Surjectivity of the zero-surgery homomorphism over arbitrary fields

Determine whether the zero-surgery homomorphism \(\mathcal{S}_F\colon\mathcal{C}\to\Omega_F(S^1\times S^2)\), from the smooth knot concordance group to the field-coefficient H-cobordism group, is surjective for every field \(F\).

Background

For every field $F$, the paper defines a homomorphism $\mathcal{S}_F$ by sending a knot concordance class $[K]$ to the distinguished zero-surgery manifold $[(S^3_0(K),\phi_K)]$. The paper also constructs a different zero-surgery map from knots in arbitrary integral homology 3-spheres and notes that this broader map is surjective.

It remains unresolved whether the map whose domain is the ordinary knot concordance group of knots in $S^3$ already surjects onto $\Omega_F(S^1\times S^2)$ for every field.