Index sets and parametric reductions

We investigate the index sets associated with the degree structures of computable sets under the parameterized reducibilities introduced by the authors. We solve a question of Peter Cholak and the first author by proving the fundamental index sets associated with a computable set A, {e : We ≤q A} for q ∈ {m, T } are 0 4 complete. We also show that FPT (≤q), that is {e : We computable and ≡q ∅}, is 0 4 complete. We also look at computable presentability of these classes.