Spaces of algebraic and continuous maps between real algebraic varieties

TL;DR

This paper investigates the relationship between algebraic and continuous maps between real algebraic varieties, establishing homotopy equivalences and approximation properties for spaces of maps, especially focusing on real projective spaces.

Contribution

It proves that the inclusion of algebraic maps into continuous maps is a homotopy equivalence for certain varieties and shows that algebraic map spaces approximate continuous map spaces via a degree filtration.

Findings

Inclusion of algebraic maps is a homotopy equivalence for certain varieties.

The algebraic map space filtration approximates the continuous map space as degree increases.

Lower bounds for approximation are established for even components of these spaces.

Abstract

We consider the inclusion of the space of algebraic (regular) maps between real algebraic varieties in the space of all continuous maps. For a certain class of real algebraic varieties, which include real projective spaces, it is well known that the space of real algebraic maps is a dense subset of the space of all continuous maps. Our first result shows that, for this class of varieties, the inclusion is also a homotopy equivalence. After proving this, we restrict the class of varieties to real projective spaces. In this case, the space of algebraic maps has a ` minimum degree\rq filtration by finite dimensional subspaces and it is natural to expect that the homotopy types of the terms of the filtration approximate closer and closer the homotopy type of the space of continuous mappings as the degree increases. We prove this and compute the lower bounds of this approximation for ` even\rq components of these spaces (more precisely, we prove a very similar and closely related result, and state this one as a conjecture).