# Symplectic Structures with Non-Isomorphic Primitive Cohomology on open 4-Manifolds

## TL;DR

This paper investigates symplectic structures on open 4-manifolds of the form $X=S^1 imes M^3$, demonstrating how primitive cohomology distinguishes inequivalent symplectic forms through explicit computation and algorithms.

## Contribution

It introduces an explicit algorithm for computing monodromies of fibrations on open 3-manifolds and shows how primitive cohomology differentiates inequivalent symplectic structures.

## Key findings

- Primitive cohomology dimensions distinguish inequivalent symplectic structures.
- Algorithm for explicit monodromy computation on certain open 3-manifolds.
- Existence of pairs of symplectic forms with different primitive cohomology dimensions.

## Abstract

We analyze four-dimensional symplectic manifolds of type $X=S^1 \times M^3$ where $M^3$ is an open $3$-manifold admitting inequivalent fibrations leading to inequivalent symplectic structures on $X$. For the case where $M^3 \subset S^3$ is the complement of a $4$-component link constructed by McMullen-Taubes, we provide a general algorithm for computing the monodromy of the fibrations explicitly. We use this algorithm to show that certain inequivalent symplectic structures are distinguished by the dimensions of the primitive cohomologies of differential forms on $X$. We also calculate the primitive cohomologies on $X$ for a class of open $3$-manifolds that are complements of a family of fibered graph links in $S^3$. In this case, we show that there exist pairs of symplectic forms on $X$, arising from either equivalent or inequivalent pairs of fibrations on the link complement, that have different dimensions of the primitive cohomologies.