Triviality of simple fiber-preserving actions of tori on Hilbert-cube-fiber bundles

Abstract

Let σ \sigma denote the standard based-free action of a torus T T on the Hilbert cube Q Q . It will be shown that every simple fiber-preserving action of T T on Q × B Q \times B , where B B is a finite-dimensional, compact metric absolute retract, is fiber-preserving equivalent to the diagonal action σ × i d B \sigma \times {\text {i}}{{\text {d}}_B} .