Abstract analogues of flux as symplectic invariants (Q2925387)

One of the well-known and interesting invariants in symplectic geometry is the flux group. It is a subgroup of \(H^1(M; \mathbb{R})\) of a closed symplectic manifold \(M\) which, roughly speaking, can be obtained from the loops of symplectic automorphisms. As a relatively recent achievement, \textit{K. Ono} proved the so-called flux conjecture saying that the flux group is always discrete by using the Floer-Novikov cohomology, see [Geom. Funct. Anal. 16, No. 5, 981--1020 (2006; Zbl 1113.53055)].NEWLINENEWLINENEWLINEAccording to the author, one of the primary motivations of the book under review is to find certain non-trivial higher-dimensional analogues of the flux group in \(H^{2k-1}(M; \mathbb{R})\) with \(k>1\) for a symplectic manifold \(M\) such that \(H^1(M; \mathbb{R})=0\). However, it seems that the author does not succeed in finding them in a classical sense (or does not deal with the problem directly). Rather, the author attempts to find more abstract (so, less clear-cut) versions of the flux group using the Fukaya category obtained from the families of Lagrangian submanifolds. One outcome is an invariant of a symplectic manifold which depends on some auxiliary data such as \(\lambda\) in the Novikov ring and an idempotent \(z\) in the quantum cohomology \(QH^0(M)\). As a consequence, the author applied his ideas to symplectic mapping tori and recovered some of the results previously proved using the flux group. NEWLINENEWLINENEWLINEThis book consists of fives chapters. Chapter 1 is devoted to explaining some families of objects together with their existence and uniqueness. It also contains basic material such as \(A_\infty\)-categories, idempotent splittings, Hochschild cohomology, and so on. Chapter 2 deals with some specific finite-dimensional algebras and \(A_\infty\)-deformations which will describe the Fukaya categories of the two-torus. In Chapter 3, the author considers the automorphism group of a symplectic manifold of dimension \(\geq 4\). To do so, he discusses the fixed point Floer cohomology and the Lagrangian Floer cohomology of graphs. As an interesting example, in Chapter 4 he also considers the symplectic mapping torus and carries out some computations for a certain Floer cohomology. Finally, Chapter 5 explains the behavior of Fukaya categories under blowups.