Injectivity of the connected-sum homomorphism over \(\mathbb{F}_2\)

Determine whether the connected-sum homomorphism \(\Sigma_{\mathbb{F}_2}\colon\Theta^3_{\mathbb{Z}}\to\Omega_{\mathbb{F}_2}(S^1\times S^2)\), sending an integral homology 3-sphere to its connected sum with \(S^1\times S^2\), is injective.

Background

The paper defines, for every field $F$, a homomorphism $\Sigma_F$ from the smooth integral homology 3-sphere homology-cobordism group to the field-coefficient H-cobordism group. Its image lies in the kernel of the smooth-to-topological map.

The authors show that $\Sigma_F$ is not injective when $\operatorname{char}(F)\neq2$, using the Brieskorn sphere $\Sigma(2,3,7)$. They explicitly leave the characteristic-two case unresolved.