Bousfield--Kan Completions of Subcontractible Presentations

Abstract

We study Bousfield--Kan completions through the interaction of free simplicial resolutions, their filtration spectral sequences, and a noncommutative arithmetic square. For every free discrete simplicial group of finite type, we express its integral pronilpotent completion as the homotopy pullback of its rational prounipotent completion and the product of its pro-$p$ completions over an explicit adelic simplicial group. The adelic entry is formed by taking restricted products at finite nilpotent stages and then their inverse limit; no nilpotency assumption on the original group of components is required. Finite subpresentations of contractible presentations provide an explicit application of this construction. Independence of the specified relators makes the positive-degree terms of the rational and mod-$p$ filtration spectral sequences vanish, with convergence verified on the quotient towers. Continuous comparison of free simplicial resolutions then realizes, in characteristic zero, the equivalence with a constant free prounipotent group by morphisms and homotopies in that category. For the corresponding presentation complex $K$ we obtain $R_\infty K\simeq K(F_R(Z),1)$ for $R=\mathbb Q,\mathbb F_p,\mathbb Z$, where $Z$ indexes a complementary basis and $F_R(Z)$ denotes, respectively, the rational points of a free prounipotent group, a free pro-$p$ group, or a free pronilpotent group. Compatible contractions at the nilpotent stages identify all four entries of the arithmetic square in this case.