Multiplicity of Continuous Mappings of Domains

Zelinskii, Yu. B.

doi:10.1007/s11253-005-0217-4

Cited by 3 publications

(2 citation statements)

References 1 publication

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The following theorem concerning estimation of the set of multiple points was received in [5,6]. Open questions.…”

Section: Remark 5 If There Is No Restriction On the Mapping Of The Cmentioning

confidence: 99%

On Mappings of Constant Multiplicity

Zelinskii¹,

Dakhil²

2017

AAN

View full text Add to dashboard Cite

The main question considered in this paper concerns constructing the mapping of constant odd multiplicity in and onto an open ball of Euclidean space, provided that on the boundary of the ball the mapping is a homeomorphism. In [1], it is shown that restriction of a continuous mapping of a closed domain on its interior is proper iff the images of the boundary and the interior of the domain do not intersect.The aim of this paper is to give a partial answer to the following [2] question: Question. Does there exist for every proper mapping domains of n-dimensional manifolds) a proper mapping g homotopic to f , such that every point of the image g(D) has no more than |degf a| + 2 preimage points (degf is the degree [1] of the mapping f )?We study a possibility of constructing a mapping of the constant odd multiplicity on an open ball of Euclidean space R n , provided that on the boundary of the ball this mapping is a homeomorphism. Next we define the functiondetermined on the open interval −1 < x < 1. Values of the function s 2r+1 at the points −1 and 1 are defined by the equalities s 2r+1 (−1) = −1, s 2r+1 (1) = 1. As a result we obtained a continuous function on the closed interval B 1 = [−1, 1] (the one-dimensional ball), which is a one-to-one mapping on the boundary and such that each point of ball interior has exactly the odd number of the preimages k = 2r + 1 ≥ 3.To proceed further we consider n-dimensional ball B n as the suspension SB n−1 [4] over the (n − 1)-dimensional ball B n−1 (for instance, two-dimensional ball B 2 = SB 1 = {(x, y)||x| + |y| ≤ 1} is the

show abstract

“…The following theorem concerning estimation of the set of multiple points was received in [5,6]. Open questions.…”

Section: Remark 5 If There Is No Restriction On the Mapping Of The Cmentioning

confidence: 99%

On Mappings of Constant Multiplicity

Zelinskii¹,

Dakhil²

2017

AAN

View full text Add to dashboard Cite

show abstract

“…In addition, we assume that this mapping realizes this minimum. In [1,2], one-sided estimates were obtained, namely, the least possible value of this minimum was established.…”

mentioning

confidence: 99%

On a mapping of a projective space into a sphere

Zelinskii

2010

Ukr Math J

Self Cite

View full text Add to dashboard Cite

We obtain an exact estimate for the minimum multiplicity of a continuous finite-to-one mapping of a projective space into a sphere for all dimensions. For finite-to-one mappings of a projective space into a Euclidean space, we obtain an exact estimate for this multiplicity for n = 2, 3. For n ≥ 4, we prove that this estimate does not exceed 4. Several open questions are formulated.The main question studied in the present paper is related to the problem of finding the minimum number that bounds the number of preimages of an arbitrary point of an image, provided that the global degree of a given mapping of two domains is a priori known. In addition, we assume that this mapping realizes this minimum. In [1, 2], one-sided estimates were obtained, namely, the least possible value of this minimum was established.We say that a mapping f : X → Y of topological spaces is finite-to-one if the preimage f y −1 of an arbitrary point y Y ∈ contains a finite or an empty set of points. In what follows, we assume that, on the considered topological spaces, the structure of a manifold is introduced and there are continuous mappings of these manifolds or their subdomains. We also assume that the topological degree of a mapping deg f [3] is defined.

show abstract

On the multiplicity of multivalued mappings of domains on manifolds

Zelinskii

Safonova

2016

J Math Sci

View full text Add to dashboard Cite

Multiplicity of Continuous Mappings of Domains

Cited by 3 publications

References 1 publication

On Mappings of Constant Multiplicity

On Mappings of Constant Multiplicity

On a mapping of a projective space into a sphere

On the multiplicity of multivalued mappings of domains on manifolds

Contact Info

Product

Resources

About