classification of Isolated Singularities in

2016

We characterize those closed 2k-manifolds admitting smooth maps into (k + 1)manifolds with only finitely many critical points, for k ∈ {2, 4}. We compute then the minimal number of critical points of such smooth maps for k = 2 and, under some fundamental group restrictions, also for k = 4. The main ingredients are King's local classification of isolated singularities, decomposition theory, low dimensional cobordisms of spherical fibrations and 3-manifolds topology.

On the triple points of singular maps

Commentarii Mathematici Helvetici, 2002

The number of triple points (mod 2) of a self-transverse immersion of a closed 2nmanifold M into 3n-space are known to equal one of the Stiefel-Whitney numbers of M. This result is generalized to the case of generic (i.e. stable) maps with singularities. Besides triple points and Stiefel-Whitney numbers, a certain linking number of the manifold of singular values with the rest of the image is involved in the generalized equation which corrects an erroneous formula in [9]. If n is even and the closed manifold is oriented then the equations mentioned above make sense over the integers. Together, the integer-and mod 2 generalized equations imply that a certain Stiefel-Whitney number of closed oriented 4k-manifolds vanishes. This Stiefel-Whitney number is in fact the first in a family which vanish on such manifolds.

Singularities with critical locus a 1-dimensional complete intersection and transversal type A1

Topology and its Applications, 1987

In this paper we study germs of holomorphic functions f: (C "+I, 0) + @ with the following two properties: (i) the critical set P off is a l-dimensional isolated complete intersection singularity (icis); (ii) the transversal singularity off in points of P-{0} is of type A, We first compute the homology of the Milnor fibre F off in terms of numbers of special points in certain deformations. Next we show that the homotopy type of the Milnor libre F off is a bouquet of spheres. There are two cases: (a) general case S" v. v S",

Singularities of Fredholm maps . . . II: Local behaviour and pointwise conditions

2014

In analogy to what happens in finite dimensions we state the Normal Form Theorem for ksingularities, introduced in the previous paper of the series. By means of that we study the local behaviour near a singularity i.e. we deduce local results of existence and multiplicity of solutions for the equation F (x) = y where F is a 0-Fredholm map and x belongs to a suitable neighbourhood of a singular point x o , once x o is identified as one of the three kinds of singularities defined in the first paper. To this end we also start to seek alternative strategies for the determination of the type of a given singularity according to our classification. Here we give a pointwise approach for lowerorder singularities which is coherent with the Thom-Boardman classification in finite dimensions. We conclude by applying the pointwise condition to a differential problem where, under suitable hypotheses, we determine a swallow's tail singularity (3-singularity).

Critical value sets of generic mappings

Pacific Journal of Mathematics, 1984

Let Y be a real analytic set. The subset of Y consisting of all points where the local dimension of Y is maximal is called the main part of Y. A subset Y' of a real analytic manifold N is called a main semi-analytic set if Y' is the main part of some analytic set in a neighborhood of each point of N. In this paper it is shown that any proper C 00 mapping between analytic manifolds can be approximated by an analytic mapping in the Whitney topology so that the critical value set is a main semi-analytic set. An analogue holds true for the algebraic case too. We see easily that a main semi-analytic set is semi-analytic [see S. Lojasiewicz [12]]. Any nowhere dense semi-analytic set is the critical value set of some analytic mapping, to say more precisely. REMARK. Let K be a semi-analytic subset of N of codimension > 0. Then there exist an analytic manifold M and an analytic mapping /: M-* N such that dim M = dim N > dim Σf and f(Σf) = K.