2023/06/26 by Pranav Chakravarthy, Chakravarthy, Pranav V., Martin Pinsonnault +1
Mathematics · #57S05 #57T20 #FOS: Mathematics #Geometric and Algebraic Topology #Geometry and complex manifolds #Homotopy and Cohomology in Algebraic Topology #Primary 53D35 #Secondary 57R17 #Symplectic Geometry (math.SG)
paper · pdf · doi:10.48550/arxiv.2306.15046
openalex publication_date 2023/06/26 · openalex created_date 2023/06/29 · openalex updated_date 2026/07/28
Let M=(M,ω) be either the product S2× S2 or the non-trivial S2 bundle over S2 endowed with any symplectic form ω. Suppose a finite cyclic group Zn is acting effectively on (M,ω) through Hamiltonian diffeomorphisms, that is, there is an injective homomorphism Zn\hookrightarrow Ham(M,ω). In this paper, we investigate the homotopy type of the group SympZn(M,ω) of equivariant symplectomorphisms. We prove that for some infinite families of Zn actions satisfying certain inequalities involving the order n and the symplectic cohomology class [ω], the actions extends to either one or two toric actions, and accordingly, that the centralizers are homotopically equivalent to either a finite dimensional Lie group, or to the homotopy pushout of two tori along a circle. Our results rely on J-holomorphic techniques, on Delzant's classification of toric actions, on Karshon's classification of Hamiltonian circle actions on 4-manifolds, and on the Chen-Wilczyński classification of smooth Zn-actions on Hirzebruch surfaces.