Classifying spaces for free actions, and the Hilbert–Smith conjecture

Abstract

T. It is shown that any free action of a zero-dimensional compact group G on the О·-dimensional Menger compactum Mn is О·-universal for free actions, and that the orbit space MnjG is В«-classifying. Nonexistence of equivariant mappings between Mn+m and Mn implies that the orbit space R/Ap has infinite dimension, where R is any compact ANR-space with free action of the group Ap of p-adic integers. Knowledge of such nonexistence would then permit proof of the Hilbert-Smith conjecture under the assumption of finite dimensionality for the orbit space.