Spaces of maps between real algebraic varieties

Abstract Given two real algebraic varieties and , we denote by the set of all regular maps from to . The set is regarded as a topological subspace of the space of all continuous maps from to endowed with the compact‐open topology. We prove, in a much more general setting than previously considered, that each path component of contains at most one path component of , and for every positive integer the inclusion map induces an isomorphism between the th homotopy groups of the corresponding path components. We also identify several cases where this inclusion map is a weak homotopy equivalence.