We study generators of the fundamental group of the group of symplectomorphisms $\mathrm{ Symp}({ \mathbb C\mathbb P}^2\#\,5\overline{ \mathbb C\mathbb P}\,\!^2, \omega)$ for some particular symplectic forms. It was observed by J. K\c{ e}dra 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 $\mathrm{ Symp}_h({ \mathbb C\mathbb P}^2\#\,k\,\overline{ \mathbb C\mathbb P}\,\!^2, \omega)$, 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 $\pi_1(\mathrm{ Symp}_h({ \mathbb C\mathbb P}^2\#\,5\overline{ \mathbb C\mathbb P}\,\!^2, \omega)).$ 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.
Categories: math.SG