Algebraic homotopy classes of rational functions

TL;DR

This paper computes naive pointed homotopy classes of endomorphisms of the projective line over a field, comparing them with motivic homotopy classes, revealing a monoid structure and its relation to group completion.

Contribution

It provides a detailed computation of naive homotopy classes of P^1 endomorphisms and relates them to motivic homotopy classes, highlighting algebraic structures involved.

Findings

Naive homotopy classes form a monoid.

Canonical map to motivic classes is a group completion.

Comparison aligns with Fabien Morel's motivic homotopy results.

Abstract

We compute the set of naive pointed homotopy classes of endomorphisms of the projective line P^1 over the spectrum of a field. Our computation compares well with Fabien Morel's one of the motivic pointed homotopy classes of endomorphisms of P^1: there is an a priori monoid structure on the set of naive homotopy classes such that canonical map from this monoid to group of motivic homotopy classes is a group completion.