Z-compactifications of ANRs
by
Craig R. Guilbault
University of Wisconsin-Milwaukee

A closed subset A of a compact ANR X is a Z-set if for every open set U of X, the inclusion of U into U is a homotopy equivalence. For example, if M is a manifold with boundary, then bdry(M) is a Z-set in M. If Y is a non-compact ANR, a Z-compactification of Y is a compact ANR, Y*, containing Y as an open subset and having the property that Y*-Y is a Z-set in Y*. In 1976, Chapman and Siebenmann gave a beautiful characterization of those Hilbert cube manifolds which may be Z-compactified. In their paper, they asked whether their conditions, when applied to an arbitrary locally compact ANR, guarantee the existence of a Z-compactification. Combining their results with a theorem of R.D. Edwards, one arrives at the equivalent question: If Y is a locally compact ANR and YxQ is Z-compactifiable (Q denotes the Hilbert cube), must Y be Z-compactifiable? We will discuss this problem.

Date received: June 3, 1998

Copyright © 1998 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 # cabi-13.