Equivalence from isometric equivariant forms for closed immersed Z-surfaces

Establish whether closed genus g immersed Z-surfaces with the same numbers of positive and negative transverse double points and with exteriors whose equivariant intersection forms are isometric must be equivalent up to homeomorphism.

Background

For embedded Z-surfaces in closed simply-connected 4-manifolds, the equivariant intersection form of the exterior determines the surface up to equivalence. This paper proves analogous uniqueness for immersed Z-surfaces with a single double point and provides partial results in other settings.

However, when double points are present in the closed case, the authors do not have a general uniqueness theorem. This question asks if isometry of equivariant intersection forms suffices to imply equivalence for closed immersed Z-surfaces with fixed double point counts.

A positive resolution would extend the embedded uniqueness paradigm to the immersed case; a negative resolution would exhibit new phenomena where the equivariant intersection form does not fully determine the topological type of the immersed surface.