Floer-Novikov fundamental group and small flux symplectic isotopies

TL;DR

This paper explores the relationship between symplectic isotopies, flux, and Novikov homology, demonstrating that under certain conditions, the Novikov fundamental group is generated by Floer moduli spaces linked to closed orbits.

Contribution

It introduces a novel connection between Floer theory, flux, and the Novikov fundamental group, extending previous relations to the fundamental group context.

Findings

Novikov fundamental group is generated by Floer moduli spaces when flux is small.

Establishes a link between symplectic isotopies and Novikov homology.

Provides conditions under which Floer theory describes fundamental group dynamics.

Abstract

Floer theory relates the dynamics of Hamiltonian isotopies and the homology of the ambient manifold. It was extended to similarly relate the dynamics of symplectic isotopies and the Novikov homology associated to their flux. We discuss this picture regarding the fundamental group, and prove that when the flux is not too big, the associated Novikov fundamental group is generated by Floer moduli spaces associated to closed orbits of the symplectic isotopy.