Z-compactifications of complexes
by
Craig Guilbault
University of Wisconsin-Milwaukee

A closed subset A of a compact ANR X is a Z-set if there is a homotopy H:X×I --> X with H₀=id_X and H_t(X) \cap A=\emptyset for all t > 0. For example, if M is a compact manifold, then \partialM is a Z-set in M. A compactification [^Y] of a noncompact ANR Y is called a Z-compactification if [^Y]-Y is a Z-set in [^Y]. In this case we call [^Y]-Y a Z-boundary for Y.

These concepts, which played a crucial role in the development of geometric topology in the 1970's, have undergone a recent rebirth due to their importance in geometric group theory and manifold topology. For example, work by Bestvina and Mess shows that if K is a finite K(G, 1) and G is a negatively curved or CAT(0) group, then the universal cover [K\tilde] may be Z-compactified. Bestvina has used this idea to extend the notion of ``boundary of a group'' to larger classes of groups.

We will discuss recent progress on some old and new problems which revolve around the question: When is a locally finite complex Z-compactifiable?

Date received: February 10, 1999

Copyright © 1999 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 # caca-31.