Fukaya Categories and Picard-Lefschetz Theory. Paul Seidel. The central objects in the book are Lagrangian submanifolds and their invariants, such as Floer homology and its multiplicative structures, which together constitute the Fukaya category. The relevant aspects of pseudo-holomorphic curve theory are covered in some detail, and there is also a self-contained account of the necessary homological algebra.

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Preface The subject of Fukaya categories has a reputation for being hard to approach. This is due to the amount of background knowledge required taken from homological algebra, symplectic geometry, and geometric analysis , and equally to the rather complicated nature of the basic definitions.

The present book is intended as a resource for graduate students and researchers whowould like to learn about Fukaya categories, and possibly use them in their own work. I have tried to focus on a rather basic subset of topics, and to describe these as precisely as I could, filling in gaps found in some of the early references.

This makes for a rather austere style for that reason, a thorough study of this book should probably be complemented by reading some of the papers dealing with applications. A second aim was to give an account of some previously unpublished results, which connect Fukaya categories to the theory of Lefschetz fibrations. This becomes predominant in the last sections, where the text gradually turns into a research monograph.

I have borrowed liberally from the work of many people, first and foremost among them Fukaya, Kontsevich, and Donaldson. On the algebraic side, he pioneered the use of the A1-version of the Yoneda embedding, which we adopt systematically. Besides that, several geometric ideas, such as the role of Pin structures, and the construction of A1-homomorphisms in terms of parametrized moduli spaces, are taken from the work of Fukaya, Oh, Ohta and Ono.

Kontsevich introduced derived categories of A1-categories, and is responsible for much of their theory, in particular the intrinsic characterization of exact triangles. He also conjectured the relation between Dehn twist and twist functors, which is one of our main results.

Finally, in joint work with Barannikov, he suggested a construction of Fukaya categories for Lefschetz fibrations; we use a superficially different, but presumably equivalent, definition. Besides his groundbreaking work on Lefschetz pencils, he introduced matching cycles, and proposed them as the starting point for a combinatorial formula for Floer cohomology, which is indeed partly realized here.

Other mathematicians have also made important contributions. Besides that, work of Fukaya—Smith on cotangent bundles provided a good testing-ground for some of the more adventurous ideas about Lefschetz fibrations. Our approach to transversality issues is the result of several conversations with Lazzarini. Fukaya Categories and Picard-Lefschetz Theory.

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## Fukaya Categories and Picard–Lefschetz Theory

Libraries and resellers, please contact cust-serv ams. See our librarian page for additional eBook ordering options. The central objects in the book are Lagrangian submanifolds and their invariants, such as Floer homology and its multiplicative structures, which together constitute the Fukaya category. The relevant aspects of pseudo-holomorphic curve theory are covered in some detail, and there is also a self-contained account of the necessary homological algebra. Generally, the emphasis is on simplicity rather than generality. The last part discusses applications to Lefschetz fibrations and contains many previously unpublished results. The book will be of interest to graduate students and researchers in symplectic geometry and mirror symmetry.

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## Fukaya Categories and Picard-Lefschetz Theory

Preface The subject of Fukaya categories has a reputation for being hard to approach. This is due to the amount of background knowledge required taken from homological algebra, symplectic geometry, and geometric analysis , and equally to the rather complicated nature of the basic definitions. The present book is intended as a resource for graduate students and researchers whowould like to learn about Fukaya categories, and possibly use them in their own work. I have tried to focus on a rather basic subset of topics, and to describe these as precisely as I could, filling in gaps found in some of the early references. This makes for a rather austere style for that reason, a thorough study of this book should probably be complemented by reading some of the papers dealing with applications. A second aim was to give an account of some previously unpublished results, which connect Fukaya categories to the theory of Lefschetz fibrations.