Recent progress on the Poincaré conjecture and the classification of 3-manifolds

Abstract

If M is a closed 3-manifold with trivial fundamental group, then is M diffeomorphic to S? The Poincare Conjecture is that the answer to this question is “Yes.” Developing 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 curvature −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 general 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 homogeneous metrics, hyperbolic or otherwise, for all manifolds; this is called Thurston’s Geometrization Conjecture for 3-manifolds. The statement of this conjecture is somewhat complicated, so it is deferred until Section 2. An important point is that Thurston’s Geometrization Conjecture includes the Poincare Conjecture as a very special case. In addition, Thurston’s conjecture has two advantages over the Poincare Conjecture: • It applies to all closed oriented 3-manifolds. • It posits a close relationship between topology and geometry in dimension three.