On LCA groups and epimorphisms of topological groups

The work presented in this thesis is divided in two very distinct parts. Chapter 1, 2 and 3 are focused around the original result presented in Theorem 2.4.8 which gives necessary and sufficient conditions for a group of algebraic homomorphisms from an abelian group G into the unitary circle in two dimensions T to be the group of all continuous morphisms of G into T for some locally compact group topology on G. In Chapter 1 I present my work on understanding compactness in Hausdorff topological groups. Two simple and original proofs of two theorems of Goto are presented in Chapter 1. These proofs clearly show that compactness in certain groups of homorphisms of topological groups can be obtained by applying Ascoli's Theorem with a group topological flavour. Asked by Stephen Watson to work a topological proof of Glicksberg's Theorem, I present the work in Chapter 1, but as I eventually found in the literature other topological proofs of the same theorem, the value of the work became more of a personal nature. Theorem 1.4.4 describes compactness in groups of type HomkHTG (G, K) where K is a compact Hausdorff abelian topological group and G is a Hausdorff abelian group with the property that the bidual map of G is continuous.; Led by the results of Theorem 2.4.8, in Chapter 3 I investigate properties of what I refer to as kk-groups. The most important original results of this chapter are Theorem 3.5.8, Theorem 3.5.9 and Theorem 3.5.23. Also, unfinished work towards finding an example that would non-trivially satisfy Theorem 3.5.23 is presented in Appendix. An interesting result that I could not find in the literature is presented in Theorem 3.4.2 .; The second part of the thesis consists of Chapter 4 and 5. Chapter 4 presents Stephen Watson's work on the epimorphism problem for Hausdorff groups. Some of the original proofs are improved. Chapter 5 presents the way I applied the results in the previous chapter in order to be able to formulate the results of Theorem 5.1.36 and Theorem 5.1.41. There are a lot of technical results presented in Chapter 5, the ones at the beginning of the chapter are well known and as the chapter progresses, the results are specifically tailored for the needs of Theorem 5.1.36, probably some of them not appearing in the literature. (Abstract shortened by UMI.)...