Spaces of maps between real algebraic varieties

Abstract

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