Motivic stable cohomotopy and unimodular rows

TL;DR

This paper explores the connection between the algebraic structure of unimodular rows over smooth algebras and the motivic cohomotopy groups, providing new insights into their relationship and comparing different motivic cohomotopy theories.

Contribution

It establishes a link between the group structure of unimodular rows and motivic cohomotopy groups, and compares two approaches to motivic cohomotopy theory via explicit morphisms between quadrics.

Findings

Relates van der Kallen's group structure to motivic cohomotopy groups.

Provides explicit morphisms between quadrics $Q_{2n+1}$ and $Q_{2n}$.

Shows equivalence of different motivic cohomotopy theories.

Abstract

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.