The knot engulfing property of 3-manifolds.

Abstract

The Freudenthal compactification of a 3-manifold M is a compactification M′ such that the end space e(M)=M′∖M is a totally disconnected compact set. The topological uniformization conjecture states that the Freudenthal compactification of the universal covering of a connected, closed 3-manifold is S3 with its end space tamely embedded. A 3-manifold M has the knot engulfing property if every contractible polyhedral simple closed curve in M lies in some polyhedral 3-cell in M. A classical result by Bing (1958) is that the only simply connected closed 3-manifold with the knot engulfing property is S3. The author proves a generalization of Bing's result: A simply connected 3-manifold M has the knot engulfing property if and only if the Freudenthal compactification M′ of M is homeomorphic to S3 with the end space e(M) tamely embedded. Thus the topological uniformization conjecture can be rephrased as stating that the universal covering of a connected, closed 3-manifold has the knot engulfing property.