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Convex Integration Theory - Solutions to the h-principle in geometry and topology (Paperback, Reprint of the 1998 Edition)
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Convex Integration Theory - Solutions to the h-principle in geometry and topology (Paperback, Reprint of the 1998 Edition)
Series: Modern Birkhauser Classics
Expected to ship within 10 - 15 working days
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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.
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