Z-COMPACTIFICATIONS OF OPEN MANIFOLDS

Suppose an open n-manifold Mn may be compactified to an ANR Mn so that Mn−Mn is a Z-set in Mn. It is shown that (when n⩾5) the double of Mn along its “Z-boundary” is an n-manifold. More generally, if Mn and Nn each admit 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 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.