Map double exact squares to motivic π_{1,0}(1) and analyze Milnor K-theory interaction

Establish a natural mapping from the generators of K1(V_k), namely double exact squares, to the motivic homotopy group π_{1,0}(1), and characterize how these generalised automorphisms interact with the Milnor K-theory component in the exact sequence 0 → K^M_2(k)/24 → π_{1,0}(1) → k^×/2 ⊕ Z/2 → 0, including a geometric interpretation of this interaction.

Background

The paper discusses lifting the motivic Euler characteristic to spectra and points to π_{1,0}(1) as a target for understanding higher structures beyond K0.

They note Morel’s identification of π{1,0}(1) and the role of Milnor K-theory but state that connecting the K1(V_k) generators (double exact squares) to π{1,0}(1) and interpreting the Milnor component geometrically remains unresolved.

References

It remains to map the double exact squares in $V_k$ to $\pi_{1,0}(1)$ in a natural way, but it is presently unclear, e.g. how these generalised automorphisms interact with the Milnor $K$-theory term, and what this means geometrically.

— $K_1(Var)$ is presented by stratified birational equivalences  (2510.20433 - Ng, 23 Oct 2025) in The Motivic Euler Characteristic (Discussion after Problem K1)

Identifying the kernel of the surjection $\pi_{1-(\star)}1_Z\to \pi_{1-(\star)}kq_Z$ with $K_{2+\star}(Z)/24$, just as in , but without inverting any prime number, requires more work. If the slices of $1_Z$ have the form conjectured by Voevodsky in (which is the case for the first three nontrivial slicesLemma 4.5, Lemma 4.7), a suitable modification of the slice convergence theorem Theorem 3.50 using local cellularity would enable one to lift the computation.

— The zeroth stable homotopy groups of motivic spheres over the integers  (2610.03069 - Röndigs, 2 Oct 2026) in Remark \ref{rem:pi1-sphere-z}, Section \ref{sec:sphere-spectrum}