Characterization of universal target homeomorphisms

Determine whether every orientation-preserving homeomorphism Φ of the sphere that preserves admissibility under postcomposition with every admissible orientation-preserving branched cover of degree at least two must be quasisymmetric.

Background

Theorem 7.5 proves that postcomposition by an orientation-preserving quasisymmetric homeomorphism preserves admissibility for the branched covers considered. The question asks whether quasisymmetry is also necessary when a target homeomorphism is required to preserve admissibility universally for every admissible branched cover of degree at least two.

The paper observes that degree-one covers cannot detect this property, since they are admissible after every target homeomorphism. It also gives examples of non-quasisymmetric homeomorphisms that preserve admissibility for particular covers, so the unresolved issue concerns universal preservation across all admissible branched covers.

References

Fix $n\ge3$. If an orientation-preserving homeomorphism $\Phi\colon\to$ makes $\Phi\circ f$ admissible for every admissible orientation-preserving branched cover $f\colon\to$ of degree at least two, must $\Phi$ be quasisymmetric?

— Pullback metrics and homological obstructions to BLD remetrization of branched covers  (2610.01087 - Zhong, 1 Oct 2026) in Question 9.5, Section 9.3, “Universal preservation under target changes”