There is no Cazanave's Theorem for punctured affine space

TL;DR

This paper demonstrates that a certain algebraic structure result for the projective line does not extend to punctured affine spaces of dimension two or higher, challenging previous assumptions in algebraic topology.

Contribution

It proves that Cazanave's theorem for the projective line does not hold for punctured affine spaces of dimension two or greater.

Findings

Cazanave's theorem does not generalize to punctured affine spaces for n ≥ 2

The monoid structure on naive -homotopy classes fails in higher dimensions

Group completion properties differ significantly from the projective line case

Abstract

In his thesis, Cazanave proved that the set of naive $\mathbb{A}^1$-homotopy classes of endomorphisms of the projective line admits a monoid structure whose group completion is genuine $\mathbb{A}^1$-homotopy classes of endomorphisms of the projective line. In this very short note we show that such a statement is never true for punctured affine space $\mathbb{A}^n\setminus\{0\}$ for $n \ge 2$ .