Surgical Methods in Rigidity

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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. ***