Tilman Bauer: Circle actions on exotic spheres

Abstract: Since the discovery of spheres with nonstandard differential structures by Milnor, the geometry of these exotic objects has been of great interest, in particular their symmetries. In the late 1960s, Brumfiel studied the question of which of the odd-dimensional exotic spheres admit free circle actions. Spheres that bound parallellizable manifolds (bp-spheres) always do by work of Hsiang, but only a finite number of non-bp examples could be constructed. Using homotopy detection results involving the homology theory of topological modular forms, I will construct infinite families of non-bp spheres that admit free actions by \(S^1\) and even by \(S^3=SU(2)\). No prior familiarity with exotic spheres or tmf will be assumed.

Preprint: arXiv:2603.23241