Vector Fields and Other Vector Bundle Morphisms — A Singularity Approach

Koschorke, Ulrich

doi:10.1007/bfb0090440

Cited by 100 publications

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“…, where T (α) is the Thom space [9] and, since T (α) is (p − 1)-connected, we conclude that η n (X) Ω n (X × BO, φ p+q ) and then this normal bordism group does not depend on α p .…”

Section: Exact Sequences Of Bordism Groupsmentioning

confidence: 65%

“…We recall that the order of the elements of the image of γ M is a power of 2 [9,13]. Therefore, γ M ([M, h]) = 0 and h is homotopic to an immersion [10].…”

Section: Proofs Of Theorems a And Bmentioning

confidence: 99%

“…For both problems, we use a normal bordism approach [9], and give an answer in terms of the induced maps of Z 2 -homology groups.…”

Section: Introductionmentioning

confidence: 99%

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Remarks on Immersions in the Metastable Dimension Range

Biasi¹,

Libardi

2004

Proceedings of the Edinburgh Mathematical Society

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In this work we present a generalization of an exact sequence of normal bordism groups given in a paper by H. A. Salomonsen (Math. Scand. 32 (1973), 87-111). This is applied to prove that ifn < 2k, is a continuous map between two manifolds and g : M n → BO is the classifying map of the stable normal bundle of h such that (h, g) * :is an isomorphism for i < n − k and an epimorphism for i = n − k, then h bordant to an immersion implies that h is homotopic to an immersion. The second remark complements the result of C. Biasi, D. L. Gonçalves and A. K. M. Libardi (Topology Applic. 116 (2001), 293-303) and it concerns conditions for which there exist immersions in the metastable dimension range. Some applications and examples for the main results are also given.

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Section: Exact Sequences Of Bordism Groupsmentioning

confidence: 65%

“…We recall that the order of the elements of the image of γ M is a power of 2 [9,13]. Therefore, γ M ([M, h]) = 0 and h is homotopic to an immersion [10].…”

Section: Proofs Of Theorems a And Bmentioning

confidence: 99%

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Remarks on Immersions in the Metastable Dimension Range

Biasi¹,

Libardi

2004

Proceedings of the Edinburgh Mathematical Society

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“…Again in view of our dimension assumption we can desuspend to get the isomorphism which extends the embedding g : ∂B = C A ֒→ X = X × {0} (compare [Ko1], figure 3.8). As in [Ko2], pp.…”

Section: Secondary Nielsen Numbersmentioning

confidence: 99%

Coincidences and secondary Nielsen numbers

Koschorke

2015

J. Fixed Point Theory Appl.

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Abstract. Let f1, f2 : X m −→ Y n be maps between smooth connected manifolds of the indicated dimensions m and n. Can f1, f2 be deformed by homotopies until they are coincidence free (i.e. f1(x) = f2(x) for all x ∈ X )? The main tool for addressing such a problem is tradionally the (primary) Nielsen number N (f1, f2) . E.g. when m < 2n − 2 the question above has a positive answer precisely if N (f1, f2) = 0 . However, when m = 2n − 2 this can be dramatically wrong, e.g. in the fixed point case when m = n = 2 . Also, in a very specific setting the Kervaire invariant appears as a (full) additional obstruction.In this paper we start exploring a fairly general new approach. This leads to secondary Nielsen numbers SecN (f1, f2) which allow us to answer our question e.g. when m = 2n − 2, n = 2 is even and Y is simply connected.Mathematics Subject Classification (2010). Primary 54H25, 55M20; Secondary 55Q40.

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“…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.…”

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confidence: 97%

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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Vector Fields and Other Vector Bundle Morphisms — A Singularity Approach

Cited by 100 publications

References 0 publications

Remarks on Immersions in the Metastable Dimension Range

Remarks on Immersions in the Metastable Dimension Range

Coincidences and secondary Nielsen numbers

Cobordism invariants of fold maps

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