Distinguishing open symplectic mapping tori via their wrapped Fukaya categories

• Main Author: Kartal, Yusuf Bariș.
• Other Authors: Paul Seidel.
• Format: Others
• Language: English
• Published: Massachusetts Institute of Technology 2019
• Subjects: Mathematics.
• Online Access: https://hdl.handle.net/1721.1/122167

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2019 === Cataloged from PDF version of thesis. === Includes bibliographical references (pages 217-223). === The main goal of this thesis is to use homological methods as a step towards the classification of symplectic mapping tori. More precisely, we exploit the dynamics of wrapped Fukaya categories to distinguish an open version of symplectic mapping torus associated to a symplectomorphism from the mapping torus of the identity. As an application, we obtain pairs of diffeomorphic Weinstein domains with the same contact boundary and symplectic cohomology, but that are different as Liouville domains. This work consists of two parts: in the first part, we define an algebraic model for the wrapped Fukaya category of the open symplectic mapping tori. This construction produces a category, called the mapping torus category, for a given dg-category over C with an autoequivalence. We then use the continuous dynamics of deformations of these categories to distinguish them under certain hypotheses. More precisely, we construct families of bimodules- analogous to flow lines- and use their different periodicity. The construction of the flow uses the geometry of the Tate curve and formal models for the graph of multiplication on G[superscript an] [subscript m,C((q))]. The second part focuses on the comparison of mapping torus categories and the wrapped Fukaya categories of the open symplectic mapping tori. For this goal, we introduce the notion of "twisted tensor product" and prove a twisted Kunneth theorem for the open symplectic mapping tori by using a count of quilted strips. In this part, we also give a large class of Weinstein domains whose wrapped Fukaya category satisfies the conditions for the theorem on mapping torus categories to hold. === by Yusuf Bariș Kartal. === Ph. D. === Ph.D. Massachusetts Institute of Technology, Department of Mathematics