Simply Connected 3-Manifolds with a Dense Set of Ends of Specified Genus

We show that for every sequence (n(i)), where each n(i) is either an integer greater than 1 or is infinity, there exists a simply connected open 3-manifold M with a countable dense set of ends {e(i)} so that, for every i, the genus of end e(i) is equal to n(i). In addition, the genus of the ends not in the dense set is shown to be less than or equal to 2. These simply connected 3-manifolds are constructed as the complements of certain Cantor sets in S-3. The methods used require careful analysis of the genera of ends and new techniques for dealing with infinite genus.