Producing Essential 2-Spheres
by
Jim Hoffman
Idaho State University
Coauthors: Daniel Matignon

Let M be an orientable 3-manifold M whose boundary is a torus, and which does not contain an essential 2-sphere. The goal is to minimize the number of slopes on the boundary of M which produce essential 2-spheres by Dehn filling, via their minimal geometric intersection number. Earlier papers in this direction are by Boyer and Zhang, Gordon and Luecke, and Wu. In 1994, Boyer and Zhang proved that the slopes on the boundary of M intersect exactly once, using the machinery of Culler and Shalen (the varieties of characters). In 1996, Gordon and Luecke gave another proof using their own machinery (the representations of types which come from the intersection of planar graphs).

This paper gives yet another proof of this result. It uses the combinatorics of intersecting planar graphs, yet avoids using the representations of all types. The combinatorial aspects of this paper are very basic.

Keywords: Dehn filling, intersection graphs, reducibility.

AMS classification: 57N10, 57M25, 57M15.

Date received: January 13, 1999

Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caca-08.