Cardinal invariants and independence results in the poset of precompact group topologies

We study the poset B(G) of all precompact Hausdorff group topologies on an infinite group G and its subposet B(G)w of topologies of weight w, extending earlier results of Berhanu, Comfort, Reid, Remus, Ross, Dikranjan, and others. We show that if B(G)w is non-empty and the power set of G/G' has the same size as the power set of G (in particular, if G is abelian) then the poset of all subsets of 2^G of size w can be embedded into B(G)w (and vice versa). So the study of many features (depth, height, width, size of chains, etc.) of the poset B(G)w is reduced to purely set-theoretical problems. We introduce a cardinal function Ded*(w) to measure the length of chains of subsets of cardinality w of a set of larger cardinality generalizing the well-known cardinal function Ded(w). We prove that Ded*(w) = Ded(w) iff the cofinality of Ded(w) is different from w^+ and we use earlier results of Mitchell and Baumgartncr to show that Ded*(aleph_1 ) = Ded(aleph_1) is independent of Zermelc+Fraenkel set theory (ZFC). We apply this result to show that it cannot be established in ZFC whether B(G)aleph_1 has chains of bigger size than those of the bounded chains. We prove that the poset B(G)aleph_0 of all Hausdorff metrizable group topologies on the group G = (direct sum of aleph_0 copies of Z/2Z) has uncountable depth, hence cannot be embedded into B(G)aleph_0. This is to be contrasted with the fact that for every infinite abelian group G the poset H(G) of all Hausdorff group topologies on G can be embedded into B(G). We also prove that it is independent of ZFC whether the poset H(G)aleph_0 has the same height as the poset B(G)aleph_0