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Distinguishing open symplectic mapping tori via their wrapped Fukaya categories

Yusuf Barış Kartal

Geometry & Topology 25 (2021) 1551–1630

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. The symplectic manifold Tϕ is another Weinstein domain and its contact boundary is independent of ϕ. We distinguish Tϕ 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. 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.

symplectic mapping torus, mapping torus category, wrapped Fukaya category, Fukaya categories of symplectic fibrations, twisted tensor products, twisted Künneth theorem, Floer homology on infinite-type Liouville domains
Mathematical Subject Classification 2010
Primary: 53D37
Secondary: 16E45, 18G99, 53D40
Received: 18 August 2019
Revised: 20 July 2020
Accepted: 22 August 2020
Published: 20 May 2021
Proposed: Leonid Polterovich
Seconded: Yasha Eliashberg, Mark Gross
Yusuf Barış Kartal
Department of Mathematics
Princeton University
Princeton, NJ
United States