# Geometrisation of 3-Manifolds

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*Laurent Bessières; Gérard Besson; Michel Boileau; Sylvain Maillot; Joan Porti*

A publication of the European Mathematical Society

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.