Lie-group characterization of compact groups admitting based-free actions

Determine whether every compact group that acts based-freely on a Keller space must be a Lie group.

Background

The paper constructs based-free actions for all nontrivial compact Lie groups and proves that allowing semifree actions on compact Hilbert cube manifolds does not enlarge the class of groups under consideration. It then identifies an unresolved question posed by Antonyan: whether the existence of a based-free action of a compact group on a Keller space forces the acting group to be Lie. The paper does not resolve this characterization problem.

References

The question of Antonyan whether a compact group acting based-freely on a Keller space must be a Lie group has remained open for over a quarter century.

— Fixed-point sets of semifree group actions on infinite-dimensional compact convex sets and their manifolds  (2610.09561 - Mill et al., 7 Oct 2026) in Section 3, subsection “Existence of Based-Free Actions”