The Bing Involution
In 1952 R.H. Bing published wild involution (it is an orientation-reversing homeomorphism which squares to the identity) of the three sphere, S^3. This example started a revolution in decomposition space theory which led to the solutions of the double-suspension problem (Edwards and Cannon in sperate work), the 4D Poincare conjecture, and the modern understanding of homology manifolds. The involution is wild in that there is no coordinate system on S^3 in which it appears smooth ( C^1). But how wild is it? Analysists have asked if it could be compatible with a Lipschitz or quasi-conformal structure. Recent work with Mike Starbird arXiv:2209.07597 shows that any topological conjugate of Bings Involution must have at least a “nearly exponential modulus of continuity”; it is far from even being Holder continuous. I will explain the involution and give an idea of the proof.