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

Ivanšić, Dubravko; Ratcliffe, John G; Tschantz, Steven T

doi:10.2140/agt.2005.5.999

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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

DOI: 10.2140/agt.2005.5.999

arXiv: math.GT/0502293

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.

Keywords

hyperbolic 4–manifolds, links in the 4–sphere, links in simply-connected closed 4–manifolds

Mathematical Subject Classification 2000

Primary: 57M50, 57Q45

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Publication

Received: 1 March 2005

Revised: 28 June 2005

Accepted: 18 July 2005

Published: 18 August 2005

Authors

Dubravko Ivanšić
Department of Mathematics and Statistics Murray State University Murray KY 42071 USA

John G Ratcliffe
Department of Mathematics Vanderbilt University Nashville TN 37240 USA

Steven T Tschantz
Department of Mathematics Vanderbilt University Nashville TN 37240 USA

Volume 5, issue 3 (2005)