Compact Structures in Descriptive Classification Theory

Abstract

An equivalence relation, E, on a Polish topological space, X, is Borel reducible to another, F, on Y, when there is a Borel-measurable function from X to Y assigning F-classes as complete invariants for E. Descriptive classification theory is the programme whose aim is to assess the complexity of naturally-arising isomorphism relations by identifying their positions in the Borel reducibility preorder. This thesis considers a class of mathematical objects, the compact metrizable structures, and their natural isomorphism relation, homeomorphic isomorphism. By encoding other objects into compact metrizable structures we can obtain upper bounds in Borel reducibility. In an application of this method, we bound the complexity of topological group isomorphism for certain classes of Polish groups. One problem of descriptive classification theory asks, if a Polish group acts continuously on a Polish space, is the orbit equivalence relation always Borel reducible to the homeomorphism relation between compact metric spaces. We answer this in the affirmative, by first establishing that all such relations are reducible to the homeomorphic isomorphism relation for an appropriate class of compact metrizable structures, and then showing that this additional structure can be eliminated by encoding it into the topology of another compact space.