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unpublished results, which connect Fukaya categories to the theory of Lefschetz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. 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. Fukaya’s foundational contribution, of

course, was to introduce A1-structures into symplectic geometry. 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. Donaldson’s influence is equally pervasive. 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. For instance, parts of our presentation of Picard–Lefschetz

theory reflect Auroux’ point of view. A result of Smith, namely that the vanishing

cycles in a four-dimensional Lefschetz pencil necessarily fill out the fibre, was crucial

in suggesting that such cycles might “split-generate” the Fukaya category. 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. Finally,

Abouzaid’s suggestions greatly improved the discussion of symplectic embeddings.