Exotic Phenomena in Dimension Four

This thesis is a comparison of the smooth and topological categories in dimension 4. We first discuss lists of homeomorphic 4-manifolds with non-equivalent “exotic” smooth structures, and show that any finite list of smooth, closed, simply-connected 4-manifolds that are homeomorphic to a given one X can be obtained by removing a single compact contractible submanifold (or cork) from X, and then regluing it by powers of a boundary diffeomorphism. Furthermore, by allowing the cork to be noncompact, the collection of all the smooth manifolds homeomorphic to X can be obtained in this way. The existence of a universal noncompact cork is also established. The results in the second half of the thesis illustrate the difference between the smooth isotopy, smooth equivalence (by a diffeomorphism), and topological isotopy of smoothly embedded surfaces in a 4-manifold. We construct infinitely many smooth oriented 4-manifolds containing pairs of homotopic, smoothly embedded 2-spheres that are not topologically isotopic, but that are equivalent by an ambient diffeomorphism inducing the identity on homology. These examples show that Gabai’s recent “Generalized” 4D Lightbulb Theorem does not generalize to arbitrary 4-manifolds. In contrast, we also show that there are smoothly embedded 2-spheres that are both equivalent and topologically isotopic, but not smoothly isotopic