Symmetry properties of singularities ofC
∞-functions

Jänich, Klaus

doi:10.1007/bf01424772

Cited by 29 publications

(31 citation statements)

References 6 publications

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“…The Rimányi-Szűcs construction uses a technical Jänich theorem (see Jänich [24] and Wall [54]), which a priori has no analogue in the positive dimension case. However, when exists, the Rimányi-Szűcs construction B J consists of classifying spaces BDiff F˛of diffeomorphism groups of fibers F˛of J -maps, ie, as a set B J is given by the disjoint union…”

Section: Remark 23mentioning

confidence: 99%

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: Remark 23mentioning

confidence: 99%

Bordism groups of solutions to differential relations

Sadykov

2009

Algebr. Geom. Topol.

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“…(Recall that this means that c is the minimal possible dimension for the representatives of OEÁ.) Let G O Á denote the maximal compact subgroup of the automorphism group of Á (in the sense of Jänich [19]; see also Wall [44] and Rimányi and the author [29]). Let Á and z Á be the representations of G O Á in the source and the target of Á, respectively (ie for 8g 2…”

Section: Definition Of the Kazarian Space By Gluingmentioning

confidence: 99%

Cobordism of singular maps

Szűcs

2008

Geom. Topol.

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Throughout this paper we consider smooth maps of positive codimensions, having only stable singularities (see Arnold, Guseȋn-Zade and Varchenko [3, Section 1.4, Chapter 1]. We prove a conjecture, due to M Kazarian, connecting two classifying spaces in singularity theory for this type of singular maps. These spaces are: 1) Kazarian's space (generalising Vassiliev's algebraic complex and) showing which cohomology classes are represented by singularity strata. 2) The space X giving homotopy representation of cobordisms of singular maps with a given list of allowed singularities as in work of Rimányi and the author [29; 34; 35].We obtain that the ranks of cobordism groups of singular maps with a given list of allowed stable singularities, and also their p -torsion parts for big primes p coincide with those of the homology groups of the corresponding Kazarian space. (A prime p is "big" if it is greater than half of the dimension of the source manifold.) For all types of Morin maps (ie when the list of allowed singularities contains only corank 1 maps) we compute these ranks explicitly.We give a very transparent homotopical description of the classifying space X as a fibration. Using this fibration we solve the problem of elimination of singularities by cobordisms. (This is a modification of a question posed by Arnold [4, page 212].) 57R45, 55P42; 57R42, 55P15In this paper all smooth maps will have positive codimensions and stable singularities. The aim of the present paper is to establish a close relationship between Kazarian's homological characteristic spectral sequence (generalising Vassiliev's complex) and the classifying space for cobordisms of maps having singularities only from a given fixed list (see the papers of the author [34] and the author with Rimányi [29]). Such a relationship was conjectured by M Kazarian, and it can be expressed as follows (in our formulation): There is a spectrum with a filtration such that the arising homological spectral sequence gives Kazarian's characteristic spectral sequence while the homotopy groups of the spectrum give the corresponding cobordism groups of singular maps (with a shift of the dimension). Hence the Hurewicz map for this spectrum induces a rational isomorphism from the cobordism groups of singular maps to the homology groups of Kazarian's space. This allows us to extend quite a few classical theorems of cobordism theory to cobordisms of singular maps. In particular (at least modulo torsion): b) We extend the definition of Conner and Floyd of the characteristic numbers of bordism classes and the statement that these numbers form a complete set of invariants. We give a complete and computable set of obstructions to elimination of singularities by cobordism.c) For any set of corank one singularities we determine explicitly the ranks of the corresponding cobordism groups.d) We give some general results beyond Morin maps, and also on the torsion groups.e) We show a "Postnikov like tower" that produces the classifying space for cobordisms of singular maps as an iterat...

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“…By [13,47] this structure group can be reduced to a maximal compact subgroup, namely to the group O(λ) × O(q + 1 − λ) in the case of 0 ≤ λ < (q +1)/2 and the group generated by the group O(λ)×O(λ) and the transformation T = 0 I λ I λ 0 in the case of λ = (q + 1)/2, see, for example, [30]. We denote this…”

Section: Cobordism Invariants Of Fold Mapsmentioning

confidence: 99%

“…We will work with bundles whose fibers and structure groups are germs and groups of automorphisms of germs, respectively, see [13].…”

Section: Cobordism Invariants Of Fold Mapsmentioning

confidence: 99%

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Cobordism invariants of fold maps

Kalmar¹

2008

Real and Complex Singularities

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Abstract. 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. IntroductionFold maps of (n + q)-dimensional manifolds into n-dimensional manifolds have the formulaas a local form around each singular point, and the subset of the singular points in the source manifold is a (q + 1)-codimensional submanifold (for results about fold maps, see, for example, [1,2,3,5,8,9,18,26,35,36]). If we restrict a fold map to the set of its singular points, then we obtain a codimension one immersion into the target manifold of the fold map. This immersion together with more detailed informations about the neighbourhood of the set of singular points in the source manifold can be used as a geometrical invariant (see Section 2) of fold cobordism classes (see Definition 1.1) of fold maps (for results about cobordisms of singular maps with completely different approach from our present paper, see, for example, [7,11,12,21,25,34,48] and the works of Ando, Sadykov, Szűcs and the author in References). In this way we obtain a geometrical relation between fold maps and immersions with prescribed normal bundles via cobordisms. In [18] we showed that these invariants describe completely the cobordisms of simple fold maps of (n + 1)-dimensional manifolds into n-dimensional manifolds and in [17] we showed that these invariants detect direct summands of the cobordism group of fold maps, namely stable homotopy groups of spheres. In this paper we extend the results of [17] and show by constructing fibrations of Morse functions over immersed manifolds 2000 Mathematics Subject Classification. Primary 57R45; Secondary 57R75, 57R42, 55Q45.

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Symmetry properties of singularities ofC ∞-functions

Cited by 29 publications

References 6 publications

Bordism groups of solutions to differential relations

Bordism groups of solutions to differential relations

Cobordism of singular maps

Cobordism invariants of fold maps

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