No abstract

No abstract

Given any set Γ, let be the family of all finite subsets of . Let f:[0, ∞) → R satisfying: (1) f(x) = 0 if and only if x = 0, (2) f is increasing, (3) f(x + y) ≧ f(x) + f(y) for all x, y ≦ 0, and (4) f is continuous at zero from the right. Such an f is called a modules. Let C be the set of all moduli, and F = {fv ∊ C:v ∊ Γ). Q(Γ) will denote the set of all such F, s. For each F ∊ Q(Γ) letthe summation is taken over Γ, and setIf Γ is countable Q(Γ) will be denoted by Q and LΓ(F) by L(F). LetNote thatsee [4, 5 and 6].

Let h be a cyclic action of period n on M, where M is either S 1 x K, K is the Klein bottle or on SI~K, the twisted Klein bottle bundle over S t, such that there is a fibering q : M ~ S 1 with fiber a Klein bottle K or a torus T with respect to which the action is fiber preserving. We classify all such actions and show that they might be distinguished by their fixed points or by their orbit spaces.