On Mappings of Constant Multiplicity

Abstract

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