Cobordism groups of immersions of oriented manifolds

Szűcs, András

doi:10.1007/bf01874122

Cited by 9 publications

(4 citation statements)

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“…A similar result holds also for n < 3k modulo finite 3 and 2 primary groups, see [16]. (In the range n < 3k the cobordism class of M n cannot be of course any cobordism class, sinceP j (M ) = 0 if j > k/2.)…”

Section: Theorem 4 Let F : Mmentioning

confidence: 58%

On the multiple points of immersions in Euclidean spaces

Szűcs

1998

Proc. Amer. Math. Soc.

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Abstract. Given a self-transverse immersion of a closed, oriented manifold in a euclidean space and a natural number i we compute the oriented cobordism class of the manifold of i-tuple points. §0. IntroductionLet f : M n → R n+k be a generic smooth immersion of a closed smooth manifold M n in the euclidean space R n+k and let ∆ i (f ) denote the manifold of i-fold points in the image. The manifold ∆ i (f ) is immersed in R n+k by a (non-generic) immersion g i ; the image of the non-multiple points of g i is the set of those points in R n+k which have exactly i preimages.The aim of this paper is to investigate the topological characteristics of these manifolds ∆ i (f ). In particular we compute their signatures and more generally all their Pontrjagin numbers, show that in many cases they have even Euler characteristics, and investigate whether the cobordism class of the manifold M n determines their cobordism class or not. It turns out that when the multiplicity i is odd, then the oriented cobordism class [∆ i (f )] of ∆ i (f ) is determined by the oriented cobordism class of M n , while for i even the oriented cobordism class [∆ i (f )] is independent of the cobordism class of M n . §1. Formulation of the resultsThe prototype of this kind of result is that of Banchoff:

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Section: Theorem 4 Let F : Mmentioning

confidence: 58%

On the multiple points of immersions in Euclidean spaces

Szűcs

1998

Proc. Amer. Math. Soc.

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“…Proof of Lemma 4. In [14] we proved the following theorem (see Theorem 1(a) and Addendum (2)(a) in [14]).…”

Section: On the Number Of Cuspsmentioning

confidence: 89%

“…Next, we recall (part of) Theorem 1 from [14]. Here, ≈ C 2 means an isomorphism modulo the class of finite 2-primary groups.…”

Section: Theorem 5 ([14]mentioning

confidence: 99%

On the Singularities of Hyperplane Projections of Immersions

Szűcs¹

2000

Bulletin of the London Mathematical Society

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Given an oriented manifold and its immersion in a euclidean space, we compute the oriented cobordism class of the manifold of Σ 1r singular points of the projection of the immersion to a hyperplane. For immersions of non-oriented manifolds, we show that the cobordism class of the domain manifold determines those of all Σ 1r singularity manifolds of the hyperplane projection. Finally, we investigate the possible (algebraic) number of cusps (that is, Σ 1,1 singular points) of generic maps of oriented 4t-manifolds in R 6t−1 . The cobordism classes of the Σ 1 r singular pointsLet M n be a closed oriented smooth manifold, and let f : M n → R n+k+1 be its generic immersion in the euclidean space R n+k+1 . Let π : R n+k+1 → R n+k be the projection onto the hyperplane R n+k . Suppose that g = π • f is a generic map. Let Σ 1 r (g) be the set of Σ 1,...,1 points (where the number of digits 1 is r) of the map g (that is, the set of A r points). This is a closed oriented manifold of dimension n − r(k + 1). Our aim is to compute the Pontrjagin numbers and the Stiefel-Whitney numbers of Σ 1 r (g), and thus to determine its oriented cobordism class. As a corollary, we shall find the possible algebraic number of Σ 1,1 points of generic maps of oriented, closed 4t-dimensional manifolds in R 6t−1 . (In these dimensions, the set of Σ 1,1 points is an oriented 0-dimensional manifold.)Remark 1. In [15] we found the Pontrjagin numbers of the manifold formed by the i-tuple points of an immersion f, denoting it by ∆ i (f). (∆ i (f) is an immersed submanifold of the target euclidean space.) Although the present paper does not rely on [15], it turns out that the Pontrjagin numbers of Σ 1 r (g) are the same as those of∆ r+1 (f) (that is, the (r + 1)-fold covering of ∆ r+1 (f)).Lemma 1. Let A a and B b be closed oriented smooth manifolds such that b−a > 0, a is divisible by 4, A ⊂ B, and A represents a homology class dual to α ∈ H b−a (B; Z). Further, let ξ be a bundle over B such that its restriction to A is isomorphic to the normal bundle of A in B.

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“…In the papers [1,2], A. Szucs introduced the notion of a prem-mapping (an abbreviation of projected embedding, see Definition 1 below). This notion turned out to be very useful in the study of the cobordism group of embeddings of smooth manifolds in the case of nonstable codimension.…”

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confidence: 99%

Prem-mappings, triple self-intersection points of oriented surfaces, and the Rokhlin signature theorem

Akhmet'ev

1996

Math Notes

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ABSTRACT. We find a connection between the Rokhlin theorem on the signature of a four-dimensional manifold and the notion of a prem-mapping that arises from the theory of embeddings of smooth manifolds.In the papers [1, 2], A. Szucs introduced the notion of a prem-mapping (an abbreviation of projected embedding, see Definition 1 below). This notion turned out to be very useful in the study of the cobordism group of embeddings of smooth manifolds in the case of nonstable codimension. Here and below a surface is the image of an arbitrary mapping s t in general position. We do not assume that a surface is embedded or immersed unless this is stated explicitly. Then for a mapping f: M 2 ~ R a of an oriented surface M 2 into three-dimensional Euclidean space R 3 there is a simple necessary and sufficient condition for .f to be a prem-mapping, see Criterion 1 below. For triple points of the surface f(M 2), there exists an additional structure (A, B-classification, see Definition 2); the criterion is related to this structure.A mapping f: M 2 ---, R 3 is not necessarily a prem-mapping. For example, consider a two-sheeted covering over the Boy surface (see [3,4]); put the covering space in general position by a small C ~176 deformation. Then this mapping is not a prem-mapping, see Proposition 2 below.Theorem 3 is the central result of the paper. This theorem is similar to the weLl-known Banchoff theorem, see [5]. It describes the relation between A-and B-structures of triple points of an oriented surface. More precisely, the condition M ~-~ S 2 implies parity restrictions for the number of triple points of type B. This theorem is a direct consequence of the Rokhlin theorem on the signature of a fourdimensional manifold in the form given in the paper [6]; also see the book [7]. On the other hand, it can be regarded as an elementary illustration of the Rokhlin theorem.Propositions 4, 5, and 6 make Theorem 3 more precise. In Proposition 4 we prove that for the torus (and thus for each surface of genus g) the statement of Theorem 3 does not hold in the general case. Proposition 5 shows that there is no direct generalization of Theorem 3 that gives integer-valued estimates. Proposition 6 detects a relationship between triple points of different types on an embedded surface. Propositions 4 and 6 are direct reformulations of the results of the paper [4]. Now let us define prem-mappings in the dimensions that we are interested in.

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Cobordism groups of immersions of oriented manifolds

Cited by 9 publications

References 11 publications

On the multiple points of immersions in Euclidean spaces

On the multiple points of immersions in Euclidean spaces

On the Singularities of Hyperplane Projections of Immersions

Prem-mappings, triple self-intersection points of oriented surfaces, and the Rokhlin signature theorem

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