Motivic stable cohomotopy and unimodular rows

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 $Q_{2n+1}\to Q_{2n}$, $Q_{2n}\times {\mathbb{G}}_m\to Q_{2n+1}$ of quadrics.