Extending Homeomorphisms over Compactifications
by
Jan J. Dijkstra
University of Alabama
Coauthors: Jan van Mill (Vrije Universiteit)

Every space is assumed to be separable and metric. The following problem is suggested by theorems of Engelking and Lelek: if X is a complete infinite-dimensional space and h: X --> X is a homeomorphism is it possible to put a bound on the dimension of the remainder of a compactification with the property that h is extendable to a homeomorphism from C to itself? Consider the topological Hilbert space s = R^Z and let \alpha stand for the coordinate shift on s.

Theorem 1. If C is a compactification of s such that \alpha extends to a continuous [(\alpha)\tilde] : C --> C, then C/s contains a strongly infinite-dimensional continua.

The question whether some of these continua must be Hilbert cubes is answered in the negative by

Theorem 2. If X is a complete space and h is an autohomeomorphism of X then there exists a compactification C of X such that h extends to an autohomeomorphism of C and every arc in C is contained in X.

Date received: March 10, 1997

Copyright © 1997 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 # caam-50.