Topology of 3-Manifolds

摘要：在过去的100年里，人们对三维流形的拓扑结构和几何结构产生了极大的兴趣。在某种程度上，因为3流形（粗略地说）是一个以我们熟悉的3维环境为模型的数学对象。一个3流形具有足够小的子集可以被3个坐标参数化的性质。表征曲面的几何和拓扑的问题，即2维流形，在近一个世纪前就已经得到了完全的解答。确定了球面、环面和多孔环面是唯一封闭的可定向连接表面。此外，这样的曲面可以给定恒定曲率的几何形状，球面为+1，圆环为0，其他曲面为-1。虽然在上个世纪取得了很大的进展，但是对于三维空间的情况还没有得到很好的理解。然而，关于3流形结构的推测图在25年前就出现了（由Thurston提出），并且有很多理论和实验证据支持他的推测。双曲- 3流形，即曲率为-1常数的流形，在这幅图中起着中心作用。在这一提议中，PI与其他数学家合作，计划研究双曲流形的结构和猜想双曲流形的结构。特别地，他们将研究双曲3-流形的微分同构群的结构，以及小体积双曲3-流形的结构。他们还将讨论闭合非球面、阿托向、3流形是否具有格罗莫夫负曲线基群。一个正解意味着一个猜想双曲的3-流形至少具有双曲3-流形的粗代数结构。

Proposal: DMS-0071852PI: David GabaiAbstract:During the last 100 years there has been tremendous interest inunderstanding the topology and geometry of 3-dimensional manifolds. Inpart, because a 3-manifold is (roughly speaking) a mathematical objectmodeled on our familiar 3-dimensional environment. A 3-manifold has theproperty that sufficiently small subsets can be parametrized by 3coordinates.The problem of characterizing the geometry and topology of surfaces, i.e.,manifolds of dimension 2, was completely answered nearly a century ago. Itwas determined that the only closed orientable connected surfaces are thesphere, torus, and the multi-holed tori. Furthermore, such surfaces canbe given geometries of constant curvature, +1 for the sphere, 0 for thetorus, and -1 for the other surfaces.Although great progress has been made during the last century, thesituation for dimension-3 is not nearly so well understood. Nevertheless aconjectural picture for the structure of 3-manifolds emerged almost 25years ago (by Thurston) and there has been much theoretical andexperimental evidence supporting his conjecture. The hyperbolic3-manifolds, i.e. the manifolds of constant -1 curvature, play a centralrole in this picture. In this proposal the PI in collaboration withvarious other mathematicians, plan to investigate the structure ofhyperbolic manifolds and the structure of manifolds which are conjecturallyhyperbolic. In particular they will study the structure of thediffeomorphism group of hyperbolic 3-manifolds, and the structure ofhyperbolic 3-manifolds of low volume. They will also address whether or nota closed aspherical, atoroidal, 3-manifold has a Gromov negatively curvedfundamental group. A positive resolution would imply that a 3-manifoldwhich is conjecturally hyperbolic, has at least the coarse algebraicstructure of a hyperbolic 3-manifold.