Real algebraic morphisms represent few homotopy classes 20 th September 2005

All real algebraic varieties that appear in the present paper are assumed to be affine (that is, isomorphic to algebraic subsets of R for some n). Morphisms between real algebraic varieties are called regular maps. For background material on real algebraic geometry the reader may consult [6]. Every real algebraic variety carries also the Euclidean topology, induced by the usual metric topology on R. Unless explicitly stated otherwise, all topological notions related to real algebraic varieties will refer to the Euclidean topology. In this paper we study the problem of representing homotopy classes of maps between real algebraic varieties by regular maps. Our results show scarcity of regular maps. Given a real algebraic variety Y , we define a numerical invariant β(Y ) to be the supremum of all nonnegative integers n with the following property: for every n-dimensional compact connected nonsingular real algebraic variety X, every continuous map from X into Y is homotopic to a regular map. If d is an integer satisfying 0 ≤ d ≤ β(Y ), then every continuous map from any d-dimensional compact connected nonsingular real algebraic variety into Y is homotopic to a regular map. Indeed, this follows from a simple fact: if A and B are real algebraic varieties and a continuous map f : A×B → Y is homotopic to a regular map, then the restriction f |A×{b} : A×{b} → Y has the same property for all b in B. It is proved below that β(Y ) ≤ dim Y , provided Y is compact, nonsingular, and dim Y ≥ 1. Of course, β(Y ) =