Overview
- Is the first book to provide a foundation of techniques of virtual fundamental chain by the originators of the theory
- Makes a timely appearance with details of the theory much needed for further development of the field
- Contains detailed explanations on orbifolds, collected in a single volume
Part of the book series: Springer Monographs in Mathematics (SMM)
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Table of contents (29 chapters)
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Abstract Theory of Kuranishi Structures, Fiber Products and Perturbations
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System of K-Spaces and Smooth Correspondences
Keywords
About this book
Part I discusses the theory on a single space equipped with Kuranishi structure, called a K-space, and its relevant basic package. First, the definition of a K-space and maps to the standard manifold are provided. Definitions are given for fiber products, differentialforms, partitions of unity, and the notion of CF-perturbations on the K-space. Then, using CF-perturbations, the authors define the integration on K-space and the push-forward of differential forms, and generalize Stokes' formula and Fubini's theorem in this framework. Also, “virtual fundamental class” is defined, and its cobordism invariance is proved.
Part II discusses the (compatible) system of K-spaces and the process of going from “geometry” to “homological algebra”. Thorough explanations of the extension of given perturbations on the boundary to the interior are presented. Also explained is the process of taking the “homotopy limit” needed to handle a system of infinitely many moduli spaces. Having in mind the future application of these chain level constructions beyond those already known, an axiomatic approach is taken by listing the properties of the system of the relevant moduli spaces and then a self-contained account of the construction of the associated algebraic structures is given. This axiomatic approach makes the exposition contained here independent of previously published construction of relevant structures.
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Authors and Affiliations
Bibliographic Information
Book Title: Kuranishi Structures and Virtual Fundamental Chains
Authors: Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, Kaoru Ono
Series Title: Springer Monographs in Mathematics
DOI: https://doi.org/10.1007/978-981-15-5562-6
Publisher: Springer Singapore
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Springer Nature Singapore Pte Ltd. 2020
Hardcover ISBN: 978-981-15-5561-9Published: 17 October 2020
Softcover ISBN: 978-981-15-5564-0Published: 17 October 2021
eBook ISBN: 978-981-15-5562-6Published: 16 October 2020
Series ISSN: 1439-7382
Series E-ISSN: 2196-9922
Edition Number: 1
Number of Pages: XV, 638
Number of Illustrations: 115 b/w illustrations, 34 illustrations in colour
Topics: Differential Geometry, Hyperbolic Geometry, Polytopes