Distinguishing open symplectic mapping tori via their wrapped Fukaya categories

Abstract

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 $T_{\varphi }$ associated to a Weinstein domain $M$, and an exact, compactly supported symplectomorphism $\varphi$. The symplectic manifold $T_{\varphi }$ is another Weinstein domain and its contact boundary is independent of $\varphi$. We distinguish $T_{\varphi }$ 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 $M_{\varphi }$ for the wrapped Fukaya category $\mathcal{W}\left(T_{\varphi }\right)$, and we have distinguished $M_{\varphi }$ from the mapping torus category of the identity. We prove $\mathcal{W}\left(T_{\varphi }\right)$ and $M_{\varphi }$ 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.