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

TL;DR

This paper investigates the homotopy types of spaces of algebraic maps from real projective spaces to complex projective spaces, improving approximation bounds and proving a special case of a conjecture relating these spaces to continuous maps.

Contribution

It refines bounds on the homotopy approximation of algebraic map spaces and proves a special case of a conjecture linking these spaces to continuous maps.

Findings

Improved bounds on homotopy approximation using new methods

Proved a special case of a conjecture relating algebraic and continuous maps

Enhanced understanding of the homotopy types of algebraic map spaces

Abstract

We study the homotopy types of certain spaces closely related to the spaces of algebraic (rational) maps from the $m$ dimensional real projective space into the $n$ dimensional complex projective space for $2\leq m\leq 2n$ (we conjecture this relation to be a homotopy equivalence). In an earlier article we proved 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 obtained bounds that describe the closeness of the approximation in terms of the degrees of the maps. Here we improve the estimates of the bounds by using new methods introduced in \cite{Mo3} and used in \cite{KY4}. In addition, in the the last section, we prove a special case ($m=1$) of the conjecture stated in \cite{AKY1} that our spaces are homotopy equivalent to the spaces of algebraic maps.