T-degrees, jump classes, and strong reducibilities

It is shown that there exist r.e. degrees other than 0 and $\mathbf{0}^{\prime}$ which have a greatest r.e. $1$-degree. This solves an old question of Rogers and Jockusch. We call such degrees $1$-topped. We show that there exist incomplete $1$-topped degrees above any low r.e. degree, but also show that no nonzero low degree is $1$-topped. It then follows by known results that all incomplete $1$-topped degrees are low$_{2}$ but not low. We also construct cappable nonzero $1$-topped r.e. degrees and examine the relationships between $1$-topped r.e. degrees and high r.e. degrees.

Finally, we give an analysis of the ``local'' relationships of r.e. sets under various strong reducibilities. In particular, we analyze the structure of r.e. ${\text{wtt-}}$ and ${\text{tt}}$-degrees within a single r.e. ${\text{T}}$-degree. We show, for instance, that there is an r.e. degree which contains a greatest r.e. ${\text{wtt-}}$-degree and a least r.e. ${\text{tt}}$-degree yet does not consist of a single r.e. ${\text{wtt}}$-degree. This depends on a new construction of a nonzero r.e. ${\text{T}}$-degree with a least ${\text{tt}}$-degree, which proves to have several further applications.