Download E-books Convex Integration Theory: Solutions to the h-principle in geometry and topology (Modern Birkhäuser Classics) PDF
By David Spring§1. ancient feedback Convex Integration thought, ?rst brought by way of M. Gromov , is one among 3 normal equipment in immersion-theoretic topology for fixing a large variety of difficulties in geometry and topology. the opposite equipment are: (i) removing of Singularities, brought by way of M. Gromov and Y. Eliashberg ; (ii) the masking homotopy approach which, following M. Gromov’s thesis , can also be often called the strategy of sheaves. The masking homotopy technique is due initially to S. Smale  who proved a vital masking homotopy lead to order to unravel the classi?cation challenge for immersions of spheres in Euclidean area. those basic tools usually are not linearly comparable within the feel that succ- sive tools subsumed the former equipment. each one strategy has its personal unique starting place, in keeping with an self sufficient geometrical or analytical perception. Con- quently, each one procedure has a number purposes to difficulties in topology which are most fitted to its specific perception. for instance, a distinguishing characteristic of ConvexIntegrationtheoryisthatitappliestosolveclosed relationsinjetspaces, together with convinced basic periods of underdetermined non-linear platforms of par- 1 tial di?erential equations. As a case of curiosity, the Nash-Kuiper C -isometric immersion theorem will be reformulated and proved utilizing Convex Integration concept (cf. Gromov ). No such effects on closed relatives in jet areas may be proved through the opposite equipment. nonetheless, many classical ends up in immersion-theoretic topology, akin to the classi?cation of immersions, are provable by means of all 3 tools.