Some Problems in Topological Rigidity

DMS-9987185F。这个项目的目标是解决关于流形结构的一些问题。流形是局部类似于欧几里得几何空间的几何对象，但在全局上可能完全不同。例如，在小范围内，地球表面可以被认为是欧几里得平面。但在大范围内，在大气层上对其进行建模更为有用。发现全球差异的一种方法是长途跋涉，回来看看是否有什么变化。例如，如果你绕着Moebius带的表面走，你会回到相反的一边。所有行进路线的集合(直到连续变形)形成一个称为流形的Poincare群的代数对象。近50年前由A.Borel提出的一个基本猜想是，闭非球面流形的庞加莱群决定了这个流形是连续的双射对应。封闭意味着没有边界，但范围有限。例如，莫比乌斯带不是闭合的，而是球体闭合的。非球面是指球体的任何连续图像都以球的图像为界。在这个定义中，必须允许尺寸大于两个的球体以及相应的更高维度的球体。例如，对于甜甜圈的表面，Borel猜想是正确的。然而，球体本身显然不是非球面的。法雷尔与L.E.琼斯合作，验证了博雷尔猜想的一些重要情况，并将继续对其进行研究。一个有趣的问题是，是否存在一个“所有可能中最好的”地图，给出了连续的双射。人们曾经认为，在两个流形都是负曲线的情况下，唯一的调和同伦等价做到了这一点。法瑞尔、琼斯和P·翁塔内达最近表明，通常情况并非如此。(Farrell，Ontaneda和M.S.Raghunathan推广了这一工作，即使是对微分同胚流形也给出了这样的例子：即与微分同胚同伦的调和映射不是单叶的。)但这种调和映射是否总是细胞的，目前还是个未知数。请注意，如果是这样的话，这将与庞加莱猜想相矛盾。另一个要研究的问题是，如果Poincare群有平凡的中心，那么Poincare群的任何有限对称群是否提升到闭非球面流形的对称群。这个问题的主旨是找到它是真的、有趣的案例。例如，曲面的情况就是科克霍夫解决的经典尼尔森问题。Farrell和Jones已经在这个问题上得到了其他一些积极的部分结果。在关于这些几何问题的工作中，Farrell和Jones提出了关于任意群环的代数K和L理论的结构的猜想。他们将继续验证这些猜想的工作。同样，这一努力的主旨是为感兴趣的群体提供积极的解决方案

DMS-9987185F. Thomas FarrellThe goal of this project is to resolve some questions about the structure of manifolds. Manifolds are geometric objects which locally resemble the space of Euclidean geometry; but could be quite different globally. For example the surface of the earth, on a small scale, can be thought of as Euclid's plane. But on a large scale, it is more useful to model it on thesphere. One way to detect global differences is to travel long distances, return and see if something has changed. For example, if you walk around the surface of a Moebius band, you'll return on the opposite side. The collection of all travel routes (up to continuous deformation) forms an algebraic object called the Poincare group of the manifold. A basic conjecture formulated almost 50 years ago by A. Borel is that the Poincare group of a closed aspherical manifold determines this manifold up to a continuous bijective correspondence. Closed means without boundary but finite in extent. The Moebius band, for example, is not closed but the sphere is. Aspherical means that any continuous image of a sphere bounds the image of a ball. Spheres of dimension bigger than two and correspondingly higher dimensional balls must be allowed in this definition. For example, the Borel conjecture is true for the surface of a doughnut. However, the sphere itself is "clearly" not aspherical.Farrell in collaboration with L.E. Jones have verified some important cases of Borel's conjecture and will continue to work on it. One interesting question is whether there is a "best of all possible" maps giving the continuous bijection. It was once thought that the unique harmonic homotopyequivalence does this in the case both manifolds are negatively curved. Farrell, Jones, and P. Ontaneda have recently shown that this is not so in general. (Farrell,Ontaneda and M.S. Raghunathan extending this work have just produced such examples even with pairs of diffeomorphic manifolds: i.e. the harmonic map homotopic to the diffeomorphism is not univalent.) But it is still open whether this harmonic map is always cellular. Note if so this would contradict the Poincare Conjecture. Another problem to be investigated is whether any finite group of symmetries of the Poincare group lifts to a group of symmetries of the closed aspherical manifold provided the Poincare group has trivial center. The thrust of this problem is to find interesting cases where it is true. The case of surfaces is, for example, the classical Nielsen problem solved by Kerkhoff. Farrell and Jones have already obtained some other positive partial results on this problem. In their work on these geometric problems, Farrell and Jones have formulated conjectures about the structure of the algebraic K and L theories of an arbitrary group ring. They will continue their work on verifying these conjectures. Again the thrust of this effort is to give positive solutions for interesting groups