Extension problem for the fundamental ideal of a twisted Grothendieck–Witt group

Determine the extension class of the exact sequence relating the rank-zero twisted Grothendieck–Witt group of a smooth real curve to the corresponding twisted fundamental-ideal cohomology groups, thereby resolving the extension problem in general.

Background

For a connected smooth real curve X and a line bundle L, Proposition \ref{prop:GW_0_curve} yields an exact sequence whose rank-zero part has the form 0 → H^1(X(R), Z/2)/(Z/2·w_1(L)) → \widehat{I}(X,L) → H^0(X,I(L)) → 0. The paper identifies the two end terms but does not determine whether this short exact sequence splits or otherwise classify its extension class.

The unresolved issue concerns the structure of the rank-zero subgroup \widehat{I}(X,L) of the twisted Grothendieck–Witt group, and is distinct from the computation of its associated graded pieces or its constituent subgroups.