Kuranishi Structures and Virtual Fundamental Chains

“Although the authors have dealt with this subject in some previous monographs and in numerous articles, it seemed necessary to have a book that could collect all the necessary and well-organized material. This is the reason why the two above-mentioned authors, together with Yong-Geun Oh and Hiroshi Ono, have decided to write this book that we review now. … this book (an authentic tour de force) it is an indispensable reference for every researcher in symplectic geometry and topology.” (Manuel de León, zbMATH 1482.53002, 2022)

• 1.Introduction.- 2.Notations and conventions.- 3.Kuranishi structure and good coordinate system.- 4.Fiber product of Kuranishi structures.- 5.Thickening of a Kuranishi structure.-  6.Multivalued perturbation.-  7.CF-perturbation and integration along the fiber (pushout).- 8.Stokes' formula.- 9.From good coordinate system to Kuranishi structure and back with CF-perturbations.- 10.Composition formula of smooth correspondences.- 11.Construction of good coordinate system.- 12.Construction of CF-perturbations.- 13.Construction of multivalued perturbations.- 14.Zero and one dimensional cases via multivalued perturbation.- 15.Introduction to Part 2.- 16.Linear K-system: Floer cohomology I: statement.- 17.Extension of Kuranishi structure and its perturbation from boundary to its neighborhood.- 18.Smoothing corners and composition of morphisms.- 19.Linear K-system: Floer cohomology II: proof.- 20.Linear K-system: Floer cohomology III: Morse case by multisection.- 21.Tree-like K-system: A1 structure I: statement.- 22.Tree-like K-system: A1 structure II: proof.- 23. Orbifold and orbibundle by local coordinate.- 24.Covering space of effective orbifold and K-space.-  25.Admissible Kuranishi structure.- 26.Stratified submersion to a manifold with corners.- 27.Local system and smooth correspondence in de Rham theory with twisted coefficients.- 28.Composition of KG and GG embeddings: Proof of Lemma 3.34.- 29.Global quotient and orbifold.