A well known principle in 4-dimensional topology asserts that homeomorphic smooth simply-connected 4-manifolds become diffeomorphic after stabilizing (connected summing with S^2 x S^2) sufficiently many times. Many explicit examples are known that require at least one stabilization, but surprisingly none of these have been shown to require more than one. Perhaps this "exotic" behavior always "dissolves" after a single stabilization.

An analogous principle holds for 2-spheres embedded in a simply-connected 4-manifold, namely, any two that are topologically isotopic become smoothly isotopic after a sufficient number of stabilizations. Until now, however, bounds on this number have not been found for any examples. In this talk, we show how to construct examples that require exactly one stabilization. This is joint work with Dave Auckly, Hee Jung Kim and Danny Ruberman.