Exotic phenomena on 4-manifolds that survive a stabilization

Starting in dimension 4, there is a significant difference between the category of smooth manifolds and the category of topological manifolds. Such phenomena are called the "exotic phenomena". In dimension 4, there is an extra complication due to the failure of the h-cobordism theorem (in the smooth category). Stabilization on 4-manifolds means doing connected-sum with S2 cross S2. This operation naturally appears when one tries to adapt the proof of h-cobordism theorem in dimension 4. In the 1960s, Wall discovered an important principle: all exotic phenomena on orientable 4-manifolds will eventually disappear after sufficiently many stabilizations. Since then, it has been a fundamental problem to search for exotic phenomena that survive one stabilization. In this talk, we will discuss relevant backgrounds and show that such phenomena actually exist by proving the following two results: (1) There exists a pair of diffeomorphisms on a 4-manifold that are topologically isotopic but not smoothly isotopic even after one stabilization. (2) There exists a pair of properly embedded surfaces in a 4-manifold with boundary which are topologically isotopic but not smoothly isotopic even after one stabilization (a part of the talk is based on the joint work with Anubhav Mukherjee).