Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Proalgebraic crossed modules of quasirational presentations   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.

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The categories of this paper are: hep-th
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The categories of this paper are: hep-ex
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The categories of this paper are: funct-an
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The categories of this paper are: adap-org
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The categories of this paper are: dg-ga
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The categories of this paper are: astro-ph
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The categories of this paper are: nucl-ex
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The categories of this paper are: q-fin
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The categories of this paper are: quant-ph
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The categories of this paper are: hep-ph
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The categories of this paper are: q-alg
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The categories of this paper are: cs
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The categories of this paper are: plasm-ph
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The categories of this paper are: math-ph
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The categories of this paper are: gr-qc
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The categories of this paper are: chem-ph