Paper detail

Chambers in the symplectic cone and stability of symplectomorphism group for ruled surface

We continue our previous work to prove that for any non-minimal ruled surface $(M,ω)$, the stability under symplectic deformations of $π_0, π_1$ of $Symp(M,ω)$ is guided by embedded $J$-holomorphic curves. Further, we prove that for any fixed sizes blowups, when the area ratio $μ$ between the section and fiber goes to infinity, there is a topological colimit of $Symp(M,ω_μ).$ Moreover, when the blowup sizes are all equal to half the area of the fiber class, we give a topological model of the colimit which induces non-trivial symplectic mapping classes in $Symp(M,ω) \cap \rm Diff_0(M),$ where $\rm Diff_0(M)$ is the identity component of the diffeomorphism group. These mapping classes are not Dehn twists along Lagrangian spheres.