R^{4} is one of the most mysterious Euclidean spaces. It admits a continuum of smooth structures, yet if n is not 4, R^{n} has only one smooth structure. Another peculiarity is that R^{4} is the only Euclidean space for which we do not know the answer to the smooth Schoenflies problem. One rather naive approach to study the geometry of R^{4} is to ask which 3-manifolds appear as smooth submanifolds of R^{4}. Hantsche and Whitney were the first to make progress on this problem, roughly 70 years ago. Only a handful of people have made progress on this problem since. I will try to bring you up to date on what is known about this problem, going into details on some of the more easily understood constructions. |