Cobordisms of maps with singularities of given class

Ando, Yoshifumi

doi:10.2140/agt.2008.8.1989

Cited by 15 publications

(29 citation statements)

References 63 publications

(134 reference statements)

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“…Corollary 14.1 was also independently observed by Szűcs [51] in the negative dimension case and by Ando [6] in the general case.…”

Section: Applicationssupporting

confidence: 56%

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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“…Corollary 14.1 was also independently observed by Szűcs [51] in the negative dimension case and by Ando [6] in the general case.…”

Section: Applicationssupporting

confidence: 56%

Bordism groups of solutions to differential relations

Sadykov

2009

Algebr. Geom. Topol.

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“…By recent results of Ando [1], Sadykov [12] and Szűcs [19], the cobordism groups of singular maps are equal to homotopy groups of spectra. Moreover, the rank, the torsion part for sufficiently high primes (and estimations for arbitrary primes) of the homotopy groups of these spectra can be computed in special and important cases in positive codimension [19] and very probably in negative codimension [8,13].…”

Section: Introductionmentioning

confidence: 92%

Fold cobordisms and stable homotopy groups

Kalmar

2009

Studia Scientiarum Mathematicarum Hungarica

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Abstract. We prove that for n ≥ 1 and q > 0 the (oriented) cobordism group of fold maps of (oriented) (n + q) -dimensional manifolds into R n contains the direct sum of ⌊(q + 1)/2⌋ copies of the (n − 1) th stable homotopy group of spheres as a direct summand. We also prove that for k ≥ 1 and q = 2k − 1 the cobordism group of fold maps of unoriented (n + q) -dimensional manifolds into R n also contains the n th stable homotopy group of the space RP ∞ as a direct summand. We have the analogous results about bordism groups of fold maps as well.

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“…In the stated form Theorem 4.1 is proved by the author in [23]. A dual version (for bordisms of τ -maps) of Theorem 4.1 with an essentially different spectrum is established by Ando in [2]. Theorem 4.1 is true for maps of negative dimension as well.…”

Section: Properties Of the Spectrum σ τmentioning

confidence: 98%

Fold maps, framed immersions and smooth structures

Sadykov

2012

Trans. Amer. Math. Soc.

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Abstract. For each integer q ≥ 0, there is a cohomology theory A 1 such that the zero cohomology group A 0 1 (N ) of a manifold N of dimension n is a certain group of cobordism classes of proper fold maps of manifolds of dimension n + q into N . We prove a splitting theorem for the spectrum representing the cohomology theory of fold maps. For even q, the splitting theorem implies that the cobordism group of fold maps to a manifold N is a sum of q/2 cobordism groups of framed immersions to N and a group related to diffeomorphism groups of manifolds of dimension q + 1. Similarly, in the case of odd q, the cobordism group of fold maps splits off (q − 1)/2 cobordism groups of framed immersions.The proof of the splitting theorem gives a partial splitting of the homotopy cofiber sequence of Thom spectra in the Madsen-Weiss approach to diffeomorphism groups of manifolds.

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Cobordisms of maps with singularities of given class

Cited by 15 publications

References 63 publications

Bordism groups of solutions to differential relations

Bordism groups of solutions to differential relations

Fold cobordisms and stable homotopy groups

Fold maps, framed immersions and smooth structures

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