Torsion in kernels over coefficient fields

Determine whether the kernel of \(\iota_F\colon\Omega_F(S^1\times S^2)\to\Omega_F^{\mathrm{top}}(S^1\times S^2)\) contains nonzero torsion elements for some coefficient field \(F\), and determine whether this occurs for every coefficient field \(F\).

Background

The paper generalizes Kawauchi H-cobordism from rational coefficients to arbitrary fields, producing smooth and topological groups $\Omega_F(S^1\times S^2)$ and $\Omega_F^{\mathrm{top}}(S^1\times S^2)$. It proves that the smooth-to-topological kernel contains an infinite cyclic subgroup for every field.

The unresolved issue is whether these kernels can also contain nonzero torsion, either for at least one field or uniformly across all fields.