Symmetries of exotic spheres via complex and quaternionic Mahowald invariants
Boris Botvinnik, J. D. Quigley
TL;DR
The paper develops a quaternionic Mahowald invariant and places it alongside the real and complex versions to detect smooth actions on exotic spheres via equivariant bordism and the Pontryagin–Thom framework. It proves that complex and quaternionic Mahowald invariants send elements in the stable stems to higher stems represented by spheres endowed with $U(1)$- or $Sp(1)$-actions with fixed points the original sphere, and that iterating these invariants yields infinite families of such symmetries. By relating these invariants to equivariant bordism and extending Stolz's $C_2$-action results to higher groups, the work provides concrete action-detection results and computes low-dimensional invariants that imply explicit transformation-group structures on selected exotic spheres. The results illuminate the interaction between chromatic phenomena in stable homotopy theory and geometric/smooth symmetry questions for exotic manifolds, with potential implications for curvature and symmetry in differential geometry.
Abstract
We use new homotopy-theoretic tools to prove the existence of smooth $U(1)$- and $Sp(1)$-actions on infinite families of exotic spheres. Such families of spheres are propagated by the complex and quaternionic analogues of the Mahowald invariant (also known as the root invariant). In particular, we prove that the complex (respectively, quaternionic) Mahowald invariant takes an element of the $k$-th stable stem $π_k^s$ represented by a homotopy sphere $Σ^k$ to an element of a higher stable stem $π_{k+\ell}^s$ represented by another homotopy sphere $Σ^{k+\ell}$ equipped with a smooth $U(1)$- (respectively, $Sp(1)$-) action with fixed points the original homotopy sphere $Σ^k\subset Σ^{k+\ell}$.