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

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

2005-02-14

arxiv.org/abs/math/0502293v2

Algebr. Geom. Topol. 5 (2005) 999-1026

Mathematics

Geometric Topology

Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol5/agt-5-41.abs.html

Scientific paper

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.

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