Generalization of the isotopy theorem to semifree actions with Z-set fixed points

Establish whether the isotopy theorem for based-free actions of compact groups on the Hilbert cube can be generalized to semifree actions whose fixed-point sets are homeomorphic Z-sets.

Background

Theorem \ref{TheoremOnIsotopies} proves that two based-free actions of a compact group on the Hilbert cube are isotopic through based-free actions, with quantitative control when the actions are sufficiently close to the trivial action. Immediately afterward, the paper asks whether an analogous result holds for semifree actions whose fixed-point sets are nontrivial Z-sets. No such generalization is established in the paper.

References

Can Theorem~\ref{TheoremOnIsotopies} be generalized to semifree actions of compact groups on $Q$ whose fixed point sets are homeomorphic Z-sets?

— 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 Question following the proof of Theorem \ref{TheoremOnIsotopies}, Section 3, subsection “Isotopies”