Some Problems in High Dimensional Manifold Topology

dms - 0305423 f。Thomas farrell这个项目的目标是研究高维流形拓扑中的某些问题以及与黎曼几何和代数K-和l -理论的联系。Borel提出的一个基本问题是确定具有同构基群的闭非球流形是否同胚。这个问题导致了两类功能性质的猜想：同构猜想和纤维同构猜想。这些猜想对这个项目的大部分都是至关重要的。除此之外，我们还将讨论Borel的其他相关几何问题。这包括：一个广义的尼尔森问题；用近似颤振代替非球面流形的映射；确定与闭负弯曲流形同态的光滑流形是否支持负弯曲的黎曼度规；看看具有同构基群的完全紧仿射平面流形是否微分同构。流形是局部类似于欧几里德几何空间的几何对象；但在全球范围内可能会大不相同。大致来说，如果流形M的局部相似性由欧几里德空间E中的点到M中的点的连续对应p全局确定，则流形M是（强）非球面的。如果该对应将E中的某个球映射到M上，则M是封闭的（注意：当M是封闭的时，对应p永远不是一对一的）。M的基本群由保持对应p的E的所有运动组成。封闭非球面流形的一些例子是圆和甜甜圈的表面。在圆的情况下，对应的p将实数x发送到点(cos(x), sin(x))，并且圆的基本群由圆周长的整数倍的所有平移组成。莫比乌斯带是另一个非球面流形的例子；但它并没有关闭。另一方面，球体是一个流形，它不是非球面的。如果一对闭流形的点能被置于连续变化的一一对应中，则它们是同胚的。众所周知，对于1维或2维的非球面流形，Borel问题的答案是肯定的；但对于三维流形来说通常是未知的。

DMS-0305423F. Thomas FarrellThe goal of this project is to investigate certain problems in high dimensional manifold topology and connections with Riemannian geometry and algebraic K- and L-theories. A basic problem posedby Borel is to determine whether closed aspherical manifolds withisomorphic fundamental groups are homeomorphic. This problem has led to two families of conjectures of a functorial nature: theIsomorphism and the Fibered Isomorphism Conjectures. Theseconjectures are of central importance to much of this project. Besides Borel's other related geometric problems will be addressed.These include: a generalized Nielsen Question; replacing a map to an aspherical manifold by an approximate fibration; determiningwhether a smooth manifold homeomorphic to a closed negativelycurved manifold supports a negatively curved Riemannian metric;and seeing if complete compact affine flat manifolds with isomorphicfundamental groups are diffeomorphic.Manifolds are geometric objects which locally resemble the space ofEuclidean geometry; but could be quite different globally. Roughlyspeaking a manifold M is (strongly) aspherical if this local resemblanceis determined globally by a continuous correspondence p from the points in Euclidean space E to the points in M. And M is closed if thiscorrespondence maps some ball in E onto M. ( Caveat: The correspondence p is never one-to-one when M is closed.) The fundamental group of M consists of all the motions of E which preserve the correspondence p. Some examples of closed aspherical manifolds arethe circle and the surface of the doughnut. In the case of the circle, thecorrespondence p sends the real number x to the point (cos(x), sin(x)) and the fundamental group of the circle consists of all translations by anintegral multiple of the circumference of the circle. The Moebius band is another example of an aspherical manifold; but it is not closed. On the other hand, the sphere is a manifold which is not aspherical. A pair of closed manifolds are homeomorphic if their points can be placed in a continuously varying one-to-one correspondence. The answer to Borel's problem is well known to be Yes for aspherical manifolds of dimensions 1 or 2; but is in general unknown even for 3-dimensional manifolds.