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 $\overline{R}\hat{\otimes }{\mathbb{Q}}_p=\underleftarrow{\lim }R/[R,R{\mathcal{M}}_n]\otimes {\mathbb{Q}}_p$. We provide the isomorphisms $\overline{R_w^{\wedge }}({\mathbb{Q}}_p)=\overline{R}\hat{\otimes }{\mathbb{Q}}_p$ and $\overline{R_u}({\mathbb{Q}}_p)=\mathcal{O}(G_u{)}^{\ast }$, where $R_w^{\wedge }$ and $R_u^{\wedge }$ stands for continuous prounipotent completions and corresponding prounipotent presentations correspondingly. We show how $\overline{R_w^{\wedge }}$ 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.

This is a corrected version of the paper which appeared in the Extended Abstracts Spring 2015, Interactions between Representation Theory, Algebraic Topology and Commutative Algebra, Research Perspectives CRM Barcelona, Vol.5, 2016

References in corpus (4)

• Pro-algebraic homotopy types
• Quasirational relation modules and p-adic Malcev completions
• Identity theorem for pro-$p$-groups
• Deligne-Beilinson Cohomology of Affine Groups

Cited by in corpus (3)

• Identity theorem for pro-$p$-groups
• Quasirationality and aspherical (pro-$p$)presentations
• Quasirationality and prounipotent crossed modules