Loops in the fundamental group of $\mathrm{Symp}(\mathbb{C}{\mathbb{P}}^2\#5\overline{\mathbb{CP}}{}^2)$ which are not represented by circle actions

Abstract

We study generators of the fundamental group of the group of symplectomorphisms $\mathrm{Symp}({\mathbb{CP}}^2\#5\overline{\mathbb{CP}}{}^2,\omega )$ 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 ${\mathrm{Symp}}_h({\mathbb{CP}}^2\#k\overline{\mathbb{CP}}{}^2,\omega )$, of symplectomorphisms that act trivially on homology, with $k\le 4$, 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 ${\pi }_1({\mathrm{Symp}}_h({\mathbb{CP}}^2\#5\overline{\mathbb{CP}}{}^2,\omega )).$ 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.

Removed section 4.5 as there was a serious issue in the proof of Proposition 4.18 in the previous version. Changed the statement of Theorem 1.4 accordingly, to a slightly weaker result. Comments welcome!