| Convex Integration Theory: Solutions to the H-Principle in Geometry and Topology Reprint of the Edition Contributor(s): Spring, David (Author) |
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ISBN: 3034800592 ISBN-13: 9783034800594 Publisher: Birkhauser OUR PRICE: $52.24 Product Type: Paperback - Other Formats Published: December 2010 |
| Additional Information |
| BISAC Categories: - Mathematics |
| Dewey: 514.72 |
| Series: Modern Birkhäuser Classics |
| Physical Information: 0.47" H x 6.14" W x 9.21" (0.70 lbs) 213 pages |
| Descriptions, Reviews, Etc. |
| Publisher Description: 1. Historical Remarks Convex Integration theory, ?rst 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 classi?cation problem for immersions of spheres in Euclidean space. These general methods are not linearly related in the sense that succ- sive methods subsumed the previous methods. Each method has its own distinct foundation, based on an independent geometrical or analytical insight. Con- 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 ConvexIntegrationtheoryisthatitappliestosolveclosed relationsinjetspaces, including certain general classes of underdetermined non-linear systems of par- 1 tial di?erential equations. As a case of interest, the Nash-Kuiper C -isometric immersion theorem can be reformulated and proved using Convex Integration theory (cf. Gromov 18]). No such results on closed relations in jet spaces can be proved by means of the other two methods. On the other hand, many classical results in immersion-theoretic topology, such as the classi?cation of immersions, are provable by all three methods. |