Multiplicity of Continuous Mappings of Domains
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.
English version (Springer): Ukrainian Mathematical Journal 57 (2005), no. 4, pp 666-670.
Citation Example: Zelinskii Yu. B. Multiplicity of Continuous Mappings of Domains // Ukr. Mat. Zh. - 2005. - 57, № 4. - pp. 554–558.