Compactifications of finite-dimensional manifolds
by
Craig Guilbault
University of Wisconsin-Milwaukee

A closed subset A of a compact ANR X is a Z-set if either of the following equivalent conditions is satisfied:

• There is a homotopy H:X×I --> X with H₀=id_X and H_t(X) \cap A=\emptyset for all t > 0.
• For every open set U of X, U\A\hookrightarrow U is a homotopy equivalence.

A Z-compactification of a noncompact ANR Y is a compactum [^Y] containing Y as an open subset and having the property that [^Y]-Y is a Z-set in [^Y]. These compactifications play and important role in geometric topology and in geometric group theory, where the universal cover of a K(G, 1) if often Z-compactified by adding on the group boundary.

In 1976 Chapman and Siebenmann gave necessary and sufficient conditions for a Hilbert cube manifold to be Z-compactifiable. It is still not known whether these conditions are sufficient to ensure Z-compactifiability for finite dimensional manifolds. Equivalently, one may ask:

Question. If M is a finite dimensional manifold and M×I^\infty is Z-compactifiable, must M be Z-compactifiable?

We will discuss the following result which relies on recent work by Guilbault and Tinsley together with a variation on G. O'Brien's manifold completion theorem.:

Theorem. If M is an n-dimensional open manifold (n >= 5) and M×I^\infty is Z-compactifiable, then M×I is Z-compactifiable.

Date received: February 28, 2003

Copyright © 2003 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 # cakw-01.