Cohomological dimension and third homology of the pronilpotent completion

Determine the cohomological dimension of the pronilpotent completion of a finitely generated non-cyclic free group and whether its third integral homology vanishes.

Background

The pronilpotent completion $\mathcal F=\varprojlim_n F/\gamma_n(F)$ of a finitely generated non-cyclic free group is itself parafree, but it is not finitely generated and has substantially different homological behavior. The paper notes that its second integral homology is uncountable and not cotorsion.

The unresolved questions concern two basic invariants of this completion: its cohomological dimension and the vanishing of its third integral homology. These questions are presented as a contrast with the finitely generated parafree groups considered in the main results.