Table Of Content

Introduction to the h-Principle Y. Eliashberg N. Mishachev iii To Vladimir Igorevich Arnold who introduced us to the world of singularities and Misha Gromov who taught us how to get rid of them iv Contents Preface XI Intrigue 1 Part 1. Holonomic Approximation Chapter 1. Jets and Holonomy 7 §1.1. Maps and sections. 7 §1.2. Coordinate definition of jets. The space Jr(Rn,Rq) 8 §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 V → X(r) 11 §1.7. Holonomic splitting 12 Chapter 2. Thom Transversality Theorem 13 §2.1. Generic Properties and Transversality 13 §2.2. Stratified sets and polyhedra 14 §2.3. Thom Transversality Theorem 15 Chapter 3. Holonomic Approximation 19 §3.1. Main theorem 19 §3.2. Holonomic approximation over a cube 21 §3.3. Fiberwise holonomic sections 22 §3.4. Inductive Lemma 22 V VI Contents §3.5. Proof of the Inductive Lemma 26 §3.6. Holonomic approximation over a cube 31 §3.7. Parametric case. 32 Chapter 4. Applications 35 §4.1. Functions without critical points 35 §4.2. Smale’s sphere eversion 36 §4.3. Open manifolds 37 §4.4. Approximate integration of tangential homotopies 39 §4.5. Directed embeddings of open manifolds 41 §4.6. Directed embeddings of closed manifolds 43 §4.7. Approximation of differential forms by closed forms 44 Part 2. Differential Relations and Gromov’s h-Principle Chapter 5. Differential Relations 49 §5.1. What is a differential relation? 49 §5.2. Open and closed differential relations 51 §5.3. Formal and genuine solutions of a differential relation 52 §5.4. Extension problem 52 §5.5. Approximate solutions to systems of differential equations 53 Chapter 6. Homotopy Principle 55 §6.1. Philosophy of the h-principle 55 §6.2. Different flavors of the h-principle 57 Chapter 7. Open DiffV-Invariant Differential Relations 61 §7.1. DiffV-invariant differential relations 61 §7.2. Local h-principle for open DiffV-invariant relations 62 Chapter 8. Applications to Closed Manifolds 65 §8.1. Microextension trick 65 §8.2. Smale-Hirsch h-principle 65 §8.3. Sections transversal to distribution 67 Part 3. Homotopy Principle in Symplectic Geometry Chapter 9. Symplectic and Contact Basics 71 §9.1. Linear symplectic and complex geometries 71 §9.2. Symplectic and complex manifolds 76 Contents VII §9.3. Symplectic stability 81 §9.4. Contact manifolds 84 §9.5. Contact stability 89 §9.6. Lagrangian and Legendrian submanifolds 91 §9.7. Hamiltonian and contact vector fields 93 Chapter 10. Symplectic and Contact Structures on Open Manifolds 95 §10.1. Classification problem for symplectic and contact structures 95 §10.2. Symplectic structures on open manifolds 96 §10.3. Contact structures on open manifolds 98 §10.4. Two-forms of maximal rank on odd-dimensional manifolds 99 Chapter 11. Symplectic and Contact Structures on Closed Manifolds 101 §11.1. Symplectic structures on closed manifolds 101 §11.2. Contact structures on closed manifolds 103 Chapter 12. Embeddings into Symplectic and Contact Manifolds 107 §12.1. Isosymplectic embeddings 107 §12.2. Equidimensional isosymplectic immersions 113 §12.3. Isocontact embeddings 116 §12.4. Subcritical isotropic embeddings 122 Chapter 13. Microflexibility and Holonomic R-approximation 125 §13.1. Local integrability 125 §13.2. Homotopy extension property for formal solutions 127 §13.3. Microflexibility 127 §13.4. Theorem on holonomic R-approximation 129 §13.5. Local h-principle for microflexible DiffV-invariant relations 130 Chapter 14. First Applications of Microflexibility 131 §14.1. Subcritical isotropic immersions 131 §14.2. Maps transversal to a contact structure 132 Chapter 15. Microflexible A-invariant Differential Relations 135 §15.1. A-invariant differential relations 135 §15.2. Local h-principle for microflexible A-invariant relations 136 Chapter 16. Further Applications to Symplectic Geometry 139 §16.1. Legendrian and isocontact immersions 139 §16.2. Generalized isocontact immersions 140 VIII Contents §16.3. Lagrangian immersions 142 §16.4. Isosymplectic immersions 143 §16.5. Generalized isosymplectic immersions 145 Part 4. Convex Integration Chapter 17. One-Dimensional Convex Integration 149 §17.1. Example 149 §17.2. Convex hulls and ampleness 150 §17.3. Main lemma 151 §17.4. Proof of the main lemma 152 §17.5. Parametric version of the main lemma 157 §17.6. Proof of the parametric version of the main lemma 158 Chapter 18. Homotopy Principle for Ample Differential Relations 163 §18.1. Ampleness in coordinate directions 163 §18.2. Iterated convex integration 164 §18.3. Principal subspaces and ample differential relations in X(1) 166 §18.4. Convex integration of ample differential relations 167 Chapter 19. Directed Immersions and Embeddings 169 §19.1. Criterion of ampleness for directed immersions 169 §19.2. Immersions into almost symplectic and almost complex manifolds 170 §19.3. Directed embeddings 172 Chapter 20. First Order Linear Differential Operators 175 §20.1. Formal inverse of a linear differential operator 175 §20.2. Homotopy principle for D-sections 176 §20.3. Non-vanishing D-sections 177 §20.4. Systems of linearly independent D-sections 178 §20.5. Two-forms of maximal rank on odd-dimensional manifolds 180 §20.6. One-forms of maximal rank on even-dimensional manifolds 181 Chapter 21. Nash-Kuiper Theorem 185 §21.1. Isometric immersions and short immersions 185 §21.2. Nash-Kuiper theorem 186 §21.3. Decomposition of a metric into a sum of primitive metrics 187 §21.4. Approximation theorem 187 Contents IX §21.5. One-dimensional approximation theorem 189 §21.6. Adding a primitive metric 190 §21.7. End of the proof of the approximation theorem 191 §21.8. Proof of the Nash-Kuiper theorem 192 Bibliography 195 Index 199 X Contents Figure 0.1. The relations between chapters of the book