Spaces of maps between real algebraic varieties

TL;DR

This paper investigates the topological relationship between regular and continuous maps between real algebraic varieties, showing under certain conditions that their homotopy groups are closely related or equivalent.

Contribution

It generalizes previous results by establishing conditions under which the space of regular maps shares homotopy properties with the space of continuous maps.

Findings

Each path component of C(X,Y) contains at most one of R(X,Y)

The inclusion induces isomorphisms on homotopy groups for all k

Several cases where the inclusion is a weak homotopy equivalence

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)-->C(X,Y) induces an isomorphism between the kth homotopy groups of the corresponding path components. We also identify several cases where this inclusion map is a weak homotopy equivalence.