On the jump classes of noncuppable enumeration degrees

Abstract. We prove that for every Σ02 enumeration degree b there exists a noncuppable Σ02 degree a> 0e such that b ′ ≤e a ′ and a′′≤e b′′. This allows us to deduce, from results on the high/low jump hierarchy in the local Turing degrees and the jump preserving properties of the standard embedding ι: DT → De, that there exist Σ02 noncuppable enumeration degrees at every possible—i.e. above low1—level of the high/low jump hierarchy in the context of De. 1