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.