Mathematical Sciences: Topological Versus Smooth Rigidity

闭非球面流形是一种紧致连通流形，它的万能覆盖是可收缩的。设M和N是闭非球面流形，F是它们的基本群的同构。Borel猜想F总是由流形的同胚f诱导的。这位研究人员与纽约州立大学石溪分校的L.E.Jones合作，打算试图解决这一猜测。调查人员和琼斯的部分结果是前一次国家安全局拨款(dms-8801312)的主要结果。该项目还涉及到微分几何和代数K-理论中的问题；例如，WH(G)是否对每个无挠群G都为零？流形是局部欧几里得空间，这使得它们成为非常自然的研究对象。例如，我们生活的物理世界有三个空间维度和一个时间维度(在某些情况下)，这使得它要么是三维的，要么是四维的流形。因此，更好地理解流形结构的即时计划甚至可能具有宇宙学意义。

A closed aspherical manifold is a compact connected manifold whose universal cover is contractible. Let M and N be closed aspherical manifolds, and let F be an isomorphism of their fundamental groups. Borel conjectured that F is always induced by a homeomorphism f of the manifolds. The investigator, in collaboration with L. E. Jones of SUNY-Stony Brook, intends to try to settle this conjecture. Partial results by the investigator and Jones were the main results of the previous N.S.F. grant (DMS-8801312). The project also relates to problems in differential geometry and algebraic K-theory; e.g., does Wh(G) vanish for every torsion-free group G? Manifolds are locally Euclidean spaces, which makes them very natural objects of study. For example, the physical world in which we live has three space dimensions and (in some contexts) a time dimension, making it either a three-dimensional or four-dimensional manifold. Thus the instant project to understand better the structure of manifolds might even have cosmological significance.