Spaces of algebraic maps from real projective spaces into complex projective spaces

TL;DR

This paper investigates the homotopy types of algebraic maps from real to complex projective spaces, showing how these approximate the space of continuous maps as the degree increases and computing homotopy groups in low dimensions.

Contribution

It establishes that the homotopy types of algebraic map spaces approximate continuous map spaces more closely with higher degrees and provides bounds and computations for homotopy groups.

Findings

Homotopy types of algebraic maps approximate continuous maps as degree increases.

Bounds on the approximation quality in terms of degree.

Homotopy groups computed in low dimensions.

Abstract

We study the homotopy types of spaces of algebraic (rational) maps from real projective spaces into complex projective spaces. In a previous paper we have shown that the inclusion of the first space into the second one is a homotopy equivalence. In this paper we prove that the homotopy types of the terms of the natural "degree" filtration approximate closer and closer the homotopy type of the space of continuous maps and obtain bounds that describe the closeness of the approximation in terms of the degree. Moreover, we compute the homotopy groups of the spaces in low dimensions.