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

Dubravko Ivanšić, John G Ratcliffe and Steven T Tschantz

Algebraic & Geometric Topology 5 (2005) 999–1026

arXiv: math.GT/0502293


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.

hyperbolic 4–manifolds, links in the 4–sphere, links in simply-connected closed 4–manifolds
Mathematical Subject Classification 2000
Primary: 57M50, 57Q45
Forward citations
Received: 1 March 2005
Revised: 28 June 2005
Accepted: 18 July 2005
Published: 18 August 2005
Dubravko Ivanšić
Department of Mathematics and Statistics
Murray State University
Murray KY 42071
John G Ratcliffe
Department of Mathematics
Vanderbilt University
Nashville TN 37240
Steven T Tschantz
Department of Mathematics
Vanderbilt University
Nashville TN 37240