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Geometrisation of 3-Manifolds
Laurent Bessières and Gérard Besson, Université Joseph Fourier, Grenoble, France, Michel Boileau, Université Paul Sabatier, Toulouse, France, Sylvain Maillot, Université Montpellier II, France, and Joan Porti, Universitat Autonoma de Barcelona, Spain
A publication of the European Mathematical Society.
cover
EMS Tracts in Mathematics
2010; 247 pp; hardcover
Volume: 13
ISBN-10: 3-03719-082-5
ISBN-13: 978-3-03719-082-1
List Price: US$64
Member Price: US$51.20
Order Code: EMSTM/13
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The geometrisation conjecture was proposed by William Thurston in the mid 1970s in order to classify compact \(3\)-manifolds by means of a canonical decomposition along essential, embedded surfaces into pieces that possess geometric structures. It contains the famous Poincaré Conjecture as a special case.

In 2002 Grigory Perelman announced a proof of the geometrisation conjecture based on Richard Hamilton's Ricci flow approach and presented it in a series of three celebrated arXiv preprints. Since then there has been an ongoing effort to understand Perelman's work by giving more detailed and accessible presentations of his ideas or alternative arguments for various parts of the proof.

This book is a contribution to this endeavor. Its two main innovations are first a simplified version of Perelman's Ricci flow with surgery, which is called Ricci flow with bubbling-off, and secondly a completely different and original approach to the last step of the proof. In addition, special effort has been made to simplify and streamline the overall structure of the argument and make the various parts independent of one another.

A complete proof of the geometrisation conjecture is given, modulo pre-Perelman results on Ricci flow, Perelman's results on the \(\mathcal L\)-functional and \(\kappa\)-solutions, as well as the Colding-Minicozzi extinction paper. The book can be read by anyone already familiar with these results or willing to accept them as black boxes. The structure of the proof is presented in a lengthy introduction which does not require knowledge of geometric analysis. The bulk of the proof is the existence theorem for Ricci flow with bubbling-off, which is treated in parts I and II. Part III deals with the long-time behaviors of Ricci flow with bubbling-off. Part IV finishes the proof of the geometrisation conjecture.

A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.

Readership

Graduate students and research mathematicians interested in the geometrisation conjecture.

Table of Contents

  • The Geometrisation conjecture
Part I. Ricci flow with bubbling-off: definitions and statements
  • Basic definitions
  • Piecing together necks and caps
  • \(\kappa\)-noncollapsing, canonical geometry and pinching
  • Ricci flow with \((r, \delta, \kappa)\)-bubbling-off
Part II. Ricci flow with bubbling-off: existence
  • Choosing cutoff parameters
  • Metric surgery and the proof of Proposition A
  • Persistence
  • Canonical neighbourhoods and the proof of Proposition B
  • \(\kappa\)-noncollapsing and the proof of Proposition C
Part III. Long-time behaviour of Ricci flow with bubbling-off
  • The thin-thick decomposition theorem
  • Refined estimates for long-time behaviour
Part IV. Weak collapsing and hyperbolisation
  • Collapsing, simplicial volume and strategy of proof
  • Proof of the weak collapsing theorem
  • A rough classification of 3-manifolds
  • Appendix A. 3-manifold topology
  • Appendix B. Comparison geometry
  • Appendix C. Ricci flow
  • Appendix D. Alexandrov spaces
  • Appendix E. A sufficient condition for hyperbolicity
  • Bibliography
  • Index
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