Topology of 3-Manifolds

建议：DMS-0071852 PI：大卫Gabai摘要：在过去的100年里，人们对三维流形的拓扑和几何有着极大的兴趣。 在某种程度上，因为三维流形（粗略地说）是一个在我们熟悉的三维环境中建模的数学对象。 一个三维流形具有足够小的子集可以用3个坐标来参数化的性质。流形的2维，完全回答了近世纪前。 它被确定为唯一的封闭定向连通曲面是曲面，环面，和多孔环面。 此外，这样的曲面可以给出常曲率的几何形状，球面为+1，θ环面为0，其它曲面为-1。尽管在上个世纪已经取得了很大的进展，但对三维的情况还没有很好的了解。 尽管如此，三维流形的结构的一个理论图出现在大约25年前（Thurston），并且有许多理论和实验证据支持他的猜想。 双曲3-流形，即常曲率流形，在这幅图中起着中心作用。 在这个提议中，PI与其他各种数学家合作，计划研究双曲流形的结构和双曲流形的结构。 特别是他们将研究双曲三维流形的同构群的结构，以及低体积的双曲三维流形的结构。他们还将解决是否关闭非球面，环面，3-流形有一个Gromov负弯曲的基本群。 一个正的分解意味着一个空间上为双曲的三维流形至少具有双曲三维流形的粗糙代数结构。

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.