2023/09/08 by Boris Botvinnik, Botvinnik, Boris, J. D. Quigley +1
Mathematics · Physics and Astronomy · #55M35 #55P42 #57R60 #57S15 #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #Black Holes and Theoretical Physics #FOS: Mathematics #Geometric Topology (math.GT) #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2309.04275
openalex publication_date 2023/09/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 πks represented by a homotopy sphere Σk to an element of a higher stable stem πk+ℓs represented by another homotopy sphere Σk+ℓ equipped with a smooth U(1)- (respectively, Sp(1)-) action with fixed points the original homotopy sphere Σk⊂ Σk+ℓ.