CYCLIC GROUP ACTIONS ON Q AND HUREWICZ FIBRATION

For a compact metric space X, let G(X) be the space of self-homeomorphisms of X with the supremum metric. Let Q denote the Hilbert cube rr ~ 1 J., J. = [-1,1] and let 1 = 1 1 Gk(Q) c G(Q) denote the subspace consisting of all period k homeomorphisms, k> 1, each having a unique fixed point ° = (0,0,---). In this paper we show that every S E Gk(Q) is joined to the standard action a by a path A in Gk(Q), and that it induces a Hurewicz fibration p: E ~ [0,1] with E a Q-manifold and the fibers p-l(t) being the orbit spaces of the non-degenerate orbits of A(t). The formulation is motivated by the following theorem of Chapman-Ferry: Theorem ([C-F]). Let p: E ~ [0,1] be a Hurewioz fiber-1 map with E a Q-manifold. If the fibers p (t) are oompaot Q-manifolds ~ then p is a trivial bundle map. To state our result more precisely, consider a map A: [0,1] ~ Gk(Q). A induces a level-preserving (l.p.) homeomorphism H: Q x [0,1] ~ Q x [0,1] by HIQx{t} = A(t). Let E denote the orbit space of non-degenerate orbits of H and let E c E be the orbits at level t. Define p: E ~ [0,1]t by p(E) = t.