Symmetries of exotic spheres via complex and quaternionic Mahowald invariants

TL;DR

This paper employs advanced homotopy-theoretic methods to demonstrate the existence of smooth circle and quaternionic actions on infinite families of exotic spheres, expanding understanding of their symmetries via complex and quaternionic Mahowald invariants.

Contribution

It introduces new homotopy-theoretic tools to establish smooth group actions on exotic spheres using complex and quaternionic Mahowald invariants, revealing novel symmetries.

Findings

Existence of smooth $U(1)$-actions on certain exotic spheres.

Existence of smooth $Sp(1)$-actions on certain exotic spheres.

Propagation of exotic spheres through Mahowald invariants.

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 $\pi_k^s$ represented by a homotopy sphere $\Sigma^k$ to an element of a higher stable stem $\pi_{k+\ell}^s$ represented by another homotopy sphere $\Sigma^{k+\ell}$ equipped with a smooth $U(1)$- (respectively, $Sp(1)$-) action with fixed points the original homotopy sphere $\Sigma^k\subset \Sigma^{k+\ell}$.