#### Algebraic and Geometric Topology 5 (2005),
paper no. 41, pages 999-1026.

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

### Dubravko Ivansic, John G. Ratcliffe and Steven T. Tschantz

**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

**AMS subject classification**.
Primary: 57M50, 57Q45.

**E-print:** `arXiv:math.GT/0502293`

**DOI:** 10.2140/agt.2005.5.999

Submitted: 1 March 2005.
(Revised: 28 June 2005.)
Accepted: 18 July 2005.
Published: 18 August 2005.

Notes on file formats
Dubravko Ivansic, John G. Ratcliffe and Steven T. Tschantz

DI: Department of Mathematics and Statistics

Murray State University, Murray, KY 42071, USA

JR and ST: Department of Mathematics, Vanderbilt University

Nashville, TN 37240, USA

Email: dubravko.ivansic@murraystate.edu, ratclifj@math.vanderbilt.edu, tschantz@math.vanderbilt.edu

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