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

TL;DR

This paper investigates the fundamental group of symplectomorphism groups of certain blow-ups of complex projective planes, revealing that some loops are not generated by circle actions, thus uncovering new topological features of these groups.

Contribution

It demonstrates that, for specific symplectic forms on P^2#5P^2, the subgroup generated by Hamiltonian circle actions is proper within the fundamental group, showing the existence of non-circle-action loops.

Findings

Some loops in the symplectomorphism group are not generated by circle actions.

The subgroup generated by circle actions is proper in the fundamental group.

The results depend on classifications of toric and Hamiltonian S^1-spaces and Seidel element computations.

Abstract

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.