Cubes with Knotted Holes

摘要

The 3-dimensional Poincare conjecture is that a compact, connected, simply connected 3-manifold without boundary is topologically a 3-sphere S sup 3. Despite efforts to prove the conjecture, it has withstood attack. It is known that every orientable 3-manifold may be obtained by removing a collection of disjoint solid tori from S sup 3 and sewing them back differently. In this paper the author examine some of the possibilities for constructing a counterexample to the Poincare conjecture by removing a single solid torus from S sup 3 and sewing it back differently. Actually, they examine not only this process but one analogous to it which they call 'attaching a pillbox to a cube with a knotted hole.' (Author)