Simplicial resolutions and spaces of algebraic maps between real projective spaces

TL;DR

This paper demonstrates that the homology of a space of algebraic maps from real projective spaces matches the homology of the space of all continuous maps up to a certain dimension, improving previous results.

Contribution

It establishes a homology equivalence between algebraic and continuous mapping spaces for real projective spaces, extending prior work with new techniques.

Findings

Homology of algebraic map space matches continuous map space up to a specific dimension.

Improves the main result of previous research by extending the dimension range.

Uses simplicial resolutions to analyze the topology of algebraic maps.

Abstract

We show that the space $\tilde{A}_{d}(m,n)$ consisting of all real projective classes of $(n+1)$-tuples of real coefficients homogeneous polynomials of degree $d$ in $(m+1)$ variables, without common real roots except zero, has the same homology as the space $ \Map(\RP^m,\Bbb \RP^n)$ of continuous maps from the $m$-dimensional real projective space $\RP^m$ into the $n$ real dimensional projective space $\RP^n$ up to dimension %in dimensions smaller than $(n-m)(d+1)-1$. This considerably improves the main result of \cite{AKY1}.