On the uniqueness of Polish group topologies

Polish groups are separable completely metrizable topological groups. A key problem in the theory of Polish groups is that of the uniqueness of a Polish group topology: under what conditions does a group admit only one Polish group topology? Closely related is the problem of automatic continuity: when is a homomorphism between Polish groups necessarily continuous? This dissertation is an investigation of these questions.The key to unlocking these problems is to determine which sets in a Polish group are definable both algebraically and topologically. By algebraically definable one has in mind sets such as the commutators, conjugacy classes, or the squares. In the context of Polish groups, topologically definable means being a Borel set. A classical uniqueness result requires algebraically definable sets that are always Borel. Unfortunately its use is sometimes limited: while algebraically definable sets are often analytic (continuous images of Borel sets), it is shown here that they are not necessarily Borel. In particular, the set of squares in the homeomorphism group of the unit circle, and the set of squares in the automorphism group of the rational circle, are not Borel.An alternative result that avoids the need for Borel sets is obtained: a Polish group with a neighborhood base at the identity consisting of sets that are always analytic has a unique Polishgroup topology. As a consequence of this result, compact, connected, simple Lie groups and finitely generated profinite groups have a unique Polish group topology