Some Problems in High Dimensional Manifold Topology

DMS-0305423F。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理论和L理论的联系。Borel提出的一个基本问题是确定基本群同构的闭非球面流形是否是同胚的。这个问题导致了两类函数性猜想：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。除了Borel的其他相关几何问题外，还将讨论其他相关的几何问题。这些问题包括：一个推广的Nielsen问题；用一个近似的纤维代替一个映射到一个非球面流形；确定一个同胚于一个闭的负曲线流形的光滑流形是否支持一个负弯曲的黎曼度量；以及是否具有同构的基本群的完全紧致仿射平坦流形是微分同构的。流形是局部类似于欧几里得几何空间的几何对象；但在全局上可能有很大的不同。粗略地说，流形M是(强)非球面的，如果这种局部相似是由从欧氏空间E中的点到M中的点的连续对应p全局确定的，并且如果这种对应将E中的某个球映射到M上，则M是闭的。(注意：当M是闭的时，这种对应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。闭非球面流形的一些例子是圆和环面。在圆的情况下，响应p将实数x送到点(cos(X)，sin(X))，并且圆的基本群由圆的周长的整数倍的所有平移组成。莫比乌斯带是非球面流形的另一个例子；但它不是闭合的。另一方面，球面是一个非非球面的流形。一对闭流形是同胚的，如果它们的点可以以连续变化的一一对应关系放置。对于1维或2维的非球面流形，Borel问题的答案是众所周知的是肯定的；但即使对于3维流形，一般情况下也是未知的。

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.