Proalgebraic crossed modules of quasirational presentations

Abstract

We introduce the concept of quasirational relation modules for discrete and pro-$p$ presentations of discrete and pro-$p$ groups and show that aspherical presentations and their subpresentations are quasirational. In the pro-$p$-case quasirationality of pro-$p$-groups with a single defining relation holds. For every quasirational (pro-$p$)relation module we construct the so called $p$-adic rationalization, which is a pro-fd-module $\overline{R}\widehat{\otimes}\mathbb{Q}_p= \varprojlim R/[R,R\mathcal{M}_n]\otimes\mathbb{Q}_p$. We provide the isomorphisms $\overline{R^{\wedge}_w}(\mathbb{Q}_p)=\overline{R}\widehat{\otimes}\mathbb{Q}_p$ and $\overline{R_u}(\mathbb{Q}_p)=\mathcal{O}(G_u)^*$, where $R^{\wedge}_w$ and $R^{\wedge}_u$ stands for continuous prounipotent completions and corresponding prounipotent presentations correspondingly. We show how $\overline{R^{\wedge}_{w}}$ embeds into a sequence of abelian prounipotent groups. This sequence arises naturally from a certain prounipotent crossed module, the latter bring concrete examples of proalgebraic homotopy types. The old-standing open problem of Serre, slightly corrected by Gildenhuys, in its modern form states that pro-$p$-groups with a single defining relation are aspherical. Our results give a positive feedback to the question of Serre.