A homotopy principle for maps with prescribed Thom-Boardman singularities

Ando, Yoshifumi

doi:10.1090/s0002-9947-06-04326-1

Cited by 6 publications

(2 citation statements)

References 22 publications

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“…The homotopy principle for Thom-Boardman bases R I s ⊂ J k (R s , R q+s ), q ≤ 0, was proved by A. du Plessis [36] and a version of the h-principle in the case of the symbol (1 − q, 0) was proved by Eliashberg [9], [10] (see also Ando [2]), and then, in the case of an arbitrary symbol I with I > (1 − q, 0), using the Eliashberg h-principle, by Ando [5]. Consequently, we derive the bordism principle for Thom-Boardman differential relations.…”

Section: Maps With Prescribed Thom-boardman Singularitiesmentioning

confidence: 95%

Bordism groups of solutions to differential relations

Sadykov

2009

Algebr. Geom. Topol.

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Let us mention that in later sections we will define the group B.W I J / for an arbitrary (not necessarily closed) manifold W . At the moment, however, we consider only maps of closed manifolds, just to simplify the exposition.Carefully choosing the set J of singularities, one may derive cobordism groups related to various objects in geometry and topology. Example 1.1 Since a proper submersion is a smooth fiber bundle, the cobordism group of submersions is closely related to diffeomorphism groups of smooth manifolds. It is also known to be related to various infinite loop spaces, moduli spaces of Riemann surfaces, the cobordism category as in Galatius et al [17] A priori J -cobordism groups do not form generalized cohomology theories since, for example, J -cobordism groups are not defined for topological spaces. In the current paper we propose a counterpart of B.W I J / that for a wide range of sets J can be used to compute B.W I J / in the same way as singular cohomology groups H n .W I R/ can be used to compute De Rham cohomology groups H

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Section: Maps With Prescribed Thom-boardman Singularitiesmentioning

confidence: 95%

Bordism groups of solutions to differential relations

Sadykov

2009

Algebr. Geom. Topol.

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“…In particular, we show the following examples of Ω(n, p): (i) Ω I (n, p) such that when n ≧ p ≧ 2, I ≧ (n − p + 1, 0) ( [8]), (ii) an open subspace consisting of all regular k-jets and a finite number of Korbits of K-simple singularities such that when n ≧ p ≧ 2, it contains all fold jets in addition ( [9]).…”

Section: Classifying Spacementioning

confidence: 97%

Cobordisms of maps with singularities of given class

Ando

2008

Algebr. Geom. Topol.

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Let N and P be smooth closed manifolds of dimensions n and p respectively. Given a Thom-Boardman symbol I, a smooth map f : N → P is called an Ω I -regular map if and only if the Thom-Boardman symbol of each singular point of f is not greater than I in the lexicographic order. We will represent the group of all cobordism classes of Ω I -regular maps of n-dimensional closed manifolds into P in terms of certain stable homotopy groups. As an application we will study the relationship among the stable homotopy groups of spheres, the above cobordism group and higher singularities.

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Characteristic classes and existence of singular maps

Kalmar¹,

Terpai²

2012

Trans. Amer. Math. Soc.

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Abstract. The existence of a corank one map of negative codimension puts strong restrictions on the topology of the source manifold. It implies many vanishing theorems on characteristic classes and often even vanishing of the cobordism class of the source manifold. Most of our results lie deeper than just vanishing of Thom polynomials of the higher singularities. We blow up the singular map along the singular set and then perturb the arising nongeneric corank one map.

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A homotopy principle for maps with prescribed Thom-Boardman singularities

Abstract: Abstract. Let N and P be smooth manifolds of dimensions n and p (n ≥ p ≥ 2) respectively. Let Ω I (N, P ) denote an open subspace of J ∞ (N, P ) which consists of all Boardman submanifolds Σ J (N, P ) of symbols J with J ≤

Cited by 6 publications

References 22 publications

Bordism groups of solutions to differential relations

Bordism groups of solutions to differential relations

Cobordisms of maps with singularities of given class

Characteristic classes and existence of singular maps

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