Cellular Covers of Abelian Groups

Abstract.

We are investigating cellular covers of abelian groups (for definition, see Preliminaries). Surjective cellular covers of divisible abelian groups have been characterized in a recent paper by Chachólski–Farjoun–Göbel–Segev [2]; see Theorem 3.1. After presenting a new, simple proof of this theorem, we concentrate on reduced groups. For reduced torsion groups the only surjective cellular covers are the trivial ones (Theorem 4.3). On the other hand, even rank 1 torsion-free groups may have cellular covers of arbitrarily large cardinalities (Theorem 5.3). Interestingly, those rank 1 torsion-free groups that do not admit such large cellular covers have only the trivial cellular covers (Theorem 5.4).

Several cases will be listed in Section 6 where the groups admit only the trivial cellular covers; e.g. they include the reduced cotorsion groups. Finally, we investigate the kernels of cellular covering maps. In Theorem 7.6 we give a full characterization of groups that may appear as kernels of such maps. They are direct sums of two groups: one is a reduced torsion-free $\hat{{\mathbb{Z}}}$ -module whose structure is described in Theorem 7.3, and the other is a cotorsion-free group that can be arbitrary (as it follows from Buckner–Dugas [1]).