Hyperbolic structure on a complement of tori in the 4-sphere
by
Dubravko Ivansic
The George Washington University

It is a familiar fact that many links in the 3-sphere have a complement that allows a hyperbolic structure. We generalize this phenomenon to one dimension higher by displaying a finite-volume noncompact hyperbolic 4-manifold M that is topologically the complement of 5 tori in the 4-sphere. The example stems from the work of J. Ratcliffe and S. Tschantz, who have used a computer to construct, via side-pairings of a polyhedron, many hyperbolic 4-manifolds. The manifold M is the orientable double cover of one of their manifolds.

Using the construction of M and the fact that it is a complement in the 4-sphere we obtain a complicated Kirby diagram of the 4-sphere with dozens of 1-, 2- and 3-handles. While this diagram is equivalent via Kirby moves to the diagram of the standard differentiable 4-sphere, its complicatedness seems to suggest that any counterexample to the differentiable Poincare conjecture in dimension 4 given by a Kirby diagram could be very involved indeed.

Date received: February 20, 2002