On a mapping of a projective space into a sphere

Abstract

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.

On Mappings of Constant Multiplicity

“…Does there exist a mapping of the n-dimensional real projective space in an n-dimensional sphere such that every point of the image has not more than two preimage points for n ≥ 4? It is known that for n = 2 and 3 such a mappings exists [8].…”