§1. Historical Remarks Convex Integration theory, first introduced by M. Gromov [17], is one of three general methods in immersion-theoretic topology for solving a broad range of problems in geometry and topology. The other methods are: (i) Removal of Singularities, introduced by M. Gromov and Y. Eliashberg [8]; (ii) the covering homotopy method which, following M. Gromov's thesis [16], is also referred to as the method of sheaves. The covering homotopy method is due originally to S. Smale [36] who proved a crucial covering homotopy result in order to solve the classification problem for immersions of spheres in Euclidean space. These general methods are not linearly related in the sense that succes­ sive methods subsumed the previous methods. Each method has its own distinct foundation, based on an independent geometrical or analytical insight. Conse­ quently, each method has a range of applications to problems in topology that are best suited to its particular insight. For example, a distinguishing feature of Convex Integration theory is that it applies to solve closed relations in jet spaces, including certain general classes of underdetermined non-linear systems of par­ tial differential equations. As a case of interest, the Nash-Kuiper Cl-isometrie immersion theorem ean be reformulated and proved using Convex Integration theory (cf. Gromov [18]). No such results on closed relations in jet spaees can be proved by means of the other two methods.

1 Introduction.- §1 Historical Remarks.- §2 Background Material.- §3 h-Principles.- §4 The Approximation Problem.- 2 Convex Hulls.- §1 Contractible Spaces of Surrounding Loops.- §2 C-Structures for Relations in Affine Bundles.- §3 The Integral Representation Theorem.- 3 Analytic Theory.- §1 The One-Dimensional Theorem.- §2 The C?-Approximation Theorem.- 4 Open Ample Relations in Spaces of 1-Jets.- §1 C°-Dense h-Principle.- §2 Examples.- 5 Microfibrations.- §1 Introduction.- §2 C-Structures for Relations over Affine Bundles.- §3 The C?-Approximation Theorem.- 6 The Geometry of Jet spaces.- §1 The Manifold X?.- §2 Principal Decompositions in Jet Spaces.- 7 Convex Hull Extensions.- §1 The Microfibration Property.- §2 The h-Stability Theorem.- 8 Ample Relations.- §1 Short Sections.- §2 h-Principle for Ample Relations.- §3 Examples.- §4 Relative h-Principles.- 9 Systems of Partial Differential Equations.- §1 Underdetermined Systems.- §2 Triangular Systems.- §3 C1-Isometric Immersions.- 10 Relaxation Theorem.- §1 Filippov's Relaxation Theorem.- §2 C?-Relaxation Theorem.- References.- Index of Notation.