On iterated inverse limits
by
Sibe Mardešić
University of Zagreb

It is well known that every compact Hausdorff space is the limit of an inverse system of compact ANR's (polyhedra). If X is the limit of a (cofinite) inverse system \boldsymbol X=(X_\lambda, p_{\lambda\lambda'}, \Lambda) of compact Hausdorff spaces X_\lambda and every X_\lambda is the limit of an inverse system \boldsymbol Y_\lambda = (Y_\lambda^\mu, q_\lambda^\mu\mu', M_\lambda) of compact ANR's (polyhedra), it is natural to ask if one can organize the collection {Y_\lambda^\mu, \lambda in \Lambda, \mu in M_\lambda} of ANR's in such a way that one obtains an inverse system whose limit is X. This question was raised by Yu.T. Lisica during the Dubrovnik 1998. Topology conference.

Lisica's question has a negative answer. This can be seen by taking a compact Hausdorff space X with dim X=1 and ind X=2 and a system \boldsymbol X of metric compacta of dimesion 1 with lim\boldsymbol X=X and by expressing each X_\lambda as the limit of a sequence of 1-dimensional compact polyhedra. Nevertheless, for every cofinite system \boldsymbol X there exist systems \boldsymbol X_\lambda of compact ANR's with limit X_\lambda, whose terms can be organized in a system with limit X. Using resolutions instead of limits, this positive result generalizes to arbitrary topological spaces X. The results of this talk are contained in a forthcomming joint paper with N. Uglesi\'c.

Date received: March 11, 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 # cadz-55.