applications in Section \[subsec:unique\]. An abstract version of our invariant is introduced in Definition \[th:per\]. Section \[sec:elliptic\] discusses the simplest example of the two-torus $T$. Of course, its Fukaya category has already been studied exhaustively, starting with [@polishchuk-zaslow98]. Still, we devote some energy to it in order to prepare for the case of mapping tori, which is the topic of Section \[sec:mapping-tori\] (following some preliminaries on Floer cohomology and Fukaya categories, in Section \[sec:automorphisms\]). At first, it will seem that our computations lead us further from the intended goal, since we choose Lagrangian submanifolds whose Floer cohomology is largely independent of the choice of automorphism underlying the mapping torus. However, we eventually do manage to recover some information about the automorphism, by a double covering trick which appears in Section \[subsec:covering-trick\]. Finally, most of Section \[sec:blowup\] is general discussion of blowups. The detailed construction of the manifolds $B$ and $B^{\mathit{triv}}$ is carried out in Section \[subsec:final\].

[**Acknowledgments.**]{} Section \[sec:blowup\] of this paper relies crucially on results of [@abouzaid10; @abouzaid-fukaya-oh-ohta-ono11; @smith10]. I am particularly indebted to Mohammed Abouzaid and Ivan Smith for explaining their insights to me. Conversations with Denis Auroux, Ludmil Katzarkov, Davesh Maulik, Tim Perutz, and Nick Sheridan were 