Topological classification of fixed‑point‑free involutions of spheres

Ascertain whether every fixed‑point‑free involution of the n‑sphere S^n is topologically standard, namely, conjugate by a homeomorphism of S^n to the antipodal map, and determine in which dimensions this holds.

Background

For S2 and S3, Brouwer–Kerekjártó and Livesay proved that all fixed‑point‑free involutions are topologically conjugate to the antipodal map. In higher dimensions, Hirsch–Milnor produced involutions that are not smoothly or PL standard (and related work shows nonstandard smooth involutions on S4), but the purely topological classification remains unclear.

Clarifying this classification impacts applications of the Borsuk–Ulam theorem and geometric arguments involving antipodal‑type symmetries, as used in this paper’s chordal Gauss map framework.

References

If the involution is topologically standard, i.e., conjugate to the antipodal map in the homeomorphism group of the sphere, then the Borsuk--Ulam theorem gives the conclusion directly, but, to the best of the authors' knowledge, it is not known whether all fixed-point-free involutions of spheres are of this type.

— Area and antipodal distance in convex hypersurfaces  (2604.02667 - Dibble et al., 3 Apr 2026) in Remark ‘Yang discussion’ (Section 3)