A co-publication of the AMS and International Press of Boston, Inc.
This is a two-volume series research monograph on the general Lagrangian Floer theory and on the accompanying homological algebra of filtered A A -algebras. This book provides the most important step towards a rigorous foundation of the Fukaya category in general context. In Volume I, general deformation theory of the Floer cohomology is developed in both algebraic and geometric contexts. An essentially self-contained homotopy theory of filtered A A? algebras and A A? bimodules and applications of their obstruction-deformation theory to the Lagrangian Floer theory are presented. Volume II contains detailed studies of two of the main points of the foundation of the theory: transversality and orientation. The study of transversality is based on the virtual fundamental chain techniques (the theory of Kuranishi structures and their multisections) and chain level intersection theories. A detailed analysis comparing the orientations of the moduli spaces and their fiber products is carried out. A self-contained account of the general theory of Kuranishi structures is also included in the appendix of this volume.
Kenji Fukaya, Kyoto University, Kyoto, Japan.
Yong-Geun Oh, University of Wisconsin, Madison, Madison, WI.
Hiroshi Ohta, Nagoya University, Nagoya, Japan.
Kaoru Ono, Hokkaido University, Sapporo, Japan.
