Distinguishing open symplectic mapping tori via their wrapped Fukaya categories

Kartal, Yusuf Barış

doi:10.2140/gt.2021.25.1551

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Distinguishing open symplectic mapping tori via their wrapped Fukaya categories

Yusuf Barış Kartal

Geometry & Topology 25 (2021) 1551–1630

DOI: 10.2140/gt.2021.25.1551

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

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Keywords

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

References

Publication

Received: 18 August 2019

Revised: 20 July 2020

Accepted: 22 August 2020

Published: 20 May 2021

Proposed: Leonid Polterovich

Seconded: Yasha Eliashberg, Mark Gross

Authors

Yusuf Barış Kartal
Department of Mathematics Princeton University Princeton, NJ United States

Volume 25, issue 3 (2021)