Upper Cones As Automorphism Bases

It is shown that the complete Turing degrees do not form an automorphism base. A class A ⊆ the Turing degrees D is an automorphism base (see Lerman [1983]) if and only if any nontrivial automorphism of D necessarily moves at least one of its elements — or, equivalently, the global action of any such automorphism is completely determined by that on A . Jockusch and Posner [1981] demonstrated the existence of a wide range of automorphism bases, and subsequent work of a number of people led eventually to Slaman and Woodin’s discovery (private communication) of upper and lower cones (in fact singletons) which are automorphism bases for the global structure. In fact, Ambos-Spies [ta] showed every nontrivial ideal of computably enumerable (c.e.) degrees to be an automorphism base, while on the other hand there were many upper cones known to be very far from being automorphism bases in that they were rigid in D — that is, all their members were invariant in D (in the sense of Rogers [1967]). The strongest such result was that of Slaman and Woodin (see Nies, Shore and Slaman [ta]): D(≥ 0) is rigid in D . Moreover, 0 turned out to be definable in D and hence invariant (see Cooper [ta1]). Below, a nontrivial Turing automorphism is constructed which only moves degrees within their atomic jump classes. The main consequence of this is that the complete Turing degrees do not form an automorphism base for D . ‡ We would like to acknowledge helpful conversations with G. E. Sacks concerning the degrees of Turing automorphisms, made possible by E.P.S.R.C. Research Grant no. GR/L63396. 1991 Mathematics Subject Classification. Primary 03D25, 03D30; Secondary 03D35.