The Geometry of the Bing Involution

by Mike Freedman (Microsoft)

One of the great examples in topology is Bing's 1952 wild involution of the 3-sphere. It opened a large field of investigation touching both on the double suspension and 4D Poincare conjectures. Mike Starbird and I have been trying to understand the analytic properties of its conjugacy class for 40 years. Our September arXiv posting gives reasonably tight bounds on both the modulus of continuity and conformal distortion, showing that the involution cannot be made Lipschitz or even quasi-conformal. I will explain the involution, and what we know about it in the colloquium. In subsequent seminars Mike Starbird and I will explain the proof.