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Multiplicity of Continuous Mappings of Domains

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Abstract

We prove that either the proper mapping of a domain of an n-dimensional manifold onto a domain of another n-dimensional manifold of degree k is an interior mapping or there exists a point in the image that has at least |k|+2 preimages. If the restriction of f to the interior of the domain is a zero-dimensional mapping, then, in the second case, the set of points of the image that have at least |k|+2 preimages contains a subset of total dimension n. In addition, we construct an example of a mapping of a two-dimensional domain that is homeomorphic at the boundary and zero-dimensional, has infinite multiplicity, and is such that its restriction to a sufficiently large part of the branch set is a homeomorphism.

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Authors and Affiliations

  1. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev, Ukraine

    Yu. B. Zelinskii

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  1. Yu. B. Zelinskii
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Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 4, pp. 554–558, April, 2005.

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Zelinskii, Y.B. Multiplicity of Continuous Mappings of Domains. Ukr Math J 57, 666–670 (2005). https://doi.org/10.1007/s11253-005-0217-4

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  • DOI: https://doi.org/10.1007/s11253-005-0217-4

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