Galatius–Randal-Williams' conjecture on the class of the homotopy sphere Sigma_Q

Galatius and Randal-Williams conjectured that the homotopy sphere class $[\Sigma_Q]$ vanishes in $\operatorname{coker}(J)_{2n+1}$ for every odd $n\geq 3$. The general assertion was open in the earlier literature, with several congruence classes already known.

Known results

• The statement records vanishing for $n=3$, $n=7$, and $n\equiv 1\pmod 4$; the 2014 treatment isolates the unresolved case $n+1=4\ell$.

2019 stable-range proof claim

A 2019 preprint claims that, for every $n>31$, the map $\pi_{8n-1}\mathbb{S}\to\pi_{8n-1}\operatorname{MO}\langle 4n\rangle$ has kernel exactly $\operatorname{Im}(J)$, equivalently proving $[\Sigma_Q]=0$ in the corresponding $\operatorname{coker}(J)$. This is substantial progress, but the retrieved source provides no independent verification or complete treatment of all odd $n\geq 3$.

Current status (as of August 2026): Vanishing is known in the listed low-dimensional and congruence cases, and a 2019 preprint claims it for $n>31$, but the full conjecture remains unverified and open for the remaining odd $n\geq 3$.