Classification of singular fibres of stable maps of 4-manifolds into 3-manifolds and its applications

Yamamoto, Takahiro

doi:10.2969/jmsj/1156342035

Cited by 11 publications

(9 citation statements)

References 12 publications

(38 reference statements)

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“…The notion of singular fibers of C ∞ maps between manifolds without boundary was first introduced in [7], where classifications of singular fibers of stable maps M → N with (dim M, dim N ) = (2, 1), (3,2) and (4,3) were obtained. Later, singular fibers of stable maps of manifolds without boundary were studied in [7,8,11,12,16,17,18], especially in connection with cobordisms. The first author [7] established the theory of universal complex of singular fibers of C ∞ maps as an analogy of the Vassiliev complex for map germs [6,15].…”

Section: Introductionmentioning

confidence: 99%

Singular fibers of stable maps of 3–manifolds with boundary into surfaces and their applications

Saeki

Yamamoto

2016

Algebr. Geom. Topol.

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Section: Introductionmentioning

confidence: 99%

Singular fibers of stable maps of 3–manifolds with boundary into surfaces and their applications

Saeki

Yamamoto

2016

Algebr. Geom. Topol.

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“…Codimension 2 strata that cannot be divided into two types. [Yamamoto 2006b] shows that, in the case κ = 3, the …”

Section: Stable Maps Of 5-manifolds Into 4-manifoldsmentioning

confidence: 99%

“…The paper is based on the author's doctoral thesis [Yamamoto 2006b]. Omitted proofs may be found there or in [Saeki 2004].…”

Section: Introductionmentioning

confidence: 99%

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Singular fibers of two-colored maps and cobordism invariants

Yamamoto

2008

Pacific J. Math.

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“…We can refine this equivalence relation by considering the singular fibers (see, for example, [23,36,37,49]) of a fold map. In this way we can obtain the notion of simple fold cobordism of simple fold maps, i.e., let τ be the set of all the singular fibers which have at most one singular point in each of their connected components.…”

Section: Introductionmentioning

confidence: 99%

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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Classification of singular fibres of stable maps of 4-manifolds into 3-manifolds and its applications

Cited by 11 publications

References 12 publications

Singular fibers of stable maps of 3–manifolds with boundary into surfaces and their applications

Singular fibers of stable maps of 3–manifolds with boundary into surfaces and their applications

Singular fibers of two-colored maps and cobordism invariants

Cobordism invariants of fold maps

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