The local Floer cohomology of indicator functions

TL;DR

This paper constructs a spectral sequence linking local Floer homology of Reeb orbits to relative symplectic cohomology for compact sets with contact boundary, with applications to SYZ fibrations and symplectic cluster manifolds.

Contribution

It introduces a functorial spectral sequence connecting local Floer homology to symplectic cohomology, applicable without the need for exact inclusions, and explores its implications in symplectic geometry.

Findings

Spectral sequence from local Floer homology to symplectic cohomology.

Functoriality with respect to non-exact inclusions.

Application to symplectic cluster manifolds in dimension 4.

Abstract

For a compact set $K$ with contact type boundary in a symplectic manifold $M$ we construct a spectral sequence from the local Floer homology of the Reeb orbits, as studied by \cite{Mclean2012}, to the relative symplectic cohomology of $K$ in $M$ over the Novikov ring. The spectral sequence is functorial with respect to inclusions which are not required to be exact. This functoriality is key to the closed string reconstruction problem near the singularity of an SYZ fibration. We illustrate this in the case of dimension $2n=4$ for symplectic cluster manifolds. In higher dimension, an additional ingredient, the locality spectral sequence, is required, and is the subject of a forthcoming work in progress.