Preface Intrigue Part 1 Holonomic Approximation Chapter 1． Jets and Holonomy §1．1． Maps and sections §1．2． Coordinate definition ofjets §1．3． Invariant definition ofjets §1．4． The space X (1) §1．5． Holonomic sections of the jet space X (r) §1．6． Geometric representation of sections of X (r) §1．7． Holonomic splitting Chapter 2． Thom Transversality Theorem §2．1． Generic properties and transversality §2．2． Stratified sets and polyhedra §2．3． Thom Transversality Theorem Chapter 3． Holonomic Approximation §3．1． Main theorem §3．2． Holonomic approximation over a cube §3．3． Fiberwise holonomic sections §3．4． Inductive Lemma §3．5． Proof of the Inductive Lemma §3．6． Holonomic approximation over a cube §3．7． Parametric case Chapter 4． Applications §4．1． Functions without critical points §4．2． Smale's sphere eversion §4．3． Open manifolds §4．4． Approximate integration of tangential homotopies §4．5． Directed embeddings of open manifolds §4．6． Directed embeddings of closed manifolds §4．7． Approximation of differential forms by closed forms Part 2 Differential Relations and Gromov's h-Principle Chapter 5． Differential Relations §5．1． What is a differential relation? §5．2． Open and closed differential relations §5．3． Formal and genuine solutions of a differential relation §5．4． Extension problem §5．5． Approximate solutions to systems of differential equations Chapter 6． Homotopy Principle §6．1． Philosophy of the h-principle §6．2． Different flavors of the h-principle Chapter 7． Open Diff V-Invariant Differential Relations §7．1． Diff V-invariant differential relations §7．2． Local h-principle for open Diff V-invariant relations Chapter 8． Applications to Closed Manifolds §8．1． Microextension trick §8．2． Smale-Hirsch h-principle §8．3． Sections transversal to distribution Part 3 The Homotopy Principle in Symplectic Geometry Chapter 9． Symplectic and Contact Basics §9．1． Linear symplectic and complex geometries §9．2． Symplectic and complex manifolds …… Part 4 Convex Integration Bibliography Index
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