LAUSR

We show that for every sequence $$(n_i)$$(ni), where each $$n_i$$ni is either an integer greater than 1 or is $$\infty $$∞, there exists a simply connected open 3-manifold M with a countable dense set of ends $$\{e_i\}$${ei} so that, for every i, the genus of end $$e_i$$ei is equal to $$n_i$$ni. 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 $$S^3$$S3. The methods used require careful analysis of the genera of ends and new techniques for dealing with infinite genus.