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Convex Integration Theory

Solutions to the h-principle in geometry and topology

  • David Spring

Part of the Monographs in Mathematics book series (MMA, volume 92)

Table of contents

  1. Front Matter
    Pages i-viii
  2. David Spring
    Pages 1-18
  3. David Spring
    Pages 19-32
  4. David Spring
    Pages 33-48
  5. David Spring
    Pages 49-69
  6. David Spring
    Pages 71-86
  7. David Spring
    Pages 87-99
  8. David Spring
    Pages 101-120
  9. David Spring
    Pages 121-164
  10. David Spring
    Pages 165-199
  11. David Spring
    Pages 201-206
  12. Back Matter
    Pages 207-213

About this book

Introduction

§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.

Keywords

Differential topology Manifold Topology differential geometry equation function geometry theorem

Authors and affiliations

  • David Spring
    • 1
  1. 1.Department of MathematicsGlendon CollegeTorontoCanada

Bibliographic information