Manifold Topology and Applications to Geometry

托马斯法雷尔本项目的目标是双管齐下。一个方向是分类同伦等价流形同胚的基本问题。外科学理论就是为此而发展起来的。但为了使这一理论对具有给定基本群的流形有效，必须计算其整群环的代数L-群和代数K-群，这些代数L-群和代数K-群出现在外科手术精确序列中。直接的代数方法一般证明这是不成功的。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，控制拓扑，动力系统和李群理论。到目前为止，使用这些方法所取得的结果相当令人鼓舞。该项目的第二个方向是找到这些流形分类结果的几何应用。一个看起来很有希望的领域是理解光滑流形上所有负弯曲黎曼度量的空间及其商模空间的拓扑。有以下相关问题。如果闭流形上的一个光滑结构支持一个负曲黎曼度量，那么是否所有其他光滑结构都支持这样一个度量？仿射几何的一个可能的应用是同胚完全闭仿射平坦流形是否必然是同胚的问题。 流形是局部类似于欧几里德几何空间的几何对象，但通常在全局上完全不同。例如，球的表面局部类似于平面;但仅在范围上是有限的。闭流形的意义是在范围上有限。光滑流形是没有角或边的流形。例如，球体的表面是光滑的，但立方体的表面虽然是流形，但并不光滑。然而，这两个流形是同胚的，即很容易在这两个曲面的点之间构造连续变化的双射对应。当流形上的点之间的距离也被考虑时，我们现在讨论的是黎曼流形，距离保持对应被称为等距。例如，鸡蛋和球表示球面上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，因为球是均匀圆的，而鸡蛋不是。然而，如果只考虑直线的概念（即测地线），我们谈论的是仿射流形和仿射等价。保持距离的对应关系将直线带到直线，但不一定反之亦然。

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.