Title:

The Local Structure of the Enumeration Degrees

This thesis discusses properties of the local structure of the enumeration degrees. We
begin with some historical background of the subject. vye give motivation for investigating
the properties of the local structure ofthe enumeration degrees and discuss the
basic concepts and methods used throughout the thesis.
Chapter 2 presents evidence that the study of the structure of enumeration degrees
can provide a richer understanding of the structure of the Thring degrees. We prove that
 '  .
there exists a II~ enumeration degree which is the bottom of a cone within which the II~
enumeration degrees cannot be cupped to O~. As a corollary we obtain a generalization
of Harrington's nonsplitting theorem for the ~g Thring degrees.
Chapters 3 and 4 are dedicated to the study of properties, specific to the properly
~g enumeration degrees. In Chapter 3 we construct a properly ~g enumeration degree
above which there is no splitting of O~. Degrees with this property can be used to define
a filter in the local structure of the enumeration degrees that consists entirely of properly
~g enumeration degrees and O~. In Chapter 4 we strengthen the result obtained by
Cooper, Li, Sorbi and Yang of the existence of a nonbounding enumeration degree
by constructing a Igeneric enumeration degree that does not bound a minimal pair.
Degrees with this propertyÃ‚Â· can be used to define an ideal consisting of properly ~g
enumeration degrees and Oe.
Chapters 5 and 6 concern the cupping properties of ~g enumeration degrees and
the subclasses of the ~g enumeration degrees related to the finite and w levels of the Ershov hierarchy. In Chapter 5 we complement a result by Cooper, Sorbi and Yi
by showing that every nonzero .6.g enumeration degree can be cupped by a partial
low .6.g enumeration degree. On the other hand we show that one cannot computably
list a sequence of degrees which contains a cupping partner for every .6.g enumeration
degree. In Chapter 6 we concentrate on the smaller subclasses, where the situation
improves. We prove that every nonzero wc.e. enumeration degree can be cupped by
a 3c.e. enumeration degree and as the 3c.e. enumeration degrees are computably
enumerable this property constitutes a difference between the .6.g enumeration degrees
and the wc.e. enumeration degrees. Furthermore we establish a structural difference
between the class of II~ enumeration degrees and the 3c.e. enumeration degrees by
proving that one cannot find a single Eg enumeration degree that cups every nonzero
3c.e. enumeration degree to O~.
Finally in Chapter 7 we show that the structure of the 3c.e. enumeration degrees is
far from trivial as there exists a Lachlan Il()nspJ~tting pair with top a II~enuIlleration
degree and bottom a 3c.e. enumeration degree.
