Abstract: 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". An important principle discovered by Wall in the 1960s states that all exotic phenomena on 4-manifolds will disappear after sufficiently many stabilizations (i.e. connected sum with the product of two spheres). Since then, it has been a long-standing open question whether there exists a pair of homeomorphic simply-connected 4-manifold that are not diffeomorphic after one stabilization. Although we are still not able to solve this problem, in this talk we will present a solution of two variations: (1) There exists a pair of diffeomorphisms on a 4-manifold that are topologically isotopic but not smoothly isotopic even after a 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 a stabilization. (based on joint work with Anubhav Mukherjee).
