Existence and Number of Maximal Precompact Topologies on Abelian Groups

The lattice PC(G) of precompact group topologies on an Abelian group G is isomorphic with the lattice SG(G*) of subgroups of the algebraic character group (Remus, 1983). Remus used this result to determine the number of precompact [Hausdorff] topologies on Abelian groups. In this paper the same tool is applied to the problems of existence and number of maximal precompact [Hausdorff] topologies on an Abelian group G, i.e. antiatoms in the lattice PC(G). It is shown that PC(G) has antiatoms iff G is not torsion-free. Further the number of maximal precompact [Hausdorff] topologies is expressed in terms of the cardinalities of the p-components of the group G.