Note on coisotropic Floer homology and leafwise fixed points

TL;DR

This paper develops a Floer homology framework for coisotropic submanifolds, linking leafwise fixed points to Floer homology, and applies it to Boothby-Wang fibrations to demonstrate the existence of translated points.

Contribution

It introduces a new Floer homology construction for coisotropic submanifolds and extends it to Boothby-Wang fibrations, providing lower bounds on leafwise fixed points and translated points.

Findings

Lower bounds on leafwise fixed points derived from Floer homology

Construction of local Floer homology for arbitrary coisotropic submanifolds

Existence of translated points in Boothby-Wang fibrations

Abstract

For an adiscal or monotone regular coisotropic submanifold $N$ of a symplectic manifold I define its Floer homology to be the Floer homology of a certain Lagrangian embedding of $N$. Given a Hamiltonian isotopy $\phi=(\phi^t)$ and a suitable almost complex structure, the corresponding Floer chain complex is generated by the $(N,\phi)$-contractible leafwise fixed points. I also outline the construction of a local Floer homology for an arbitrary closed coisotropic submanifold. Results by Floer and Albers about Lagrangian Floer homology imply lower bounds on the number of leafwise fixed points. This reproduces earlier results of mine. The first construction also gives rise to a Floer homology for a Boothby-Wang fibration, by applying it to the circle bundle inside the associated complex line bundle. This can be used to show that translated points exist.