Generalize composition-splitting to arbitrary piecewise automorphisms

Determine whether the composition of arbitrary pairs of piecewise automorphisms in the Assembler framework for varieties splits in K1 without an obstruction term by extending the explicit splitting argument shown for coproduct inclusions and automorphisms, and ascertain the conditions under which such splitting holds universally.

Background

The authors compare their pCGW-based K1 relations with Zakharevich’s Assembler presentation and give a specific example where the obstruction term vanishes, yielding splitting of composition in K1.

They note that while the example works, extending the argument to arbitrary pairs of piecewise automorphisms is not straightforward and remains to be clarified.