Complements of tori and Klein bottles in the 4–sphere that have hyperbolic structure

Abstract

Many noncompact hyperbolic 3–manifolds are topologically complements of links in the 3–sphere. Generalizing to dimension 4, we construct a dozen examples of noncompact hyperbolic 4–manifolds, all of which are topologically complements of varying numbers of tori and Klein bottles in the 4–sphere. Finite covers of some of those manifolds are then shown to be complements of tori and Klein bottles in other simply-connected closed 4–manifolds. All the examples are based on a construction of Ratcliffe and Tschantz, who produced 1171 noncompact hyperbolic 4–manifolds of minimal volume. Our examples are finite covers of some of those manifolds.