Mathematical Conversations

The idea of corrugation goes back to Whitney, who proved that homotopy classes of immersed curves in the plane are classified by their rotation number. Generalizing this result, Smale and Hirsch proved that the space of immersions of a manifold X into a manifold Y is (weakly) homotopy equivalent to the space of injective bundle maps from TX to TY, whenever dim(X) < dim(Y). One obtains surprising consequences, such as Smale's eversion of the sphere. The key insight is that one can wiggle X in the extra dimensions dim(Y)-dim(X), using the extra room for the corrugation. When dim(X)=dim(Y) there is no extra room and we cannot hope for such a result. For example, every real valued function on a circle has at least two critical points. Nevertheless, one can always wrinkle X back and forth upon itself to create the extra room. This inevitably produces some singularities, namely folds along the wrinkle, however these singularities are very simple. This basic principle underlies deep results of many mathematicians, including M. Gromov, Y. Eliashberg, N. Mishachev, K. Igusa and E. Murphy among others. In this talk we will learn how to wiggle and wrinkle manifolds, with a focus on folded mappings from the sphere to the plane. This will lead us to consider the surgery of folds and a beautiful picture discovered by J. Milnor.