Classification of 2-knot groups with non-trivial normal bundles

Determine which groups can arise as fundamental groups of complements of smoothly embedded 2-spheres (2-knots) with non-trivial normal bundles in closed 4-manifolds, without imposing exoticness, thereby classifying 2-knot groups in the non-trivial normal bundle case.

Background

The authors’ construction yields 2-links with trivial normal bundles and is carried out in ambient manifolds of the form Z # (S^2 × S^2), which have trivial Seiberg–Witten invariants. They remark that when the normal bundle of a 2-knot is non-trivial, the landscape of possible fundamental groups of complements (2-knot groups) is not fully understood.

This observation points to an incomplete classification problem, with constraints and partial results informed by gauge theory (e.g., Kronheimer–Mrowka) and recent developments (e.g., Hughes–Ruberman). The remark situates this gap explicitly and motivates further work to characterize all realizable 2-knot groups in the non-trivial normal bundle setting.