# Distinguishing open symplectic mapping tori via their wrapped Fukaya categories

## TL;DR

This paper advances the classification of symplectic mapping tori by analyzing their wrapped Fukaya categories, distinguishing non-trivial cases from trivial ones, and providing new examples of Weinstein domains with identical invariants but different structures.

## Contribution

It constructs a symplectic manifold associated to a Weinstein domain and a symplectomorphism, and proves the wrapped Fukaya category of this manifold is derived equivalent to a categorical model, enabling distinction of certain symplectic structures.

## Key findings

- Distinguished symplectic mapping tori using wrapped Fukaya categories.
- Proved derived equivalence between the wrapped Fukaya category and a categorical model.
- Provided examples of Weinstein domains with identical invariants but different Liouville structures.

## Abstract

In this paper, we present partial results towards a classification of symplectic mapping tori using dynamical properties of wrapped Fukaya categories. More precisely, we construct a symplectic manifold $T_\phi$ associated to a Weinstein domain $M$, and an exact, compactly supported symplectomorphism $\phi$. $T_\phi$ is another Weinstein domain and its contact boundary is independent of $\phi$. In this paper, we distinguish $\phi$ from $T_{1_M}$, under certain assumptions (Theorem 1.1). As an application, we obtain pairs of diffeomorphic Weinstein domains with the same contact boundary and whose symplectic cohomology groups are the same, as vector spaces, but that are different as Liouville domains. To our knowledge, this is the first example of such pairs that can be distinguished by their wrapped Fukaya category. Previously, we have suggested a categorical model $M_\phi$ for the wrapped Fukaya category $\mathcal{W}(T_\phi)$, and we have distinguished $M_\phi$ from the mapping torus category of the identity. In this paper, we prove $\mathcal{W}(T_\phi)$ and $M_\phi$ are derived equivalent (Theorem 1.9); hence, deducing the promised Theorem 1.1. Theorem 1.9 is of independent interest as it preludes an algebraic description of wrapped Fukaya categories of locally trivial symplectic fibrations as twisted tensor products.