A symplectic Gysin sequence

TL;DR

This paper develops a symplectic Floer homology analogue of the Gysin sequence for sphere-bundles using pseudo-holomorphic quilts, drawing parallels with Seiberg-Witten monopole Floer homology for 3-manifolds.

Contribution

It introduces a symplectic Gysin sequence in Floer homology, establishing a new link between symplectic topology and gauge theory via pseudo-holomorphic quilts.

Findings

Established a symplectic Gysin sequence using pseudo-holomorphic quilts.

Demonstrated the sequence's analogy to Seiberg-Witten monopole Floer homology.

Provided examples illustrating the sequence's applications.

Abstract

We use the theory of pseudo-holomorphic quilts to establish a counterpart, in symplectic Floer homology, to the Gysin sequence for the homology of a sphere-bundle. In a motivating class of examples, this "symplectic Gysin sequence" is precisely analogous to an exact sequence describing the behaviour of Seiberg-Witten monopole Floer homology for 3-manifolds under connected sum.