EMS Tracts in Mathematics 2010; 247 pp; hardcover Volume: 13 ISBN10: 3037190825 ISBN13: 9783037190821 List Price: US$64 Member Price: US$51.20 Order Code: EMSTM/13
 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 bubblingoff, 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 prePerelman results on Ricci flow, Perelman's results on the \(\mathcal L\)functional and \(\kappa\)solutions, as well as the ColdingMinicozzi 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 bubblingoff, which is treated in parts I and II. Part III deals with the longtime behaviors of Ricci flow with bubblingoff. 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 bubblingoff: definitions and statements  Basic definitions
 Piecing together necks and caps
 \(\kappa\)noncollapsing, canonical geometry and pinching
 Ricci flow with \((r, \delta, \kappa)\)bubblingoff
Part II. Ricci flow with bubblingoff: 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. Longtime behaviour of Ricci flow with bubblingoff  The thinthick decomposition theorem
 Refined estimates for longtime behaviour
Part IV. Weak collapsing and hyperbolisation  Collapsing, simplicial volume and strategy of proof
 Proof of the weak collapsing theorem
 A rough classification of 3manifolds
 Appendix A. 3manifold topology
 Appendix B. Comparison geometry
 Appendix C. Ricci flow
 Appendix D. Alexandrov spaces
 Appendix E. A sufficient condition for hyperbolicity
 Bibliography
 Index
