文档介绍：BULLETIN (New Series) OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 42, Number 1, Pages 57–78
S 0273-0979(04)01045-6
Article electronically published on October 29, 2004
RECENT PROGRESS ON THE POINCARE´ CONJECTURE AND
THE CLASSIFICATION OF 3-MANIFOLDS
JOHN W. MORGAN
Introduction
Motivated by what was well-known at the time about surfaces, and after a long
topological study of 3-manifolds, at the very end of his 1904 paper Poincar´e[18]
states that there remains one question to treat, namely (reformulated in modern
language):
If M is a closed 3-manifold with trivial fundamental group, then is M
diffeomorphic1 to S3?
The Poincar´eConjectureis that the answer to this question is “Yes.” De-
veloping tools to attack this problem formed the basis for much of the work in
3-dimensional topology over the last century, including for example, the proof of
Dehn’s lemma and the loop theorem and the study of surgery on knots and links.
In the 1980’s Thurston developed another approach to 3-manifolds, see [24] and
[4]. He considered 3-manifolds with riemannian metrics of constant negative cur-
vature −1. These manifolds, which are locally isometric to hyperbolic 3-space,
are called hyperbolic manifolds. There are fairly obvious obstructions showing
that not every 3-manifold can admit such a metric. Thurston formulated a gen-
eral conjecture that roughly says that the obvious obstructions are the only ones;
should they vanish for a particular 3-manifold then that manifold admits such a
metric. His proof of various important special cases of this conjecture led him
to formulate a more general conjecture about the existence of locally homoge-
neous metrics, hyperbolic or otherwise, for all manifolds; this is called Thurston’s
Geometrization Conjecture for 3-manifolds. The statement of this conjecture
is plicated, so it is deferred until Section 2. An important point is
that Thurston’s Geometrization Conjecture includes the Poincar´e Conjecture as a