We relate the group structure of van der Kallen on orbit sets of unimodular rows with values in a smooth algebra $A$ over a field $k$ with the motivic cohomotopy groups of the spectrum of $A$ with coefficients in $\mathbb{ A}^n\setminus 0$ in the sense of Asok and Fasel. In the last section, we compare the motivic cohomotopy theory studied in this paper and defined by $\mathbb{ A}^{ n+1}\setminus 0$ or, equivalently, by an $\mathbb{ A}^1$-weakly equivalent quadric $Q_{ 2n+1}$ to that considered by Asok and Fasel, defined by a quadric $Q_{ 2n}$, by means of explicit morphisms $Q_{ 2n+1}\rightarrow Q_{ 2n}$, $Q_{ 2n}\times\mathbb{ G}_m\rightarrow Q_{ 2n+1}$ of quadrics.
Categories: math.KT math.AG
