Unlike in higher dimensions, most exotic phenomena on simply-connected 4-manifolds are unstable; they become non-exotic after finitely many stabilizations. While we now know that some of them survive one stabilization, nothing is known about their behavior when we stabilize them more than once. In this talk, we present the first example of an exotic diffeomorphism on a smooth contractible 4-manifold, given as a boundary Dehn twist along its (nontrivial) boundary, which stays exotic after two stabilizations. This is an ongoing joint work with JungHwan Park and Masaki Taniguchi.
