Fold cobordisms and stable homotopy groups

Kalmar, Boldizsar

doi:10.1556/sscmath.2009.1100

Cited by 3 publications

(2 citation statements)

References 16 publications

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“…The cobordism groups of special generic maps are related to diffeomorphism groups of spheres and exotic smooth structures on spheres [45], [41]. The cobordism groups of fold maps are known to be related to stable homotopy groups of spheres and other interesting objects [8], [1], [26].…”

Section: Introductionmentioning

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: Introductionmentioning

confidence: 99%

Bordism groups of solutions to differential relations

Sadykov

2009

Algebr. Geom. Topol.

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“…Cobordism groups of various versions of Morin maps of non-negative dimension are studied in a series of papers by Ando [2], [3], [4], Kalmar [23], [24], [25], [26], Saeki [52], [51], Ikegami-Saeki [20], [21], Ikegami [19], and the author [48]. For a study in the case d < 0 we refer an interested reader to the paper of Szucz [54].…”

Section: Cobordism Groups Of Morin Mapsmentioning

confidence: 99%

Invariants of singular sets of smooth maps

Sadykov¹

2010

Preprint

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A singular point of a smooth map f : M → N of manifolds of dimensions m and n respectively is a point in M at which the rank of the differential df is less than the minimum 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 [ S] ∈ 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 S may not possess the fundamental class. The simplest new invariant is defined by carefully choosing the fundamental class of the intersection of 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 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 ≥ 1 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 ksingularities for each k ≥ 1.

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Fold cobordisms and a Poincaré–Hopf-type theorem for the signature

Kalmar

2022

Algebr. Geom. Topol.

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Fold cobordisms and stable homotopy groups

Cited by 3 publications

References 16 publications

Bordism groups of solutions to differential relations

Bordism groups of solutions to differential relations

Invariants of singular sets of smooth maps

Fold cobordisms and a Poincaré–Hopf-type theorem for the signature

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