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Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms Selected Title s i n This Serie s 199 Shigeyuk i Morita , Geometr y of characteristic classes, 2001 198 V . A. Smirnov, Simplicia l and operad methods in algebraic topology, 2001 197 Kenj i Ueno, Algebrai c geometry 2: Sheaves and cohomology, 2001 196 Yu . N. Lin'kov, Asymptoti c statistical methods for stochastic processes, 2001 195 Minor u Wakimoto, Infinite-dimensiona l Lie algebras, 2001 194 Valer y B. Nevzorov, Records : Mathematical theory, 2001 193 Toshi o Nishino, Functio n theory in several complex variables, 2001 192 Yu . P. Solovyov and E. V. Troitsky, C*-algebra s and elliptic operators in differential topology, 2001 191 Shun-ich i Amar i and Hiroshi Nagaoka, Method s of information geometry, 2000 190 Alexande r N . Starkov , Dynamica l systems on homogeneous spaces, 2000 189 Mitsur u Ikawa, Hyperboli c partial differential equations and wave phenomena, 2000 188 V . V. Buldygin and Yu. V. Kozachenko, Metri c characterization of random variables and random processes, 2000 187 A . V. Fursikov, Optima l control of distributed systems. Theory and applications, 2000 186 Kazuy a Kato, Nobushige Kurokawa , an d Takeshi Saito, Numbe r theory 1: Fermat's dream, 2000 185 Kenj i Ueno, Algebrai c Geometry 1: From algebraic varieties to schemes, 1999 184 A . V. Mel'nikov, Financia l markets, 1999 183 Hajim e Sato , Algebrai c topology: an intuitive approach, 1999 182 I . S. Krasil'shchik and A. M. Vinogradov, Editors , Symmetrie s and conservation laws for differential equations of mathematical physics, 1999 181 Ya . G. Berkovich and E. M. Zhmud' , Character s of finite groups. Part 2, 1999 180 A . A. Milyutin and N. P. Osmolovskii, Calculu s of variations and optimal control, 1998 179 V . E. Voskresenskii, Algebrai c groups and their birational invariants, 1998 178 Mitsu o Morimoto, Analyti c functionals on the sphere, 1998 177 Sator u Igari, Rea l analysis—with an introduction to wavelet theory, 1998 176 L . M. Lerman and Ya. L. Umanskiy, Four-dimensiona l integrabl e Hamiltonian systems with simple singular points (topological aspects), 1998 For a complete list of titles in this series, visit th e AMS Bookstore at www.ams.org/bookstore/ . Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms This page intentionally left blank Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms (f) Translation s o f 10.1090/mmono/199 i l MATHEMATICAL < MONOGRAPH S S ^ Volum e 19 9 fVi Geometry of O Characteristi c Classes CO Shigeyuki Morita J^* Mg^^f e America n Mathematical Society *<L il l *= i) 5 p ence, Rhode Island rovid Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms Editorial Boar d Shoshichi Kobayashi (Chair) Masamichi Takesaki ftttauamt CHARACTERISTIC CLASSES AND GEOMETRY by Shigeyuki Morit a Copyright © 199 9 by Shigeyuki Morit a Originally published in Japanes e by Iwanami Shoten, Publishers, Tokyo, 199 9 Translated from the Japanese by the autho r 2000 Mathematics Subject Classification. Primar y 57R20, 57R32, 55P62; Secondary 55R40, 20F38, 32G15. Library of Congress Cataloging-in-Publication Dat a Morita, S. (Shigeyuki), 1946- [Tokuseirui to kikagaku. English ] Geometry of characteristic classes / Shigeyuk i Morita ; [translated from th e Japanese by the author]. p. cm. — (Translations of mathematical monographs, ISSN 0065-9282 ; 199) (Iwanami series in modern mathematics ) Includes bibliographical references and index. ISBN 0-8218-2139-3 (alk. paper) 1. Characteristic classes. I . Title. II . Series. III . Series: Iwanami series in modern mathematics. QA613.618.M67 200 1 514/.72—dc21 00-05431 2 © 200 1 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. @ Th e paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at URL: http://www.ams.org/ 10 9 8 7 6 5 4 3 2 1 0 6 05 04 03 02 01 Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms Contents Preface to the English Edition vi i Preface i x Summary and Purpose of This Volume x i Chapter 1. D e Rham Homotopy Theory 1 1.1. Postniko v decomposition and rational homotopy type 1 1.2. Minima l models of differential graded algebras 1 8 1.3. Th e main theorem 3 4 1.4. Fundamenta l groups and de Rham homotopy theory 4 3 Chapter 2. Characteristi c Classes of Flat Bundles 4 9 2.1. Fla t bundles 4 9 2.2. Cohomolog y of Lie algebras 6 0 2.3. Characteristi c classes of flat bundles 6 6 2.4. Gel'fand-Fuk s cohomology 7 5 Chapter 3. Characteristi c Classes of Foliations 8 7 3.1. Foliation s 8 7 3.2. Th e Godbillon-Vey class 9 1 3.3. Canonica l forms on frame bundles of higher orders 10 0 3.4. Bot t vanishing theorem and characteristic classes of foliations 11 3 3.5. Discontinuou s invariants 11 9 3.6. Characteristi c classes of flat bundles II 12 6 Chapter 4. Characteristi c Classes of Surface Bundles 13 5 4.1. Mappin g class group and classification of surface bundles 13 5 4.2. Characteristi c classes of surface bundles 14 4 4.3. Non-trivialit y of the characteristic classes (1) 15 1 4.4. Non-trivialit y of the characteristic classes (2) 16 0 Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms vi CONTENT S 4.5. Application s of characteristic classes 17 1 Directions and Problems for Future Research 17 5 Bibliography 17 9 Index 18 3 Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms Preface t o th e Englis h Editio n This is a translation of my book originally published in Japanese by Iwanami Shoten, Publishers. It s aim is to introduce the reade r to various theories of characteristic classes which have been initiated and develope d durin g the last thre e decades. The y include chara - teristic classes of flat bundles, characteristic classes of foliations and characteristic classes of surface bundles. I would lik e to thank th e America n Mathematica l Societ y fo r publishing this English edition and their staff for providing excellent support. Shigeyuki Morita February 2001 Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms This page intentionally left blank Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms Preface The purpose of the present volume is to give expositions on the basics o f some ne w theories concernin g geometr y o f characteristi c classes which have appeared since the end of the 1960's. Among characteristic classes, there are Stiefel-Whitney classes , Euler classes or Pontrjagin classes and Chern classes as typical exam- ples. These classes were introduced during the 1930's and 1940's, and since then they have played fundamental roles in the classification as well as analysis of the structure of manifolds. On the other hand , fro m th e end of the 1960' s onward, ther e arose a few theories which treat finer structures o f manifolds tha n before. Thes e includ e Gel'fand-Fuk s theory , Chern-Simon s theor y and als o the theory o f characteristic classe s of flat bundles, whic h are closely related t o each other. Thes e new characteristic classe s are defined when the above mentioned classical characteristic classes vanish (partially ) s o that sometime s they ar e called the secondar y classes. Among other things, the characteristic classes of flat bundles have been shown to have intimate relations with not only geometry but also algebraic geometry and number theory. I t is plausible that they will play more and more crucial roles in future mathematics . The theory of characteristic classes of fiber bundles, whose structure group s ar e infinite dimensiona l group s such a s the diffeomor - phism group s of manifolds, remai n largel y unknown . However , i n the case of diffeomorphism group s of surfaces, quite detailed studies have been made. This is the theory of characteristic classes of surface bundles which began in the 1980's. As will be mentione d i n "Direction s an d Problem s fo r Futur e Research" at the end of this book, studies of the above new theories are continuing from various points of view. Although the description of this book is limited to the foundational part, the author would be happy if readers are interested in these theories. ix Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms