Convex Integration Theory : Solutions to the h-principle in geometry and topology

David Spring

ISBN 10: 3034800592 / ISBN 13: 9783034800594

Editore: Springer, Basel, Springer Basel, Birkhäuser, 2010

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Druck auf Anfrage Neuware - Printed after ordering - 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.

Codice articolo 9783034800594

Dati bibliografici

• Titolo: Convex Integration Theory : Solutions to the h-principle in geometry and topology
• Editore: Springer, Basel, Springer Basel, Birkhäuser
• Data di pubblicazione: 2010
• Lingua: Inglese
• ISBN 10: 3034800592
• ISBN 13: 9783034800594
• Rilegatura: Taschenbuch
• Condizione: Neu
• Dimensioni: 236 L x 157 A x 13 P (millimetri) 236 millimetri di larghezza per 157 millimetri di altezza per 13 millimetri di profondità
• Peso: 344 grammi

Descrizione articolo

Riassunto

§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.

Informazioni sull?autore

David Spring is a Professor of mathematics at the Glendon College in Toronto, Canada.