Back to Departmental Colloquium: Spring 2012
Departmental Colloquium
Date: Thursday, Feb 2, 2012
Time: 4:15PM
Location: JWB 335
John Morgan
Stony Brook University
Title |
Geometrization of 3-manifolds |
Abstract |
The Geometrization Conjecture for 3-manifolds was formulated by Thurston in the early 1980s. `Geometric’ manifolds are defined to be smooth manifolds that admit complete, finite-volume locally homogeneous Riemannian metrics. Such manifolds are local homogeneous spaces and thus can be enumerated in terms of Lie groups and finite co-volume lattices in them. Thurston’s geometrization conjecture states that every compact 3-manifold is constructed from geometric ones by simple geometric operations. This conjecture is a vast generalization of the Poincare Conjecture, which can be reformulated as saying that every closed, simply connected 3-manifold is geometric, which means that it has a round (constant positive curvature) metric. Such a manifold is easily seen to be homeomorphic to the 3-sphere. Perelman used Ricci flow techniques to study 3-manifolds and to prove the geometrization conjecture. There are two completely separate parts of the argument: Ricci flow with surgery deforms the metric and does connected sum decompositions. Eventually (i.e., for sufficiently large time) under this flow the metric on a 3-manifold becomes one that decomposes into 2 pieces: one piece on which the metric is converging smoothly to a complete hyperbolic metric and another piece on which the metric is locally volume-collapsed on a scale set by the negative part of the Riemannian curvature. These pieces are connected via incompressible tori. All of this is analytic/geometric relying on deep estimates about Ricci flow. To complete the proof of the geometrization conjecture, one must deal with the piece of the second type – the locally volume-collapsed piece. For that one uses a wider class of spaces on which curvature bounded below makes sense, the so-called Alexandrov spaces. The local Alexandrov limits of the locally volume-collapsed piece are Alexandrov balls of dimension 1 or 2 and are completely understood. From this one constructs nice local models which mesh well on the overlaps and from these one can show that the second piece is a disjoint union of pieces that fiber (including Seifert fiber) over one- and two-dimensional manifolds, completing the proof of the Geometrization Conjecture In this talk we will describe the precise statement of the Geometrization Conjecture, briefly summarize how Ricci flow on a 3-manifold behaves as time goes to infinity. We then pass to a description of the locally volume-collapsed pieces in terms of their weak geometric limits and show how this is enough information to determine the topological type of the collapsing piece of the manifold. |
Representation Theory
Topology