Infinite-rank kernel of the smooth-to-topological H-cobordism map

Determine whether the kernel of the natural homomorphism \(\iota\colon\Omega(S^1\times S^2)\to\Omega^{\mathrm{top}}(S^1\times S^2)\) contains a subgroup isomorphic to \(\mathbb{Z}^{\infty}\), and determine whether this kernel contains an element that cannot be represented in the form \([Y\#(S^1\times S^2)]\) for any integral homology 3-sphere \(Y\).

Background

The paper constructs an infinite cyclic subgroup in the kernel of the forgetful map from the smooth H-cobordism group of distinguished homology handles to its topological counterpart. The authors then ask whether the kernel is substantially larger, specifically whether it has infinite rank.

The second part asks whether every element detected in this kernel arises from the connected-sum homomorphism that sends an integral homology 3-sphere $Y$ to the distinguished homology handle $Y\#(S^1\times S^2)$.