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

Abstract

We study the 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. The monotone case was studied by P. Seidel and J. Evans: they proved that this group is homotopy equivalent to the group of orientation-preserving diffeomorphisms of $S^2$ preserving 5 points. Recently J. Li, T. J. Li and W. Wu completely determined the Torelli symplectic mapping class group as well as the rank of the fundamental group of the group \mathrm{Symp}_{h}(\mathbb C\mathbb P^2\#\,5\overline{\mathbb C\mathbb P}\,\!^2) of symplectomorphisms that act trivially on homology, for any given symplectic form. In general, we expect the generators of the fundamental group to be given by circle actions on the manifold. However, we show that in some particular cases, there are loops in the fundamental group which cannot be realized by circle actions. This is a new phenomenon since in \pi_1 (\mathrm{Symp}_h(\mathbb C\mathbb P^2\#\,k\overline{\mathbb C\mathbb P}\,\!^2)), with $k \leq 4$, all the generators can be represented by circle actions on the manifold. Our work depends on Delzant classification of toric symplectic manifolds, Karshon's classification of Hamiltonian circle actions and the computation of Seidel elements of some circle actions.Comment: 27 figure