Dynamical invariants of mapping torus categories

TL;DR

This paper develops homological algebra tools to construct and analyze the mapping torus category, aiding in the classification of symplectic mapping tori and distinguishing them from trivial cases.

Contribution

It introduces the mapping torus category $M_$ for auto-equivalences of dg categories, providing a framework to understand symplectic mapping tori.

Findings

Defined a family of bimodules on a deformation of $M_$

Characterized the bimodules uniquely

Distinguished $M_$ from the identity mapping torus

Abstract

This paper describes constructions in homological algebra that are part of a strategy whose goal is to understand and classify symplectic mapping tori. More precisely, given a dg category and an auto-equivalence, satisfying certain assumptions, we introduce a category $M_\phi$-called the mapping torus category that describes the wrapped Fukaya category of an open symplectic mapping torus. Then we define a family of bimodules on a natural deformation of $M_\phi$, uniquely characterize it and using this, we distinguish $M_\phi$ from the mapping torus category of the identity. The proof of the equivalence of $M_\phi$ with wrapped Fukaya category is proven in a different paper (arXiv:1907.01156).