Manifold Topology and Applications to Geometry

流形拓扑及其在几何中的应用

• 批准号: 0602298
• 负责人: F. Thomas Farrell
• 金额: $ 13.6万
• 依托单位: SUNY at Binghamton
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-07-01 至 2009-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0602298&HistoricalAwards=false
• 关键词: Manifold; Topology; Applications; Geometry

ABSTRACTDMS-0602298 F.Thomas Farrell The goal of this project is two pronged. One direction is the basic problem of classifying homotopically equivalent manifolds up to homeomorphism. Surgery theory was developed for this purpose. But in order to make this theory effective for manifolds with a given fundamental group, it is necessary to calculate the algebraic L- and K-groups of its its integral group ring which occur in the surgery exact sequence. Direct algebraic methods have generally provenunsuccessful for this. The case of finite fundamental groups being a major exception. The techniques to be used instead will come from differential geometry, controlled topology, dynamical systems and Lie group theory. The results obtained so far using these methods have been quite encouraging. The second direction of the project is to find applications of these manifold classification results to geometry. One area that looks promising is to understand the topology of the space of all negatively curved Riemannian metrics on a smooth manifold and its quotient moduli space. There is the related following question. If one smooth structure on a closed manifold supports a negatively curved Riemannian metric, then does every other smooth structure support such a metric? And a possible application to affine geometry is to the problem of whether homeomorphic complete closed affine flat manifolds are necessarily diffeomorphic. Manifolds are geometric objects which locally resemble the space of Euclidean geometry but are usually quite different globally. For example the surface of the sphere locally resembles the plane; but is only finite in extent. Being finite in extent is what is meant by a closed manifold. A smooth manifold is one without corners or edges. For example the surface of the sphere is smooth but the surface of a cube, although a manifold, is not smooth. However these two manifolds are homeomorphic; i.e. it is easy to construct a continuously varying bijective correspondence between the points of these two surfaces. When the distance between points on a manifold is also considered, we are now talking about Riemannian manifolds and distance preserving correspondences are called isometries. For example the egg and the ball represent two different positively curved Riemannian metrics on the surface of the sphere. And these two metrics even represent different points in the moduli space of such metrics since the ball is uniformly round while the egg is not. However if only the notion of straight line is being considered (i.e. geodesic ) we are talking about affine manifolds and affine equivalences. Distance preserving correspondences take straight lines to straight lines but not necessarily vice versa.

AbstractDMS-0602298 F.Thomas Farrell该项目的目标是两个脚步。一个方向是将同型等效流形分类为同态形态的基本问题。为此，开发了手术理论。但是，为了使该理论有效地对具有给定基本组的流形有效，有必要计算其积分组环的代数L和k组，该基置组发生在手术精确序列中。直接代数方法通常已经对此有效。有限基本团体的情况是一个主要例外。相反，要使用的技术将来自差分几何，受控拓扑，动力学系统和谎言组理论。到目前为止，使用这些方法获得的结果令人鼓舞。该项目的第二个方向是找到这些歧管分类结果对几何形状的应用。一个看起来很有希望的领域是了解所有负面弯曲的Riemannian指标的拓扑拓扑，并在平滑的歧管及其商模量空间上。有以下问题。如果封闭的歧管上的一个平滑结构支持负弯曲的riemannian度量，那么其他所有平滑结构都支持这样的度量吗？仿射几何形状的可能应用是在同质形态完全闭合封闭式扁平歧管是否必然是差异的问题。 歧管是局部类似于欧几里得几何空间的几何对象，但通常在全球范围内完全不同。例如，球的表面局部类似于平面；但仅是有限的。在范围内，有限的是封闭的多种多样的含义。光滑的歧管是没有角或边缘的一个。例如，球的表面是光滑的，但是立方体的表面虽然是歧管，但并不光滑。但是，这两个歧管是同构的。即，很容易构建这两个表面的点之间连续变化的两种射对对应关系。当还考虑了歧管上点之间的距离时，我们现在正在谈论riemannian歧管，并且保留对应关系的距离称为异构体。例如，鸡蛋和球代表球体表面上的两个不同弯曲的riemannian指标。而且，这两个指标甚至代表了此类指标模量空间中的不同点，因为球均匀圆形而鸡蛋却没有。但是，如果仅考虑直线的概念（即，我们都在谈论仿射歧管和仿射等价。保留对应关系的距离对直线采取直线，但不一定是副词。