This final issue of GT Volume 12 is devoted to Grigori Perelman's Geometrization Theorem for 3-manifolds and related topics.

Perelman's Theorem was originally conjectured by Bill Thurston in 1978¹ and opens the door to a full classification of compact 3–manifolds. A special case of the theorem answers a question raised by Henri Poincaré in 1904: "Est-il possible que le groupe fondamental de V [from context, a closed 3–manifold] se réduise à la substitution identique, et que pourtant V ne soit pas simplement connexe? .... simplement connexe au sens propre du mot, c'est-à-dire homéomorphe à l'hypersphere"². The negative answer to Poincaré's question, namely that every closed simply-connected 3–manifold is homeomorphic to S³, has since been called the Poincaré Conjecture, and it is this that Perelman's theorem implies.

Perelman's arguments are contained in essence in three articles available only from the arXiv³. The first two contain the arguments which prove his Geometrization Theorem and the third provides a shortcut to the Poincaré Conjecture. The article 'Notes on Perelman's papers' by Bruce Kleiner and John Lott, in this issue, provides the details for Perelman's proof of the Geometrization Theorem. The paper is intended to be read alongside Perelman's papers. The electronic version is hyperlinked to the Perelman arXiv papers, but the reader may well find it convenient to print copies of the Perelman papers in order to have them in hand whilst reading this paper.

The other two papers in this issue are by Toby Colding and Bill Minicozzi. They concern width and sweepouts, with the second providing an exposition, suitable for a nonspecialist reader, of the authors' alternative shortcut to the Poincaré Conjecture which appeared in JAMS⁴.

¹ Colloquium, Berkeley, California, January 1978
² Cinquiéme complément à l'analysis situs, Ouvres de Poincaré, Ed. Gauthier-Villars, Tome VI, Paris (1953) 435–498: Originally published: Rendiconti de Circolo Matematica di Palermo, 18 (1904) 45–110
³ arxiv:math.DG/0211159, arxiv:math.DG/0303109, arxiv:math.DG/0307245
⁴ J. Amer. Math. Soc. 18 (2005) 561–569

^{GP at MIT (114K image)}