Equivariant hyperspaces and Banach-Mazur compacta
by
Sergey Antonyan
Universidad Nacional Autonoma de Mexico

Let G be a compact Lie group, X be a metric G-space and exp X be the hyperspace of all nonvoid compact subsets of X endowed with the Hausdorff metric topology and with the induced action of G. In this work we prove that the following three assertions are mutually equivalent: (a) X is locally continuum-connected (resp., connected and locally continuum-connected); (b) exp X is a G-ANR (resp., a G-AR); (c) the orbit space (exp X)/G is an ANR (resp., an AR). This is applied to obtain the following generalization of the classical Curtis-Schori-West Hyperspace Theorem: If P is a compact metric G-space on which G acts continuously and nontransitively, then the orbit space (exp P)/G is a Hilbert cube iff P is a nondegenerate Peano continuum. By applying these results to the Banach-Mazur compacta we establish in particular that for every n >= 2, the Banach-Mazur compactum BM(n) is homeomorphic to the orbit space (exp S^n-1)/O(n). Furthermore, BM(2n+1) is homeomorphic to (exp S²ⁿ)/SO(2n+1) for all n >= 1. Other related results and some open problems will be discussed as well.

Date received: June 23, 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 # caeu-47.