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

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