2019/07/31 by Yusuf Barış Kartal
Mathematics · #math.KT #math.SG #msc:16E45 #msc:18G99 #msc:53D37 #msc:53D40
paper · pdf · doi:10.2140/gt.2021.25.1551
published as Geom. Topol. 25 (2021) 1551-1630 · Published at Geometry & Topology. 66 pages, 24 figures
arxiv created 2021/07/10 · arxiv updated 2021/07/13
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 Tϕ associated to a Weinstein domain M, and an exact, compactly supported symplectomorphism ϕ. Tϕ is another Weinstein domain and its contact boundary is independent of ϕ. In this paper, we distinguish ϕ from T1M, 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 Mϕ for the wrapped Fukaya category W(Tϕ), and we have distinguished Mϕ from the mapping torus category of the identity. In this paper, we prove W(Tϕ) and Mϕ 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.