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 该项目的目标有两个方面。一个方向是将同伦等价流形分类到同胚的基本问题。外科理论就是为此目的而发展的。但为了使该理论对给定基本群的流形有效，有必要计算其整数群环的代数L-群和K-群，这些代数L-群和K-群在手术精确序列中出现。直接代数方法在这方面通常被证明是不成功的。有限基本群的情况是一个主要的例外。所使用的技术将来自微分几何、受控拓扑、动力系统和李群理论。迄今为止使用这些方法获得的结果非常令人鼓舞。该项目的第二个方向是寻找这些流形分类结果在几何中的应用。一个看起来很有前途的领域是理解光滑流形上所有负弯曲黎曼度量的空间拓扑及其商模空间。有以下相关问题。如果闭流形上的一个平滑结构支持负弯曲黎曼度量，那么其他所有平滑结构是否都支持这样的度量？仿射几何的一个可能应用是解决同胚完全闭仿射平面流形是否必然微分同胚的问题。 流形是局部类似于欧几里得几何空间的几何对象，但通常在全局上有很大不同。例如，球体的表面局部类似于平面；但范围有限。封闭流形的含义是范围有限。光滑流形是没有角或边缘的流形。例如，球体的表面是光滑的，但立方体的表面（尽管是流形）并不光滑。然而，这两个流形是同胚的；即很容易在这两个表面的点之间构造连续变化的双射对应。当还考虑流形上点之间的距离时，我们现在讨论的是黎曼流形，并且距离保持对应关系称为等距。例如，鸡蛋和球代表球体表面上的两个不同的正弯曲黎曼度量。这两个度量甚至代表了这些度量的模空间中的不同点，因为球是均匀的圆形，而鸡蛋不是。然而，如果仅考虑直线的概念（即测地线），我们正在讨论仿射流形和仿射等价。保持距离的对应关系将直线与直线联系起来，但不一定反之亦然。