Loops in the fundamental group of \(\mathrm{Symp}(\mathbf{C}P^2\sharp 5\overline{\mathbf{C}P^2},\omega)\) which are not represented by circle actions
Published in Canadian Journal of Mathematics. Journal Canadien de Mathématiques, 2023
Recommended citation: Anjos, S., Barata, M., Pinsonnault, M., Reis, A. Loops in the fundamental group of $\mathbf{Symp}(\overline{\mathbf{C}P^2\sharp 5\mathbf{C}P^2},\omega)$ which are not represented by circle actions. Canad. J. Math.75 (2023), no.4, 1226–1271. https://doi.org/10.4153/s0008414x22000323
We study generators of the fundamental group of the group of symplectomorphisms $\mathrm{Symp}(\mathbf{C}P^2\sharp 5\overline{\mathbf{C}P^2},\omega)$ for some particular symplectic forms. It was observed by J. Kȩdra that there are many symplectic 4-manifolds $(M,\omega)$, where $M$ is neither rational nor ruled, that admit no circle action and $\pi_1(\mathrm{Ham}(M,\omega))$ is nontrivial. On the other hand, it follows from previous results that the fundamental group of the group $\mathrm{Symp}(\mathbf{C}P^2\sharp k\overline{\mathbf{C}P^2},\omega)$, of symplectomorphisms that act trivially on homology, with $k\leq 4$, is generated by circle actions on the manifold. We show that, for some particular symplectic forms $\omega$, the set of all Hamiltonian circle actions generates a proper subgroup in $\pi_1(\mathrm{Symp}(\mathbf{C}P^2\sharp 5\overline{\mathbf{C}P^2},\omega))$. Our work depends on Delzant classification of toric symplectic manifolds, Karshon’s classification of Hamiltonian $S^1$-spaces and the computation of Seidel elements of some circle actions.