Universal Covers of Aspherical Manifolds with Geometric Fundamental Groups
by
Craig Guilbault
University of Wisconsin-Milwaukee
Coauthors: Fredric Ancel

Suppose an open n-manifold M may be compactified to an ANR M* so that M*-M is a Z-set in M*. We show that (when n>4) the double of M* along its "Z-boundary'' is an n-manifold. More generally, if M and N each admit Z-compactifications with homeomorphic Z-boundaries, then their union along this common boundary is an n-manifold. This result is used to show that in many cases Z-compactifiable manifolds are determined by their Z-boundaries. For example, contractible open n-manifolds (n>4) with homeomorphic Z-boundaries are homeomorphic. As an application, some special cases of a weak Borel Conjecture are verified. Specifically, it is shown that closed aspherical n-manifolds (n>4) having isomorphic fundamental groups which are either word hyperbolic or CAT(0) have homeomorphic universal covers.

Date received: April 13, 1998