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
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
Amer. Math. Soc. 18 (2005) 561–569
GP at MIT (114K image)