Surgical Methods in Rigidity

[701746]法雷尔F. T.法雷尔与纽约州立大学石溪分校的L. E.琼斯合作，正在研究流形的结构，特别强调非球面流形。一个激励问题是Borel猜想，它假设了闭合非球面流形的拓扑刚性。他们还在额外的几何假设下研究了相关的光滑刚性问题。特别是，正在研究的关系之间的外科和谐波映射方法的刚性问题。如果一个几何物体不能变形，它就是刚性的。例如，球体不改变曲率就不能变形。在过去的35年中，人们对以非正曲率几何物体为中心的刚性问题产生了浓厚的兴趣。（欧几里得平面为零曲率，球面为正曲率，非欧几里得平面为负曲率。）直到最近，用于研究这些问题的技术主要来自分析（微积分）以及微分和综合几何。在过去的15年里，从剪切和粘贴拓扑（外科理论）的技术已经找到了这些问题的应用。法雷尔和琼斯从拓扑学中解决刚性问题，但也使用了几何学中的技术。* * *

9701746 Farrell F. T. Farrell, in collaboration with L. E. Jones of SUNY-Stony Brook, is investigating the structure of manifolds, with particular emphasis on aspherical manifolds. A motivating problem is Borel's Conjecture, which posits topological rigidity for closed aspherical manifolds. They also study related smooth rigidity questions under extra geometric assumptions. In particular, the relationship is being studied between the surgical and harmonic map approaches to rigidity problems. A geometric object is rigid if it can't be deformed. For example, a sphere cannot be deformed without changing its curvature. During the last 35 years, there has been intense interest in rigidity questions centering on geometric objects with non-positive curvature. (The Euclidean plane has zero curvature, the sphere has positive curvature, and the non-Euclidean plane has negative curvature.) The techniques used until recently to investigate these questions have mostly come from analysis (calculus) together with differential and synthetic geometry. In the last 15 years, techniques from cut and paste topology (surgery theory) have found application to these questions. Farrell and Jones approach rigidity problems from topology but also use techniques from geometry. ***