Symmetries of exotic spheres via complex and quaternionic Mahowald invariants

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