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Compacta with the shape of finite complexes: a new look at the Edwards-Geoghegan-Wall obstruction
by
Craig R. Guilbault
University of Wisconsin-Milwaukee
In the 1970's D.A. Edwards and R. Geoghegan solved two fundamental problems in shape theory-both related to the issue of ``stability''. Roughly speaking, these problems ask when a ``bad'' space has the same shape as a ``good'' space. For simplicity, we focus on the following versions of these problems:
Problem A. Give necessary and sufficient conditions for a connectd finite dimensional pointed compactum Z to have the pointed shape of a CW complex.
Problem B. Give necessary and sufficient conditions for a connected finite dimensional pointed compactum Z to have the pointed shape of a finite CW complex.
One particularly nice version of the solution to Problem A states: Z has the pointed shape of a CW complex if and only if each of its homotopy pro-groups is stable. By combining the solution to Problem A with C.T.C. Wall's famous work on finite homotopy types, Edwards and Geoghegan then provided the following solution to Problem B: Z has the pointed shape of a finite CW complex if and only if each of its homotopy pro-groups is stable and an intrinsically defined Wall obstruction \omega( Z, z) in [K\tilde]0( \check\pi1( Z, z) ) vanishes. In order to understand Edwards and Geoghegan's solution to Problem B, it is then necessary to understand two things-the solution to Problem A, and Wall's work on the finiteness obstruction. In this talk, we will discuss a more direct, and (we believe) simpler solution to Problem B.
If time permits, we will take a quick look at the issues which led us reconsider Problem B. Let X be a connected non-compact finite dimensional complex which admits a Z-compactification [^X]=X \cup Z. If the Z-boundary Z is a ``bad'' space, one may wish to choose a new Z-compactification X \cup Z' where Z' is a ``good'' space. For example, one may ask whether X admits a Z-compactification with a CW-complex or a compact ANR as a Z-boundary. Although two Z-boundaries of the same space may be quite different, it is well known that they always have the same shape. Hence, this question is intimately tied to Problem B.
Date received: February 10, 2000
Copyright © 2000 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 # cady-40.