There is no Cazanave's Theorem for punctured affine space

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 $n\ge 2$ .

3 pages, comments welcome!