Introduction to the h-Principle = )PMEWLFIVK 2 MMWLEGLIZ GVEHYEXI SXYHMIW MR MEXLIQEXMGW :SPYQI 48 AQIVMGER MEXLIQEXMGEP SSGMIX] http://dx.doi.org/10.1090/gsm/048 Titles in This Series 48 Y. Eliashberg and N. Mishachev, Introduction to the h-principle, 2002 47 A. Yu. Kitaev, A. H. Shen, and M. N. Vyalyi, Classical and quantum computation, 2002 46 Joseph L. Taylor, Several complex variables with connections to algebraic geometry and Lie groups, 2002 45 Inder K. Rana, An introduction to measure and integration, second edition, 2002 44 Jim Agler and John E. McCarthy, Pick interpolation and Hilbert function spaces, 2002 43 N. V. Krylov, Introduction to the theory of random processes, 2002 42 Jin Hong and Seok-Jin Kang, Introduction to quantum groups and crystal bases, 2002 41 Georgi V. Smirnov, Introduction to the theory of differential inclusions, 2002 40 Robert E. Greene and Steven G. Krantz, Function theory of one complex variable, 2002 39 Larry C. Grove, Classical groups and geometric algebra, 2002 38 Elton P. Hsu, Stochastic analysis on manifolds, 2002 37 HershelM.FarkasandIrwinKra, Theta constants, Riemann surfaces and the modular group, 2001 36 Martin Schechter, Principles of functional analysis, second edition, 2002 35 JamesF.DavisandPaulKirk, Lecture notes in algebraic topology, 2001 34 Sigurdur Helgason, Differential geometry, Lie groups, and symmetric spaces, 2001 33 Dmitri Burago, Yuri Burago, and Sergei Ivanov, A course in metric geometry, 2001 32 Robert G. Bartle, A modern theory of integration, 2001 31 Ralf Korn and Elke Korn, Option pricing and portfolio optimization: Modern methods of financial mathematics, 2001 30 J. C. McConnell and J. C. Robson, Noncommutative Noetherian rings, 2001 29 Javier Duoandikoetxea, Fourier analysis, 2001 28 Liviu I. Nicolaescu, Notes on Seiberg-Witten theory, 2000 27 Thierry Aubin, A course in differential geometry, 2001 26 Rolf Berndt, An introduction to symplectic geometry, 2001 25 Thomas Friedrich, Dirac operators in Riemannian geometry, 2000 24 Helmut Koch, Number theory: Algebraic numbers and functions, 2000 23 Alberto Candel and Lawrence Conlon, Foliations I, 2000 22 G¨unter R. Krause and Thomas H. Lenagan, Growth of algebras and Gelfand-Kirillov dimension, 2000 21 John B. Conway, A course in operator theory, 2000 20 Robert E. Gompf and Andr´as I. Stipsicz, 4-manifolds and Kirby calculus, 1999 19 Lawrence C. Evans, Partial differential equations, 1998 18 Winfried Just and Martin Weese, Discovering modern set theory. II: Set-theoretic tools for every mathematician, 1997 17 Henryk Iwaniec, Topics in classical automorphic forms, 1997 16 Richard V. Kadison and John R. Ringrose, Fundamentals of the theory of operator algebras. Volume II: Advanced theory, 1997 15 Richard V. Kadison and John R. Ringrose, Fundamentals of the theory of operator algebras. Volume I: Elementary theory, 1997 14 Elliott H. Lieb and Michael Loss, Analysis, 1997 13 Paul C. Shields, The ergodic theory of discrete sample paths, 1996 For a complete list of titles in this series, visit the AMS Bookstore at www.ams.org/bookstore/. Introduction to the h-Principle Introduction to the h-Principle Y. Eliashberg N. Mishachev Graduate Studies in Mathematics Volume 48 American Mathematical Society Providence, Rhode Island Editorial Board Walter Craig Nikolai Ivanov Steven G. Krantz David Saltman (Chair) 2000 Mathematics Subject Classification. Primary 58Axx. Abstract. The book is devoted to topological methods for solving differential equations and inequalities. Its content significantly overlaps with Gromov’s book “Partial differential relations”. However, the exposition is more elementary and intended for a broader mathematical audience, including graduate, and even advanced undergraduate students. Library of Congress Cataloging-in-Publication Data Eliashberg, Y., 1946– Introduction to the h-principle / Y. Eliashberg and N. Mishachev. p. cm. — (Graduate studies in mathematics, ISSN 1065-7339 ; v 48) Includes bibliographical references. ISBN 0-8218-3227-1 (alk. paper) 1. Geometry, Differential. 2. Differentiable manifolds. 3. Differential equations–Numerical solutions. I. Mishachev, N (Nikolai M.) 1952– II. Title. III. Series. QA641.E62 2002 516.36–dc21 2002019347 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02940-6248 USA. Requests can also be made by e-mail to [email protected]. c 2002 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. ∞ The 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 07 06 05 04 03 02 To Vladimir Igorevich Arnold who introduced us to the world of singularities and Misha Gromov who taught us how to get rid of them Contents Preface xv Intrigue 1 Part 1. Holonomic Approximation Chapter 1. Jets and Holonomy 7 §1.1. Maps and sections 7 §1.2. Coordinate definition of jets 7 §1.3. Invariant definition of jets 9 §1.4. The space X(1) 10 §1.5. Holonomic sections of the jet space X(r) 11 §1.6. Geometric representation of sections of X(r) 12 §1.7. Holonomic splitting 12 Chapter 2. Thom Transversality Theorem 15 §2.1. Generic properties and transversality 15 §2.2. Stratified sets and polyhedra 16 §2.3. Thom Transversality Theorem 17 Chapter 3. Holonomic Approximation 21 §3.1. Main theorem 21 §3.2. Holonomic approximation over a cube 23 §3.3. Fiberwise holonomic sections 24 §3.4. Inductive Lemma 25 ix x Contents §3.5. Proof of the Inductive Lemma 28 §3.6. Holonomic approximation over a cube 33 §3.7. Parametric case 34 Chapter 4. Applications 37 §4.1. Functions without critical points 37 §4.2. Smale’s sphere eversion 38 §4.3. Open manifolds 40 §4.4. Approximate integration of tangential homotopies 41 §4.5. Directed embeddings of open manifolds 44 §4.6. Directed embeddings of closed manifolds 45 §4.7. Approximation of differential forms by closed forms 47 Part 2. Differential Relations and Gromov’s h-Principle Chapter 5. Differential Relations 53 §5.1. What is a differential relation? 53 §5.2. Open and closed differential relations 55 §5.3. Formal and genuine solutions of a differential relation 56 §5.4. Extension problem 56 §5.5. Approximate solutions to systems of differential equations 57 Chapter 6. Homotopy Principle 59 §6.1. Philosophy of the h-principle 59 §6.2. Different flavors of the h-principle 62 Chapter 7. Open Diff V -Invariant Differential Relations 65 §7.1. Diff V -invariant differential relations 65 §7.2. Local h-principle for open Diff V -invariant relations 66 Chapter 8. Applications to Closed Manifolds 69 §8.1. Microextension trick 69 §8.2. Smale-Hirsch h-principle 69 §8.3. Sections transversal to distribution 71 Part 3. The Homotopy Principle in Symplectic Geometry Chapter 9. Symplectic and Contact Basics 75 §9.1. Linear symplectic and complex geometries 75 §9.2. Symplectic and complex manifolds 80 Contents xi §9.3. Symplectic stability 85 §9.4. Contact manifolds 88 §9.5. Contact stability 94 §9.6. Lagrangian and Legendrian submanifolds 95 §9.7. Hamiltonian and contact vector fields 97 Chapter 10. Symplectic and Contact Structures on Open Manifolds 99 §10.1. Classification problem for symplectic and contact structures 99 §10.2. Symplectic structures on open manifolds 100 §10.3. Contact structures on open manifolds 102 §10.4. Two-forms of maximal rank on odd-dimensional manifolds 103 Chapter 11. Symplectic and Contact Structures on Closed Manifolds 105 §11.1. Symplectic structures on closed manifolds 105 §11.2. Contact structures on closed manifolds 107 Chapter 12. Embeddings into Symplectic and Contact Manifolds 111 §12.1. Isosymplectic embeddings 111 §12.2. Equidimensional isosymplectic immersions 118 §12.3. Isocontact embeddings 121 §12.4. Subcritical isotropic embeddings 128 Chapter 13. Microflexibility and Holonomic R-Approximation 129 §13.1. Local integrability 129 §13.2. Homotopy extension property for formal solutions 131 §13.3. Microflexibility 131 §13.4. Theorem on holonomic R-approximation 133 §13.5. Local h-principle for microflexible Diff V -invariant relations 133 Chapter 14. First Applications of Microflexibility 135 §14.1. Subcritical isotropic immersions 135 §14.2. Maps transversal to a contact structure 136 Chapter 15. Microflexible A-Invariant Differential Relations 139 §15.1. A-invariant differential relations 139 §15.2. Local h-principle for microflexible A-invariant relations 140 Chapter 16. Further Applications to Symplectic Geometry 143 §16.1. Legendrian and isocontact immersions 143 §16.2. Generalized isocontact immersions 144 xii Contents §16.3. Lagrangian immersions 146 §16.4. Isosymplectic immersions 147 §16.5. Generalized isosymplectic immersions 149 Part 4. Convex Integration Chapter 17. One-Dimensional Convex Integration 153 §17.1. Example 153 §17.2. Convex hulls and ampleness 154 §17.3. Main lemma 155 §17.4. Proof of the main lemma 156 §17.5. Parametric version of the main lemma 161 §17.6. Proof of the parametric version of the main lemma 162 Chapter 18. Homotopy Principle for Ample Differential Relations 167 §18.1. Ampleness in coordinate directions 167 §18.2. Iterated convex integration 168 §18.3. Principal subspaces and ample differential relations in X(1) 170 §18.4. Convex integration of ample differential relations 171 Chapter 19. Directed Immersions and Embeddings 173 §19.1.