Loops in the fundamental group of \Symp ( mathbb C mathbb\n P² # ,5\ mathbb C mathbb P , !²) which are not represented by\n circle actions

2019/10/07 by Sı́lvia Anjos, Anjos, Sílvia, Miguel Barata +5
Mathematics · #53D35 (Primary) 57R17 #57S05 #57T20 (Secondary) #FOS: Mathematics #Geometric and Algebraic Topology #Geometry and complex manifolds #Homotopy and Cohomology in Algebraic Topology #Symplectic Geometry (math.SG)

paper · pdf · doi:10.48550/arxiv.1910.02796

openalex publication_date 2019/10/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study generators of the fundamental group of the group of\nsymplectomorphisms \Symp( mathbb C mathbb P² # ,5\ mathbb\nC mathbb P , !², \ω) for some particular symplectic forms. It was\nobserved by J. K cedra that there are many symplectic 4-manifolds (M,\n\ω), where M is neither rational nor ruled, that admit no circle action\nand \π₁ (\Ham (M,\ω)) is nontrivial. On the other hand, it\nfollows from previous results that the fundamental group of the group\n\Symp_h( mathbb C mathbb P² # ,k ,\ mathbb C mathbb\nP , !², \ω), of symplectomorphisms that act trivially on homology, with\nk \≤ 4, is generated by circle actions on the manifold. We show that, for\nsome particular symplectic forms \ω, the set of all Hamiltonian circle\nactions generates a proper subgroup in \π₁(\Symp_h( mathbb C mathbb\nP² # ,5\ mathbb C mathbb P , !², \ω)). Our work depends on\nDelzant classification of toric symplectic manifolds, Karshon's classification\nof Hamiltonian S¹-spaces and the computation of Seidel elements of some\ncircle actions.\n

Related

• Centralizers of Hamiltonian circle actions on rational ruled surfaces
• Periodic symplectic and Hamiltonian diffeomorphisms on irrational ruled surfaces
• Centralizers of Hamiltonian finite cyclic group actions on rational ruled surfaces
• The homotopy Lie algebra of symplectomorphism groups of 3-fold blow-ups of (S² × S², σ_std ⊕ σ_std)
• Triviality of symplectic SU(2)-actions on homology