central filtration of the free discrete group $\Phi(X)$) have finite dimension
(Se, , Theorem 1, Chapter I,4.6), hence the quotients $S(k)\cap
L_{i}(k)/S(k)\cap L_{i+1}(k)$ are also of finite dimension and, therefore,
taking their lifts in $S(k)$ we get a convergent basis. As a consequence, we
see that the quotients $\overline{S}_{i}(k)=\overline{S(k)/S(k)\cap L_{i}(k)}$
are in fact finite-dimensional vector spaces over $k$.

We want to show that each $\overline{S}_{i}(k)$ with the conjugation-induced
action of $(F/L_{i})(k)$ (we take into account that conjugation action of
$L_{i}(k)$ on $\overline{S}_{i}(k)$ is trivial) is a topological
$\mathcal{O}(F/L_{i})^{*}$-module. By Definition 8, this is equivalent to
finding a continuous $k$-linear mapping

$\mu_{i}:\overline{S}_{i}(k)\widehat{\otimes}\mathcal{O}(F/L_{i})^{*}\rightarrow\overline{S}_{i}(k),$

arising naturally from the conjugation action and included into the necessary
diagrams.

By (HM2003, , Theorem A.2., Corollary A.4.) $\Phi(X)/C_{i}\subset
F(k)/L_{i}(k)$ and hence $\Phi(X)/C_{i}$ also acts on $S_{i}(k)=S(k)/S(k)\cap
L_{i}(k)$ by conjugations. We continue this action by linearity to the group
ring $k[\Phi(X)/C_{i}]$:

$\widetilde{\mu}_{i}:\overline{S}_{i}(k)\otimes
k[\Phi(X)/C_{i}]\rightarrow\overline{S}_{i}(k),$

given by the rule $r\otimes\sum a_{j}f_{j}\mapsto\sum a_{j}r^{f_{j}},a_{j}\in
k,f_{j}\in\Phi(X)/C_{i},r\in\overline{S}_{i}(k)$.

The required mapping $\mu_{i}$ will be obtained as a completion of this map.
We start from Quillen’s formulae
$\mathcal{O}(F/L_{i})^{*}\cong\varprojlim_{n}k[\Phi(X)/C_{i}]/I_{\Phi}^{n}$
and as $\Phi(X)/C_{i}$ is torsion-free nilpotent, then $\cap I_{\Phi}^{n}=0$,
where $I_{\Phi}$ is the augmentation ideal of $k[\Phi(X)/C_{i}]$, and hence we
get the decomposition
$\overline{S}_{i}(k)\cong\varprojlim_{n}\overline{S}_{i}(k)/\overline{S}_{i}(k)I^{n}_{\Phi},$
where $\overline{S}_{i}(k)I^{n}_{\Phi}$ denotes the $k$-linear subspace of
$\overline{S}_{i}(k)$ generated by elements of the form $a\cdot f,$ where
$a\in\overline{S}_{i}(k)$, $f\in I^{n}_{\Phi}$.

$F$ acts prounipotently on $\overline{S}$. Indeed, $F$ acts trivially on
graded quotients $S\cap L_{i}/S\cap L_{i+1}$, and we take their images under
abelianization of $S/S\cap L_{i}$ to obtain the quotients $\overline{S/S\cap
L_{i}}$ of the prounipotent filtration of $\overline{S}$.

Let $\widetilde{G}=\overline{S}_{i}(k)\leftthreetimes F(X)/L_{i}$, by Lemma 6
$\widetilde{G}$ is a unipotent group and we have well defined, by Quillen’s
formulae, closed subspaces
$\overline{S}_{i}(k)\cdot\widehat{I}^{m}\subseteq\overline{S}_{i}(k)$, where
$\widehat{I}$ is the augmentation ideal of $\mathcal{O}(F/L_{i})^{*}$. As
$\overline{S}_{i}(k)$ is of finite dimension, then there is $m\in\mathbb{N}$
such that $\overline{S}_{i}(k)\cdot\mathfrak{I}^{m}=0$, where
$\mathfrak{I}\subset\mathcal{O}(\widetilde{G})^{*}$ is the augmentation ideal,
but
$\cap\overline{S}_{i}(k)I^{n}_{\Phi}\subseteq\cap\overline{S}_{i}(k)\widehat{I}^{n}\subseteq\cap\overline{S}_{i}(k)\mathfrak{I}^{n}=0$
and therefore
$\overline{S}_{i}(k)I^{n}_{\Phi}=\overline{S}_{i}(k)\widehat{I}^{n}$ for each
$n\in\mathbb{N}$.
$(\overline{\widetilde{S}}_{i}\leftthreetimes\Phi(X)/C_{i})^{\wedge}_{u}(k)\cong\overline{S}_{i}(k)\leftthreetimes
F/L_{i}(k)$, by Remark 4, where $\overline{\widetilde{S}}(k)$ is taken
finitely generated and dense in $\overline{S}_{i}(k)$, now we see that the
conjugation action of $F/L_{i}(k)$ on $\overline{S}_{i}(k)$ coincides with the
action defined by completion, and therefore the last one is independent of the
chosen basis $X$ in $F/L_{i}(k)$. Since
$\overline{S}_{i}(k)/\overline{S}_{i}(k)I^{n}_{\Phi}\cong\overline{S}_{i}(k)/\overline{S}_{i}(k)\widehat{I}^{n}$
and $\mathcal{O}(F)^{*}/\widehat{I}^{n}\cong k[\Phi(X)/C_{i}]/I^{n}_{\Phi}$,
for each $n\in\mathbb{N}$ we have the $k$-linear maps of finite dimensional
vector spaces

$\overline{S}_{i}(k)/\overline{S}_{i}(k)\widehat{I}^{n}\otimes\mathcal{O}(F/L_{i})^{*}/\widehat{I}^{n}\rightarrow\overline{S}_{i}(k)/\overline{S}_{i}(k)\widehat{I}^{n},$

which are completely defined by the conjugation action of $F/L_{i},$ since by
(Qui5, , Proposition 3.6 (a)) for $n>i$ the canonical map
$\Phi(X)/C_{i}\rightarrow k[\Phi(X)/C_{i}]/I^{n}_{\Phi}$ is injective and
$\Phi(X)/C_{i}$ defines this action. Passing to the inverse limit over $n$, we
obtain the required $k$-linear mapping $\mu_{i}$, which fits into the required
diagrams, since the mappings $\widetilde{\mu}_{i}$ are taken from the
associative group ring action with unit and, therefore, $\overline{S}_{i}(k)$
is the topological $\mathcal{O}(F/L_{i})^{*}$-module. Passing to the inverse
limit over $i$ we get $\mathcal{O}(F)^{*}$-module structure on
$\overline{S}(k)$.

Let $X$ be a chosen basis in $(F/[F,F])^{\vee}$, then we present
$F=F(X)(k)=F(\widehat{X},*)(k)$ as the inverse limit
$F\cong\varprojlim_{\lambda}F(X_{\lambda},*)$ over finite pointed spaces
$(X_{\lambda}^{*},*)$ as in (10). As a result, we obtain the decomposition
$S(k)\cong\varprojlim S_{\lambda}(k)$ into the inverse limit of
$S_{\lambda}(k)=\phi_{\lambda}(S)(k),$ where $\phi_{\lambda}:F\rightarrow
F(X_{\lambda}^{*},*)$ \- the natural projections and, therefore, the
decomposition $\overline{S}=\varprojlim\overline{S}_{\lambda}$. For each
$\lambda$ we have $\mathcal{O}(F(X_{\lambda}^{*},*))^{*}$-module structure
$\mu_{\lambda}:\overline{S}_{\lambda}(k)\widehat{\otimes}\mathcal{O}(F_{\lambda})^{*}\rightarrow\overline{S}_{\lambda}(k)$,
induced by the conjugation action of $F_{\lambda}=F(X_{\lambda}^{*},*)(k)$ on
$S_{\lambda}(k)$ and, as the conjugation action commute with factorization, we
can take $\mu=\varprojlim\mu_{\lambda}$ in order to obtain the structure of
$\mathcal{O}(F)^{*}$-topological module on $\overline{S}(k)$. By construction,
taking in mind Proposition 6, this action is also independent of the chosen
basis $X$.

As a byproduct of ideas in the proof, we obtain the following useful
corollaries

###### Corollary 2

Let $M$ and $N$ be topological $\mathcal{O}(G)^{*}$-modules and let
$f:M\rightarrow N$ be a $G(k)$-homomorphism of linearly-compact vector spaces,
then $f$ is a homomorphism of topological $\mathcal{O}(G)^{*}$-modules.

###### Corollary 3

Let (1) be a prounipotent presentation, then the linearly compact $k$-vector
space $\overline{R}(k)=R/[R,R](k)$ can be naturally endowed with the structure
of a topological $\mathcal{O}(G)^{*}$-module with the action induced by the
conjugations of $F$ on $R$. We call such a topological
$\mathcal{O}(G)^{*}$-module the relation module of a prounipotent
presentation.

###### Proof

$R\lhd F$ lies in the kernel of the conjugation action of $F$ on
$\overline{R}$ and hence the action factors through the action of $G$ and,
therefore, $\overline{R}(k)$ is topological $\mathcal{O}(G)^{*}$-module.

###### Proposition 12

Suppose we are given a prounipotent presentation in the simplicial form (12).
Then $\overline{C}(k)$ inherits structure of a topological
$\mathcal{O}(G)^{*}$-module.

###### Proof

By Proposition 11, $\overline{Ker\;d_{0}}(k)$ \- the abelianization of
$Ker\;d_{0}$ is naturally a $\mathcal{O}(F)^{*}$-module. It contains
$\langle[Ker\;d_{0},Ker\;d_{1}](k)\rangle$ \- the
$\mathcal{O}(F)^{*}$-submodule of $\overline{Ker\;d_{0}}(k)$ generated by the
image of $[Ker\;d_{0},Ker\;d_{1}](k)$ in $\overline{Ker\;d_{0}}(k)$.

Since
$\overline{C}(k)\cong\overline{Ker\;d_{0}}(k)/\langle[Ker\;d_{0},Ker\;d_{1}](k)\rangle$
we get that $\overline{C}(k)$ is a topological $\mathcal{O}(F)^{*}$-module.
The rest is to show, that $R(k)$ lies in the kernel of this action. Lets take
$d_{1}(x)\in R(k)=d_{1}(Ker\;d_{0}(k))$, $x,y\in Ker\;d_{0}(k)$ by CM 2) of
Definition 3 we have $x^{-1}yx=y^{d_{1}(x)}$ and hence
$x^{-1}yxy^{-1}=y^{d_{1}(x)}y^{-1}$ \- the image of $y^{d_{1}(x)}y^{-1}$ in
$\overline{C}(k)$ is zero.

###### Definition 17

Let $G$ be a prounipotent group, we say that the mapping
$\mu:(\widehat{X}^{*},*)\rightarrow L$ of a pointed profinite space
$(\widehat{X}^{*},*)$ into a topological $\mathcal{O}(G)^{*}$-module $L$ is
convergent if there exists a decomposition $L\cong\varprojlim L_{\lambda}$
into the inverse limit of finite dimensional $\mathcal{O}(G)^{*}$-modules
$L_{\lambda}$, such that for all but finite $x\in\widehat{X}^{*}$ we have
$\phi_{\alpha}\mu(x)=1$, where $\phi_{\lambda}:L\rightarrow L_{\lambda}$ is
the projection. We also call the topological $\mathcal{O}(G)^{*}$-module $M$
free over the pointed profinite space $i:(\widehat{X}^{*},*)\rightarrow M$,
$i(*)=0$, we denote it $(\mathcal{O}(G)^{*})^{(\widehat{X}^{*},*)}$, if it has
the following universal property: for any toplogical
$\mathcal{O}(G)^{*}$-module $L$ and a convergent mapping
$\mu:(\widehat{X}^{*},*)\rightarrow L$ there exists a homomorphism
$\widehat{\mu}:M\rightarrow L$ of topological $\mathcal{O}(G)^{*}$-modules
such that $\mu=\widehat{\mu}\circ i$.

###### Lemma 7

For any pointed profinite space $(\widehat{X}^{*},*)$ there is a free
topological $\mathcal{O}(G)^{*}$-module
$M=(\mathcal{O}(G)^{*})^{(\widehat{X}^{*},*)}$ over $(\widehat{X}^{*},*)$.

###### Proof

Let $(\widehat{X}^{*},*)\cong\varprojlim(\widehat{X_{\lambda}}^{*},*)$, it is
easy to check that the toplogical $\mathcal{O}(G)^{*}$-module
$\varprojlim(\mathcal{O}(G)^{*})^{(\widehat{X_{\lambda}}^{*},*)}$ possesses
the required universal properties, its dual $M^{\vee}\cong\mathcal{O}(G)^{X}$
is an induced $G$-module.

###### Remark 12

By Proposition 16 (below) $H^{1}(R,k_{a})$ is endowed with a structure of
$\mathcal{O}(G)$-comodule, and $H^{1}(R,k_{a})$ can be identified with
$\overline{R}(k)^{\vee}$ (Mikh2017, , Proposition 10) and, therefore, by
duality, $\overline{R}(k)$ is a topological $\mathcal{O}(G)^{*}$-module. Such
$\mathcal{O}(G)^{*}$-module structure is the same as $k\langle\langle
G\rangle\rangle=End_{G}(\mathcal{O}(G)),\mathcal{O}(G))$-module structure on
relation modules of Mag , LM2 , since by (4) $k\langle\langle
G\rangle\rangle\cong\mathcal{O}(G)^{*}$ and by (Mag, , Lemma 8(1)) the action
of $G(k)$ on $\overline{R}(k)$ is induced by conjugations.

It is also proved in Proposition 16 that $H^{1}(R,k_{a})$ is naturally
embedded into its injective hull, which is $\mathcal{O}(G)^{Y}$, and,
therefore, by duality as well, we have a natural epimorphism of
$\mathcal{O}(G)^{*}$-modules
$(\mathcal{O}(G)^{*})^{(\widehat{Y}^{*},*)}\rightarrow\overline{R}(k)$.

###### Remark 13

Let $C(k)\xrightarrow{\overline{d}_{1}}F(k)$ be the free prounipotent crossed
module of Section 4.6. Then $\overline{d}_{1}$ induces a homomorphism, say
$\partial:\overline{C}(k)=(\mathcal{O}(G)^{*})^{(\widehat{Y}^{*},*)}\rightarrow\overline{R}(k)$,
of topological $\mathcal{O}(G)^{*}$-modules, now
$\partial_{G}:\overline{C}(k)_{G}=k^{(\widehat{Y}^{*},*)}\rightarrow\overline{R}(k)_{G}=k^{(\widehat{Y}^{*},*)}$
\- the induced homomorphism on coinvariants is the “fixed” isomorphism of
topological vector spaces of Remark 10.

###  5.2 Non-proper presentations and their relation modules

The following prouniptent analog of the Gaschütz Lemma, which was originally
proved by Gaschütz for finite groups and later extended by Fried-Jarden to
pro-finite groups (FJ, , Lemma 17.7.2), enlightens the roots of the Gaschütz
result. We do not need this result in the rest of the paper, however its
analog is crucial in the study of non-proper pro-finite presentations Lub .

###### Proposition 13

Let $f:G\rightarrow H$ be an epimorphism of prounipotent groups with
$rank(G)\leq e$. Let $h_{1},...,h_{e}$ be a system of generators of $H$. Then
there exists a system of generators $g_{1},...,g_{e}$ of $G$ such that
$f(g_{i})=h_{i}$, $i=1,...,e$.

###### Proof

As we have mentioned this statement is already proven for profinite groups and
in particular it is true for pro-$p$-groups, therefore, we need to consider
the case, when $char(k)=0$. Let, for first, $f:G\rightarrow H$ be a
homomorphism of unipotent groups. By the functorial correspondence Example 3
between unipotent groups over a field $k$ of characteristics 0 and nilpotent
Lie algebras over $k$, this $f$ gives rise to the epimorphism
$\mathfrak{f}:\mathfrak{g}\rightarrow\mathfrak{h}$ of the nilpotent Lie
algebras. Since $f$ is a Lie algebra homomorphism, then
$f([\mathfrak{g},\mathfrak{g}])=[\mathfrak{h},\mathfrak{h}]$ and, therefore,
we obtain the following commutative diagram

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
5.5pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
29.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
29.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\mathfrak{u}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
77.83336pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
35.27779pt\raise-19.72221pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.75pt\hbox{$\scriptstyle{\tau_{\mathfrak{g}}|_{\mathfrak{u}}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
35.27779pt\raise-29.22221pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
77.83336pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\mathfrak{g}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
108.93753pt\raise 6.1111pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.75pt\hbox{$\scriptstyle{f}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 138.94449pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
83.33336pt\raise-19.72221pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-0.6875pt\hbox{$\scriptstyle{\tau_{\mathfrak{g}}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
83.33336pt\raise-28.94444pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
138.94449pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\mathfrak{h}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
144.72227pt\raise-19.72221pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-0.8125pt\hbox{$\scriptstyle{\tau_{\mathfrak{h}}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
144.72227pt\raise-28.94444pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
187.8334pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
187.8334pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{0}$}}}}}}}{\hbox{\kern-5.5pt\raise-39.44444pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
29.5pt\raise-39.44444pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
29.5pt\raise-39.44444pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\widetilde{\mathfrak{u}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
65.05557pt\raise-39.44444pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
65.05557pt\raise-39.44444pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathfrak{g}/[\mathfrak{g},\mathfrak{g}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
108.25003pt\raise-32.83333pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-3.61111pt\hbox{$\scriptstyle{\widetilde{\mathfrak{f}}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
125.61115pt\raise-39.44444pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
125.61115pt\raise-39.44444pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathfrak{h}/[\mathfrak{h},\mathfrak{h}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
187.8334pt\raise-39.44444pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
187.8334pt\raise-39.44444pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{0}$}}}}}}}\ignorespaces}}}}\ignorespaces.$

Consider a vector space decomposition
$\mathfrak{g}\cong\mathfrak{h}_{1}\oplus\mathfrak{u}$, where
$\mathfrak{f}|_{\mathfrak{h}_{1}}:\mathfrak{h}_{1}\rightarrow\mathfrak{h}$ is
the isomorphism of vector spaces, since
$[\mathfrak{u},\mathfrak{u}]\subseteq[\mathfrak{g},\mathfrak{g}]\cap\mathfrak{u}$,
it follows, that nilpotent Lie algebra generators of $\mathfrak{u}$ cover a
basis of $\widetilde{\mathfrak{u}}$.

Since $\mathfrak{f}$ induces the epimorphism of abelianizations
$\overline{\mathfrak{f}}:\mathfrak{g}/[\mathfrak{g},\mathfrak{g}]\rightarrow\mathfrak{h}/[\mathfrak{h},\mathfrak{h}]$
and since $h_{1},...,h_{e}$ generate $H$, then we can choose
$\widetilde{h}_{1}=log(h_{1}),...,\widetilde{h}_{l}=log(h_{l})$, where
$l=dim_{k}(\mathfrak{h}/[\mathfrak{h},\mathfrak{h}])$, - a minimal subset,
which generate $\mathfrak{h}$ as a nilpotent Lie algebra, i.e. a minimal
subset of $Im\;\tau_{\mathfrak{h}}(\widetilde{h}_{i})$ which form a basis in
$\mathfrak{h}/[\mathfrak{h},\mathfrak{h}]$. Let
$\widetilde{g}_{1},..,\widetilde{g}_{l}\in\mathfrak{h}_{1}$, such that
$\mathfrak{f}(\widetilde{g}_{i})=\widetilde{h}_{i},i=1,..,l$, and there are
exist
$\widetilde{g}^{\prime}_{l+1},..,\widetilde{g}^{\prime}_{d}\in\mathfrak{u}$,
where $d=d(\mathfrak{g})=dim_{k}(\mathfrak{g}/[\mathfrak{g},\mathfrak{g}])$,
such that, $\tau_{\mathfrak{g}}|_{\mathfrak{u}}(\widetilde{g}^{\prime}_{i})$
generate $\widetilde{\mathfrak{u}}$. Now we define
$\widetilde{g}_{l+1},..,\widetilde{g}_{d}\in\mathfrak{g}$ by the rule
$\widetilde{g}_{i}=\widetilde{g}^{\prime}_{i}+r_{i}$, where
$r_{i}\in\mathfrak{h}_{1}$, such that $\mathfrak{f}(r_{i})=\widetilde{h}_{i}$,
$i=l+1,..,d$. For $i>d$ we take arbitrary $\widetilde{g}_{i}$, such that
$\mathfrak{f}(\widetilde{g}_{i})=\widetilde{h}_{i}$. By construction
$\widetilde{g}_{i}$ generate $\mathfrak{g}$ and
$\mathfrak{f}(\widetilde{g}_{i})=\widetilde{h}_{i}$ and hence the result
follows for unipotent groups as the elements $g_{i}=exp(\widetilde{g}_{i})\in
G(k)$ posses the required properties.

For $\mathfrak{g}$ general pronilpotent Lie algebra we will follow the ideas
in the proof of (FJ, , Lemma 17.7.2). Present
$\mathfrak{f}:\mathfrak{g}\rightarrow\mathfrak{h}$ as an inverse limit of
epimorphisms of nilpotent Lie algebras
$\mathfrak{f}_{i}:\mathfrak{g}_{i}\rightarrow\mathfrak{h}_{i}$. Specifically,
if $j\geq i$, then there are epimorphisms
$\psi_{ji}:\mathfrak{g}_{j}\rightarrow\mathfrak{g}_{i}$ and
$\phi_{ji}:\mathfrak{h}_{j}\rightarrow\mathfrak{h}_{i}$ such that
$\mathfrak{f}_{i}\circ\psi_{ji}=\phi_{ji}\circ\phi_{j}$. In addition, there
are epimorphisms $\kappa_{i}:\mathfrak{g}\rightarrow\mathfrak{g}_{i}$ and
$\rho_{i}:\mathfrak{h}\rightarrow\mathfrak{h}_{i}$ such that
$\mathfrak{f}_{i}\circ\kappa_{i}=\rho_{i}\circ\mathfrak{f}$.

Let $\textbf{h}\in\mathfrak{h}^{e}$ be a $e$-tuple, that generate
$\mathfrak{h}$ as the pronilpotent Lie algebra. For each $i\in I$ denote the
vector subspace generated by $e$-tuples $\textbf{x}\in\mathfrak{g}^{e}_{i}$
that generate $\mathfrak{g}_{i}$ and satisfy
$\mathfrak{f}_{i}(x)=\rho_{i}(\textbf{h})$ by $\mathfrak{A}_{i}$. By the case
where $\mathfrak{g}$ is unipotent, $\mathfrak{A}_{i}$ is nonempty. In
addition, $\mathfrak{A}_{i}$ is of finite dimension. If $j\geq i$ and
$\textbf{y}\in\mathfrak{A}_{j}$ , then
$\psi_{ji}(\textbf{y})\in\mathfrak{A}_{i}$ and as the inverse limit
$\mathfrak{A}$ of the inverse system $(A_{i},\psi_{ji})_{i,j\in I}$ (since by
2. of Definition 3.4
$\mathfrak{A}\cong\cap(\pi_{i}^{e})^{-1}(\mathfrak{A}_{i})\neq 0$) is nonempty
closed subspace of the linearly-compact vector space
$\mathfrak{g}^{e}\cong\varprojlim\mathfrak{g}_{i}^{e}$. Each element in
$\mathfrak{A}$ defines a system of generators $\textbf{g}=(g_{1},...,g_{e})$
of $\mathfrak{g}$ with $\mathfrak{f}(\textbf{g})=\textbf{h}$ and again
$exp(\textbf{g})$ is the desired element in $G(k)$.

We will need the following prounipotent analog of (Lub, , Proposition 2.2.)
and (Lub, , Proposition 2.4.), it has a slighter weaker form than Lub  as we
do not limit to only finitely generated prounipotent groups.

###### Lemma 8

Let we are given $1\rightarrow R\rightarrow F\xrightarrow{\pi}G\rightarrow 1$
\- a non-proper prounipotent presentation, then there is a basis
$\widehat{S}^{*}=\widehat{X}^{*}\cup\widehat{Z}^{*}\subseteq F(k)$, such that
$\widehat{Z}^{*}\subseteq R(k)$ and $1\rightarrow R_{1}\rightarrow
F(X)\xrightarrow{\widehat{\pi}}G\rightarrow 1$ is the proper presentation,
where $R_{1}$ is the image of $R$ under the natural epimorphism
$F(\widehat{X}^{*}\cup\widehat{Z}^{*})\xrightarrow{\theta}F(\widehat{X}^{*})$,
which sends $x\mapsto x$ if $x\in\widehat{X}^{*}$ and $z\mapsto*$ if
$z\in\widehat{Z}^{*}$.

Then $\theta$ gives rise to an exact sequence of topological
$\mathcal{O}(G)^{*}$-modules as follows
$0\rightarrow(\mathcal{O}(G)^{*})^{(\widehat{Z}^{*},*)}\rightarrow\overline{R}(k)\xrightarrow{\widetilde{\theta}}\overline{R_{1}}(k)\rightarrow
0$.

###### Proof

We prove in the case $char(k)=0$, the pro-$p$-case is similar. Since the
presentation is not proper, the induced epimorphism of linearly-compact
topological vector spaces $\widetilde{\pi}:F/[F,F](k)\rightarrow G/[G,G](k)$
has a non-zero kernel $Ker\;\widetilde{\pi}\cong R[F,F]/[F,F](k)\cong
R(k)/(R(k)\cap[F,F](k))$ and therefore we get the decomposition

$F/[F,F](k)\cong G/[G,G](k)\oplus R[F,F]/[F,F](k).$

Let $Z$ be a basis of $(R[F,F]/[F,F](k))^{\vee}$ and let $X$ be a basis of
$(G/[G,G](k))^{\vee}$. We choose the union uplifts $\widehat{Z}^{*}$ to $R(k)$
(along the projection $R(k)\rightarrow R(k)/(R(k)\cap[F,F](k))$) and
$\widehat{X}^{*}$ in $F(k)$ (in the pro-$p$-case, as always, we use
$F^{p}[F,F]$ and $G^{p}[G,G]$ instead $[F,F]$ and $[G,G]$ and the uplifts are
continuous sections, which exist by (Se4, , Proposition 1)). By construction,
$\widehat{X}^{*}\cup\widehat{Z}^{*}$ has the desired properties.

Let $N=Ker\;\theta$, then, by previous discussion, we have an exact sequence
$1\rightarrow N\rightarrow R\rightarrow R_{1}\rightarrow 1$ of free
prounipotent groups. The corresponding low degree 5-exact sequence in the
Lyndon-Hochschild-Serre spectral sequence of $G$-modules takes the form
$0\rightarrow H^{1}(R_{1},k_{a})\rightarrow H^{1}(R,k_{a})\rightarrow
H^{1}(N,k_{a})^{R}\rightarrow 0,$ which is equivalent by duality to
$0\rightarrow\overline{R}_{1}(k)^{\vee}\rightarrow\overline{R}(k)^{\vee}\rightarrow(\overline{N}(k)^{\vee})^{R}\rightarrow
0$ and, therefore, again, by duality, we get an exact sequence of topological
$\mathcal{O}(G)^{*}$-modules

$0\rightarrow\overline{N}(k)_{R(k)}\rightarrow\overline{R}(k)\rightarrow\overline{R_{1}}(k)\rightarrow
0.$

By Proposition 10 $(N,F(\widehat{X}^{*}),d_{1}|_{N})$, where
$d_{1}(\widehat{Z}^{*})=1$ and $Y=Z$ in our environment, is a free
prounipotent pre-crossed module, and since $Ker\;d_{0}=Ker\;d_{1}$ we see
$(C=Ker\;d_{0}/[Ker\;d_{0},Ker\;d_{1}],F(X),\overline{d}_{1})\cong(\overline{N}(k),F(X),\overline{d}_{1})$
is the free crossed module. Its universal property, using Corollary 2, imply
that $\overline{N}(k)\cong(\mathcal{O}(F(d))^{*})^{(\widehat{Z}^{*},*)}$ is a
free topological $\mathcal{O}(F(d))^{*}$-module (BH1, , Proposition 7) and,
therefore, taking coinvariants, as $\mathcal{O}(F)^{R}\cong\mathcal{O}(G)$, we
obtain that
$\overline{N}(k)_{R}\cong(\mathcal{O}(G)^{*})^{(\widehat{Z}^{*},*)}$.

###  5.3 Relation modules and aspherical presentations

When $G$ is a prounipotent group we can obtain a precise description of
injective hulls.

###### Proposition 14

(LM, , 1.11) Let $G$ be a prounipotent group, and let $V$ be a $G$-module.
Then $V^{G}\otimes\mathcal{O}(G)$ is an injective $G$-module containing $V$.
Each injective $G$-module containing $V$ contains a copy of
$V^{G}\otimes\mathcal{O}(G)$.

Due to the previous statement we can define the _injective hull_ of a
$G$-module $V$ by the formula $\mathcal{E}_{0}(V)=V^{G}\otimes\mathcal{O}(G).$
Put $\mathcal{E}_{-1}(V)=V$, and let
$d_{-1}:\mathcal{E}_{-1}(V)\rightarrow\mathcal{E}_{0}(V)$ be the corresponding
inclusion.

###### Lemma 9

Let $A$ be a $G$-module over a prounipotent group $G$ then the following
statements are equivalent:

A1

$A^{*}$ is a free topological $\mathcal{O}(G)^{*}$-module over a pointed
profinite space $(\widehat{X}^{*},*)$, i.e.
$A^{*}\cong(\mathcal{O}(G)^{*})^{(\widehat{X}^{*},*)}$

A2

$A$ is an induced $G$-module over $X$, i.e.
$A\cong(\oplus_{X}k_{a})\uparrow_{1}^{G}$

A3

$H^{1}(G,A)=0$

###### Proof

By duality $A1\Leftrightarrow A2$, also $A2\Rightarrow A3$, so we must prove
$A3\Rightarrow A2$.

By Proposition 14, $A$ is a submodule of its injective hull
$V=A^{G}\otimes\mathcal{O}(G)\cong(A^{G})\uparrow_{1}^{G}$, that is,
$A\xrightarrow{j}V=A^{G}\otimes\mathcal{O}(G)$. Let $C=Coker\;(j)$, then we
have the exact sequence $0\rightarrow A^{G}\rightarrow V^{G}\rightarrow
C^{G}\rightarrow H^{1}(G,A)\rightarrow 0$. Since $H^{1}(G,A)=0$, then
$C^{G}=0$ and hence $C=0$ and, therefore, $A\cong(A^{G})\uparrow_{1}^{G}$.

###### Proposition 15

(LM, , 1.12) Let $G$ be a prounipotent group and $V$ be a $G$-module. Define
the minimal resolution $\mathcal{E}_{i}(V)$ and
$d_{i}:\mathcal{E}_{i}(V)\rightarrow\mathcal{E}_{i+1}(V)$ inductively,

$\mathcal{E}_{i+1}(V)=\mathcal{E}_{0}(\frac{\mathcal{E}_{i}(V)}{d_{i-1}(V))})\quad
d_{i}=\mathcal{E}_{i}(V)\rightarrow\frac{\mathcal{E}_{i}(V)}{d_{i-1}(\mathcal{E}_{i-1}(V))}\rightarrow\mathcal{E}_{i+1}.$

Then $\\{\mathcal{E}_{i}(V),d_{i}\\}$ is an injective resolution of $V$ and
$H^{i}(G,V)=\mathcal{E}_{i}(V)^{G}.$

###### Proof

$\mathcal{E}_{i}(V)=\mathcal{E}_{0}(Ker\;d_{i})=(Ker\;d_{i})^{G}\otimes\mathcal{O}(G)$.
By construction and Proposition 14 $\\{\mathcal{E}_{i}(V),d_{i}\\}$ is an
injective resolution, hence $\mathcal{E}_{i}(V)^{G}=(Ker\;d_{i})^{G}$, as
$\\{\mathcal{E}_{i}(V)^{G}\\}$ has zero differentials and, therefore,
$H^{i}(G,V)=\mathcal{E}_{i}(V)^{G}$.

###### Proposition 16

Assume we are given a prounipotent presentation (12) of a pro-unipotent group
$G$. Then $cd(G)=2$ if and only if there is a non-canonical isomorphism of
right topological $\mathcal{O}(G)^{*}$-modules
$\overline{R}(k)\cong(\mathcal{O}(G)^{*})^{(\widehat{Y}^{*},*)}$.

###### Proof

First of all, note that the proof of the isomorphism of right topological
$\mathcal{O}(G)^{*}$-modules
$\overline{R}(k)\cong(\mathcal{O}(G)^{*})^{(\widehat{Y}^{*},*)}$, by
continuous duality, is equivalent to the proof of the isomorphism
$Hom_{cts}(\overline{R}(k),k)\cong\mathcal{O}(G)^{Y}$ of right
$\mathcal{O}(G)$-comodules. (Mikh2017, , Proposition 10) asserts that there is
an isomorphism of $\mathcal{O}(G)$-comodules $H^{1}(R,k_{a})\cong
Hom(\overline{R},k_{a})\cong\overline{R}(k)^{\vee}$. Thus, we must prove the
isomorphism of right $\mathcal{O}(G)$-comodules
$H^{1}(R,k_{a})\cong\mathcal{O}(G)^{Y}$.

Let us study the minimal injective $\mathcal{O}(G)$-resolution of the trivial
$\mathcal{O}(G)$-comodule $k_{a}$ (i.e. we consider a proper presentation).
Since the cohomological dimension of $F$ equals one, then the minimal
$\mathcal{O}(F)$-resolution of $k_{a}$ takes the form

$0\rightarrow
k_{a}\rightarrow\mathcal{O}(F)\rightarrow\mathcal{O}(F)^{X}\rightarrow 0.$

By (Jan, , I, 4.12, 3.3) we can regard this resolution as injective
$\mathcal{O}(R)$-comodule resolution. Applying the $R$-fixed points functor
and taking into account that $R$-fixed points of $\mathcal{O}(F)$ coincide
with $\mathcal{O}(G)$ (Wat, , 16.3), we get the exact sequence $0\rightarrow
k_{a}\rightarrow\mathcal{O}(G)\rightarrow\mathcal{O}(G)^{X}\rightarrow
H^{1}(R,k_{a})\rightarrow 0.$ It follows from Proposition 14 that the
injective hull of $H^{1}(R,k_{a})$ coincides with
$H^{1}(R,k_{a})^{G}\otimes\mathcal{O}(G)\cong\mathcal{O}(G)^{Y}$, since
$H^{2}(G,k_{a})\cong H^{1}(R,k_{a})^{G}\cong k^{Y}$. A composition of the map
$\mathcal{O}(G)^{X}\rightarrow H^{1}(R,k_{a})$ with the inclusion of
$H^{1}(R,k_{a})$ in its injective hull give rise to minimal injective
$\mathcal{O}(G)$-resolution $0\rightarrow
k_{a}\rightarrow\mathcal{O}(G)\rightarrow\mathcal{O}(G)^{X}\rightarrow\mathcal{O}(G)^{Y}$
of the trivial comodule $k_{a}$. In particular, one has an isomorphism of
$\mathcal{O}(G)$-comodules $H^{1}(R,k_{a})\cong
Im\;(\mathcal{O}(G)^{X}\rightarrow\mathcal{O}(G)^{Y}).$ From Proposition 15 it
follows that $\\{\mathcal{E}_{i}(V)^{G}\\}$ has zero differentials and,
therefore, since nontrivial modules over prounipotent groups have nontrivial
fixed points, we see that conditions $H^{1}(R,k_{a})\cong\mathcal{O}(G)^{Y}$
and $cd(G)=2$ are equivalent.

If the presentation is not proper, then, by Lemma 8 and above considerations,
we get a short exact sequence of topological $\mathcal{O}(G)^{*}$-modules

$0\rightarrow(\mathcal{O}(G)^{*})^{(\widehat{Z}^{*},*)}\rightarrow\overline{R}(k)\xrightarrow{\widetilde{\theta}}(\mathcal{O}(G)^{*})^{(\widehat{Y\setminus
Z}^{*},*)}\rightarrow 0,$

where we use a slightly non-precise notations, identifying $Y$ with a special
new basis in $R/[R,F](k)$, but this is not important (see also Proposition 17
below). Indeed, the cardinality of $Y$ is the sum of cardinalities of the
basis of $(R[F,F]/[F,F](k))^{\vee}$ and the cardinality of a basis of the
complement to $(R[F,F]/[F,F](k))^{\vee}\hookrightarrow(R/[R,F](k))^{\vee}$.
This sequence splits along the embedding $(\widehat{Y\setminus
Z}^{*},*)\hookrightarrow(\widehat{Y}^{*},*)$ and therefore
$\overline{R}(k)\cong(\mathcal{O}(G)^{*})^{(\widehat{Y}^{*},*)}$. We only need
to check, that $Y\setminus Z$ has the same cardinality as a basis of
$(R_{1}/[R_{1},F(X)](k))^{\vee}$. By construction $1\rightarrow
R_{1}\rightarrow F(X)\xrightarrow{\widehat{\pi}}G\rightarrow 1$ is the proper
presentation and therefore $(R_{1}/[R_{1},F(X)](k))^{\vee}\cong
H^{1}(R,k_{a})^{F(X)}\cong H^{2}(G,k_{a}).$ The 5-exact sequence in the
Lyndon-Hochschild-Serre spectral sequence of a presentation $1\rightarrow
R\rightarrow F\xrightarrow{\pi}G\rightarrow 1$ takes the form

$0\rightarrow H^{1}(G,k_{a})\xrightarrow{Inf}H^{1}(F,k_{a})\rightarrow
H^{1}(R,k_{a})^{F}\rightarrow H^{2}(G,k_{a})\rightarrow 0.$

By construction $Coker\;(Inf)=k^{Z}$, and, therefore, we have $0\rightarrow
k^{Z}\rightarrow(R/[R,F](k))^{\vee}\rightarrow(R_{1}/[R_{1},F(X)](k))^{\vee}\rightarrow
0$ \- the exact sequence of vector spaces, which imply that $k^{Y\setminus Z}$
($Y\setminus Z$ \- in the sense of cardinality of sets) is the basis of
$(R_{1}/[R_{1},F(X)](k))^{\vee}$ and hence (by Lemma 9)
$R_{1}/[R_{1},F(X)](k)\cong k^{(\widehat{Y\setminus Z}^{*},*)}$. But since
$\overline{R_{1}}(k)$ is free we have
$\overline{R_{1}}(k)\cong(\mathcal{O}(G)^{*})^{(\widehat{Y\setminus
Z}^{*},*)}$.

Proposition 17 below says that a presentation (12) is aspherical if and only
if its relation module is free. It is clear from this result that
$(R/[R,F](k))^{\vee}\cong k^{Y}$ if $u_{2}(X|R)=0$.

###### Proposition 17

Assume we are given a simplicial prounipotent presentation (12) of a
prounipotent group $G$. Then $u_{2}(X|R)=0$ if and only if there is an
isomorphism of topological $\mathcal{O}(G)^{*}$-modules
$\overline{R}(k)\cong(O(G)^{*})^{(\widehat{Y}^{*},*)},$ i.e. $\overline{R}(k)$
is a free topological $\mathcal{O}(G)^{*}$-module over the pointed profinite
space $(\widehat{Y}^{*},*)$.

###### Proof

Since $Im(\overline{d}_{1})(k)=R(k)$ is the subgroup of $F(k)$ it is also free
(LM, , Corollary 2.10) and, by Lemma 5, $\overline{d}_{1}(k)$ has a section,
say $s$, and therefore $C(k)\cong u_{2}(X|R)\times sR(k)$. Indeed,
$u_{2}(X|R)$ is central in $C$, as for $p\in C(k),a\in u_{2}(X|R)$ the
following equalities hold $p^{-1}a^{-1}pa=p^{-1}p^{\partial a}a=1$. But since
$u_{2}(X|R)$ is abelian it follows that $[C(k),C(k)]=[sR(k),sR(k)]$ and
therefore $[C(k),C(k)]\cap u_{2}(X|R)=1$ and, therefore,
$u_{2}(X|R)=Ker\;({\overline{C}(k)\rightarrow\overline{R}(k)})$. Now it
follows from Proposition 10 that
$\overline{C}(k)\cong(\mathcal{O}(G)^{*})^{(\widehat{Y}^{*},*)}$ (see also
(Mikh2016, , Corollary 3.17)) and the result is clear if the presentation is
proper.

Let us consider a non-proper presentation. As $\overline{R}(k)$ is a free
topological $\mathcal{O}(G)^{*}$-module, then its dual
$(\overline{R}(k))^{\vee}$ is the injective $G$-module and, therefore, the
image of the dual homomorphism
$\overline{d}_{1}^{\vee}:(\overline{R}(k))^{\vee}\rightarrow(\overline{C}(k))^{\vee}$
is a direct summand, i.e. there is a $G$-module decomposition
$(\overline{C}(k))^{\vee}\cong(\overline{R}(k))^{\vee}\oplus B$. By Remark 10,
the induced homomorphism on fixed points
$(\overline{d}_{1}^{\vee})^{G}:((\overline{R}(k))^{\vee})^{G}\rightarrow((\overline{C}(k))^{\vee})^{G}$
is the isomorphism, hence $B^{G}=0$. By Lemma 2, $B=0$ and therefore
$\overline{d}_{1}^{\vee}$ is the isomorphism, i.e. $u_{2}(X|R)=0$.

###  5.4 Tsvetkov’s criterion

Consider the diagram (15) of prounipotent groups arising from a prounipotent
presentation (12) of a prounipotent group $G$ with $cd(G)=2$ and its
subpresentation (14) of a prounipotent group $G_{0}$. In the case of positive
characteristics we always assume that the base field $k=\mathbb{F}_{p}$ and
$G$ is an affine group scheme, defined as in Example 5.

|
---|---
$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{G\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1}$$\textstyle{F\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho}$$\scriptstyle{\pi\circ\rho}$$\textstyle{R\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{R_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
(15)

The inclusion of normal subgroups $R_{0}\subset R$ of $F=F(X)$ induces a
homomorphism of relation modules

$\phi:\overline{R}_{0}(k)\rightarrow\overline{R}(k),$

that is, a homomorphism of topological $\mathcal{O}(G_{0})^{*}$-modules with
the module structure of Corollary 3. Since $R_{0}$ is normal in $F$,
$Im\;\phi$ is a topological $\mathcal{O}(G)^{*}$-submodule of
$\overline{R}(k)$. We remind that in the pro-$p$-case we always assume
$\overline{R}=R/R^{p}[R,R]$ is a topological $\mathbb{F}_{p}G$-module (i.e.
$\mathcal{O}(G)^{*}=\mathbb{F}_{p}G$).

Short exact sequence of groups $1\rightarrow R_{0}\rightarrow R\rightarrow
H\rightarrow 1$ gives rise to a low degree 5-exact sequence in the Lyndon-
Hochschild-Serre spectral sequence

$0\rightarrow
H^{1}(H,k_{a})\xrightarrow{Inf}H^{1}(R,k_{a})\xrightarrow{Res}H^{1}(R_{0},k_{a})^{R/R_{0}}\xrightarrow{Tg}H^{2}(H,k_{a})\rightarrow
0,$

where $H^{2}(R,k_{a})=0$ since $R$ is free.

The image of the restriction map $Res:H^{1}(R,k_{a})\rightarrow
H^{1}(R_{0},k_{a})$ in the 5-exact sequence coincide with $Im\;\phi^{\vee}$
and in particular, since $Im(Res)\subseteq H^{1}(R_{0},k_{a})^{R/R_{0}}$,
$Im(Res)$ is a $G$-module. Consider two exact sequences of $G$-modules

$0\rightarrow H^{1}(H,k_{a})\xrightarrow{Inf}H^{1}(R,k_{a})\rightarrow
Im\;\phi^{\vee}\rightarrow 0$ (16)

and

$0\rightarrow Im\;\phi^{\vee}\rightarrow
H^{1}(R_{0},k_{a})^{R/R_{0}}\xrightarrow{Tg}H^{2}(H,k_{a})\rightarrow 0$ (17)

###### Proposition 18

Suppose we are given a diagram (15) of prounipotent groups, then the following
conditions are equivalent:

(i)

$cd\;G_{0}\leq 2$

(ii)

$cd\;H\leq 1$

and

(I)

$Im\;\phi$ is the free topological $\mathcal{O}(G)^{*}$-module

(II)

$H^{0}(G,Im\;\phi^{\vee})\cong H^{0}(G_{0},H^{1}(R_{0},k_{a}))$ \- the map
arising by applying the fixed points functor to (17) is the isomorphism

###### Proof

First, we show that the conditions $(I)$ and $(II)$ are equivalent to:

(*)

$H^{1}(G,Im\;\phi^{\vee})=0;$

(**)

$H^{0}(G,Im\;\phi^{\vee})\cong H^{0}(G,H^{1}(R_{0},k_{a})^{R/R_{0}})$ \- the
map arising by applying the fixed points functor to (17) is the isomorphism

For any $G$-module $A$ and a normal subgroup $H\lhd G$ one has
$(A^{H})^{G/H}=A^{G}$, taking $R/R_{0}\lhd G_{0}$, we get
$H^{0}(G_{0},H^{1}(R_{0},k_{a}))\cong H^{0}(G,H^{1}(R_{0},k_{a})^{R/R_{0}})$
and therefore $(**)$ is equivalent to $(II)$. By Lemma 9 (I) is equivalent to
$H^{1}(G,(Im\;\phi)^{\vee})=0$ and, as
$Im\;\phi^{\vee}\cong(Im\;\phi)^{\vee}$, (I) is equivalent to (*).

Now we prove that (*) and (**) are equivalent to:

(a)

$H^{1}(G,H^{1}(H,k_{a}))\rightarrow H^{1}(G,H^{1}(R,k_{a}))\mbox{ is the
epimorphism};$

(b)

$H^{2}(G,H^{1}(H,k_{a}))\rightarrow H^{2}(G,H^{1}(R,k_{a}))\mbox{ is the
monomorphism};$

(c)

$H^{0}(G,H^{2}(H,k_{a}))=0$.

This follows from the long exact sequences of cohomology of (16) and (17):

$0\rightarrow H^{0}(G,H^{1}(H,k_{a}))\rightarrow
H^{0}(G,H^{1}(R,k_{a}))\rightarrow H^{0}(G,Im\;\phi^{\vee})$ $\rightarrow
H^{1}(G,H^{1}(H,k_{a}))\rightarrow
H^{1}(G,H^{1}(R,k_{a}))\rightarrow\mathbf{H^{1}(G,Im\;\phi^{\vee})}$
$\rightarrow H^{2}(G,H^{1}(H,k_{a}))\rightarrow
H^{2}(G,H^{1}(R,k_{a}))\rightarrow H^{2}(G,Im\;\phi^{\vee})\rightarrow\ldots$

and

$0\rightarrow\mathbf{H^{0}(G,Im\;\phi^{\vee})}\rightarrow\mathbf{H^{0}(G,H^{1}(R_{0},k_{a})^{R/R_{0}})}\rightarrow
H^{0}(G,H^{2}(H,k_{a}))$
$\rightarrow\mathbf{H^{1}(G,Im\;\phi^{\vee})}\rightarrow
H^{1}(G,H^{1}(R_{0},k_{a})^{R/R_{0}})\rightarrow
H^{1}(G,H^{2}(H,k_{a}))^{R/R_{0}}\rightarrow\ldots$

The next step is to show that conditions (a), (b), (c) are equivalent to:

(d)

$d_{2}^{1,1}:H^{1}(G,H^{1}(R,k_{a}))\rightarrow H^{3}(G,k_{a})\mbox{ is the
epimorphism};$

(e)

$d_{2}^{2,1}:H^{2}(G,H^{1}(R,k_{a}))\rightarrow H^{4}(G,k_{a})\mbox{ is the
monomorphism};$

(f)

$cd\;(H)\leq 1.$

By Lemma 2, (c) is equivalent to $H^{2}(H,k_{a})=0$ and hence to $cd\;H\leq 1$
(i.e. $H$ is free), therefore (c) is equivalent to (f). We arrange (a) and (b)
into the commutative diagram

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
39.61235pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{H^{p}(G,H^{1}(H,k_{a}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
80.21507pt\raise 6.57639pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.21529pt\hbox{$\scriptstyle{Inf^{*}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 141.27602pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
35.3599pt\raise-28.46887pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-3.61111pt\hbox{$\scriptstyle{\widetilde{d}_{2}^{p,1}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 64.87296pt\raise-29.5pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern
87.44418pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 141.27602pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{H^{p}(G,H^{1}(R,k_{a}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
132.28047pt\raise-27.2883pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.43056pt\hbox{$\scriptstyle{d_{2}^{p,1}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 115.90176pt\raise-29.5pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
63.61235pt\raise-41.13776pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{H^{p+2}(G,k_{a})}$}}}}}}}\ignorespaces}}}}\ignorespaces,$

where $Inf^{*}$ is induced by the inflation map $Inf:H^{1}(H,k_{a})\rightarrow
H^{1}(R,k_{a})$, $Inf^{*}$ coincides with (a), when $p=1$, and with (b), when
$p=2$. The maps $\widetilde{d}_{2}^{p,1}$ and $d_{2}^{p,1}$ are the
differentials in the spectral sequences $\widetilde{E}_{2}^{p,q}$,
$E_{2}^{p,q}$ of group extensions $1\rightarrow H\rightarrow G_{0}\rightarrow
G\rightarrow 1$ and $1\rightarrow R\rightarrow F\rightarrow G\rightarrow 1,$
respectively. Lets take $G_{0}$-module resolution (8) of the trivial module
$k_{a}$ and $F$-module resolution (8) of the trivial module $k_{a}$, these
resolution are also injective $H$ and $R$-module resolutions of the trivial
module $k_{a}$. Then $Inf$ is induced on the cohomology by the inclusion of
the complexes (Jan, , 6.10)

$(k_{a}\otimes\mathcal{O}(G_{0})^{\otimes(q+1)})^{H}\rightarrow(k_{a}\otimes\mathcal{O}(F)^{\otimes(q+1)})^{R}.$

Let $\gamma_{0}:\widetilde{E}_{0}^{p,q}\rightarrow E_{0}^{p,q}$ be the induced
inclusion of the double complexes

$(k_{a}\otimes\mathcal{O}(G_{0})^{\otimes(q+1)})^{H}\otimes\mathcal{O}(G)^{\otimes
p}\rightarrow(k_{a}\otimes\mathcal{O}(F)^{\otimes(q+2)})^{R}\otimes\mathcal{O}(G)^{\otimes
p},$

then $\gamma_{0}$ commute with $d_{0}^{p,q}$, $\widetilde{d}_{0}^{p,q}$,
$d_{1}^{p,q}$, $\widetilde{d}_{1}^{p,q}$ and, therefore, by (HSJ, , VIII,9),
$\gamma_{0}$ induces a homomorphism of spectral sequences
$\gamma_{n}:\widetilde{E}_{n}^{p,q}\rightarrow E_{n}^{p,q}$ and in particular
$\gamma_{2}:H^{p}(G,H^{1}(H,k_{a}))\rightarrow H^{p}(G,H^{1}(R,k_{a}))$. By
construction, $\gamma_{2}$ coincides with $Inf^{*}$ (an explicit mapping is
given in Remark 2 and therefore $\widetilde{d}^{p,1}_{2}=d_{2}^{p,1}\circ
Inf^{*}$.

The Lyndon-Hochschild-Serre spectral sequence of the group extension

$1\rightarrow R\rightarrow F\rightarrow G\rightarrow 1$

degenerates (since $R,F$ are free), hence $d_{2}^{p,1}$ is an isomorphism and
therefore conditions (a), (b) are equivalent to (d), (e).

Let us also consider the Lyndon-Hochschild-Serre spectral sequence
$E_{2}^{p,q}=H^{p}(G,H^{q}(H,k_{a}))$ of the group extension $1\rightarrow
H\rightarrow G_{0}\rightarrow G\rightarrow 1$ which also degenerates for
$q>1$, since $H$ is free and, therefore, $E_{3}^{p,q}\cong E_{\infty}^{p,q}.$

A filtration of $H^{p+q}(G_{0},k_{a})$ provided by $E_{\infty}^{p,q}$ is given
by the formulae as follows

$E_{\infty}^{p,q}\cong
gr_{p}H^{p+q}(G,k_{a})=F^{p}H^{p+q}(G,k_{a})/F^{p+1}H^{p+q}(G,k_{a})$

and therefore (d),(e),(f) are equivalent to $H^{3}(G_{0},k_{a})=0,$ $cd\;H\leq
1,$ i.e. conditions (i), (ii) of Proposition 18.

###  5.5 Proof of Theorem 4.1

We use notations of sections 4.5, 5.4, i.e.
$R_{0}(k)=\langle\tau(\widehat{Y}_{1}^{*})\rangle_{F(k)}$, where $Y_{1}\subset
Y$.

###### Proof

The inclusion of normal subgroups $R_{0}\subset R$ induces an abelianization
homomorphism $\phi:\overline{R_{0}}(k)\rightarrow\overline{R}(k)$. First of
all, note that $Im\;\phi\cong(\mathcal{O}(G)^{*})^{(\widehat{Y_{1}}^{*},*)}$
is a free topological module on the profinite space $(\widehat{Y_{1}}^{*},*)$.
By Proposition 17,
$\overline{R}(k)\cong(\mathcal{O}(G)^{*})^{(\widehat{Y}^{*},*)}$ and
$Im\;\phi$ is a closed submodule of $\overline{R}(k)$ generated as a
topological $\mathcal{O}(G)^{*}$-submodule by $(\widehat{Y_{1}}^{*},*)$. Now
we want to show, that (17) induces the isomorphism
$H^{0}(G,Im\;\phi^{\vee})\cong H^{0}(G_{0},H^{1}(R_{0},k_{a})).$

The last isomorphism is equivalent to the isomorphism
$(Im\;\phi^{\vee})^{G}\cong H^{1}(R_{0},k_{a})^{G_{0}}$ of $k$-spaces arising
from the embedding of (17) and by continuous duality to the isomorphism
$((Im\;\phi^{\vee})^{G})^{*}\cong((H^{1}(R_{0},k_{a}))^{G_{0}})^{*}$. By Lemma
3 that is equivalent to the isomorphism
$(Im\;\phi^{\vee})^{*}_{G}\cong(H^{1}(R_{0},k_{a})^{*})_{G_{0}}.$ But
$Im\;\phi^{\vee}\cong(Im\;\phi)^{\vee}$ and so we must check that taking
$G$-coinvariants of $\phi$ imply the isomorphism $(Im\;\phi)_{G}\cong
R_{0}/[R_{0},F]$. Indeed,
$(Im\;\phi)_{G}\cong((\mathcal{O}(G)^{*})^{(\widehat{Y_{1}}^{*},*))}_{G}\cong
k^{(\widehat{Y_{1}}^{*},*)}$ and $R_{0}/[R_{0},F]\cong
k^{(\widehat{Y_{1}}^{*},*)}$, as $R_{0}$ is taken from (14).

We are now ready to apply Proposition 18 (which shows that $cd\;G_{0}\leq 2$)
and Propositions 16, 17 in order to show that $(X|R_{0})$ is aspherical.

##  6 An application: discrete group presentations

In the recent paper Berrick and Hillman have obtained the following beautiful
consequence (BH, , Corollary 4.8).

###### Proposition 19

The Whitehead Conjecture is equivalent to the conjecture that the fundamental
group of a finite subcomplex of a contracnible 2-complex has rational
cohomological dimension at most 2.

This result eliminate the crucial role of finite subcomplexes of contractible
2-complexes in the study of asphericity.

From now and on, we use the standard notation of the section 4.5 with the only
difference that for brevity we will identify in the notation the normal
subgroup and the relations generating it. As always, we replace arbitrary
contractible 2-complex by a complex of a presentation $(X|Y)$ of the trivial
group. As in Section 2.2 consider its combinatorial model $S_{1}K(X|Y)$. In
what follows we shall prove that the Bousfield-Kan $R$-completion
$R_{\infty}S_{1}K(X|Y)$ is an aspherical space.

We remind (see Section 1) that $R_{\infty}S_{1}K(X|Y)$ may be obtained as the
classifying space $\overline{W}((GS_{1}K(X|Y))^{\wedge}_{R})$ of the
dimension-wise $R$-nilpotent completion of the Kan loop-group $G$ of
$S_{1}K(X|Y)$. Recal, by Section 2.2 the geometric realization of
$GS_{1}K(X|Y)$ is homotopy equivalent to $\Omega K(X|Y)$. Since
$(X_{1}|Y_{1})$ is a finite presentation it follows that the dimension-wise
$R$-nilpotent completion $(GS_{1}K(X_{1}|Y_{1}))^{\wedge}_{R}$ is homotopy
equivalent to the corresponding pro-unipotent simplicial group presentation
$(X_{1}|Y_{1})_{u}$ (Mikh2016, , (3.3)), see Section 4.4.

###### Corollary 4

Let $(X_{1}|Y_{1})$ be a finite subpresentation of a contractible presentation
$(X|Y)$ and $R=k$ is a field of zero characteristics or $R=\mathbb{F}_{p}$,
then $R_{\infty}S_{1}K(X_{1}|Y_{1})$ is aspherical.

###### Proof

Since $(X|Y)$ is contractible it is also aspherical, in our case $R=\Phi(X)$
and $R/[R,R]$ is forced to be isomorphic to $\mathbb{Z}G^{Y}$ (see Section
2.1), therefore we obtain a non-canonical isomorphism
$\mathbb{Z}^{X}\cong\Phi(X)/[\Phi(X),\Phi(X)]\cong\mathbb{Z}^{Y}.$ Now we
consider the corresponding prounipotent presentation $(\widehat{X}^{*}|Y)$ of
the trivial prounipotent group, so $R_{u}=\langle
Y\rangle_{F}=F(\widehat{X}^{*},*)$ is the Zariski closure of defining
relations $r_{y}\in\Phi(X),y\in Y$ in the the free prounipotent group
$F=F(X)(k)$.

Since the images of $r_{y},y\in Y$ generate $F/[F,F]$ and are linearly
independent, it follows from free prounipotent group construction (see
Sections 4.1 and 4.3, Proposition 2) that they are generators of $F$, i.e
$F(X)(k)=F(Y)(k)$. By assumption, $X_{1}$ and $Y_{1}$ are finite, so we may
choose elements of $Y_{1}$ as generators of $F(X_{1})(k)$ and therefore
$F(X_{1})/(Y_{1})_{u}\cong F(\widetilde{Y}_{1})$ for some
$\widetilde{Y}_{1}\subseteq X_{1}$, i.e. $(X_{1}|Y_{1})_{u}$ is a presentation
of a free prounipotent group, hence $cd\;F(X_{1})/(Y_{1})_{u}\leq 1$. By
Propositions 16, 17 (or by Theorem 4.1), $(X_{1}|Y_{1})_{u}$ is an aspherical
simplicial prounipotent presentation and therefore
$\overline{W}(X_{1}|Y_{1})_{u}$ is an aspherical simplicial set in the common
sense. But $X_{1}$ and $Y_{1}$ are finite and, as mentioned above,
$R_{\infty}S_{1}K(X_{1}|Y_{1})\simeq\overline{W}(X_{1}|Y_{1})_{u}$ is a
homotopy equivalence and the result follows.

The arithmetic square argument imply the following consequence.

###### Corollary 5

Let $(X_{1}|Y_{1})$ be a finite subpresentation of a contractible presentation
$(X|Y)$ then $\mathbb{Z}_{\infty}S_{1}K(X_{1}|Y_{1})$ is aspherical.

###### Remark 14

In BDH  it was shown (see also D ) that any homotopy type $X$ can be
represented by a pair of discrete groups $(G_{X},P_{X})$, where $P_{X}$ is a
normal perfect subgroup in $G_{X}$ (a group $P$ is said to be perfect if
$P=[P,P]$). In more detail, $X$ can be recovered in the homotopy category from
the Quillen plus-construction $(K(G_{X},1),P_{X})^{+}$. Thus, the category of
connected $CW$-complexes and pointed homotopy classes of maps is equivalent to
the category of fractions of the category of pairs $(G,P)$, where $G$ is a
group and $P$ is a perfect normal subgroup.

One can think of Bousfield-Kan $R$-completion as a homotopically reasoned
functorial way of replacing the usual homotopy type with a homotopy type that
escapes perfect subgroups. Corollaries 4 and 5 show that “$R$-extraction” of
perfect subgroups turns finite subrepresentations of contractible
presentations into aspherical ones and thus pushing Whitehead’s conjecture now
to the plane of perfect subgroups. It is noteworthy that in the category of
prounipotent groups, where there are no nontrivial perfect subgroups,
asphericity is completely determined, as it should be, with the help of
$K(G,1)$ (namely Propositions 16,17) by the cohomological dimension.

The author is sincerely gratefull to the referee for insisting that I not omit
an application of the main result. The author would also like to thank the
referee for the useful reference (CCH, , Proposition 2.6.), which suggests
another way to get some of the paper’s results in the pro-$p$-case.

## References

  * (1) Eiichi Abe.  Hopf algebras, volume 74 of Cambridge Tracts in Mathematics.  Cambridge University Press, Cambridge-New York, 1980.  Translated from the Japanese by Hisae Kinoshita and Hiroko Tanaka.
  * (2) David J. Anick.  A rational homotopy analog of Whitehead’s problem.  In Algebra, algebraic topology and their interactions (Stockholm, 1983), volume 1183 of Lecture Notes in Math., pages 28–31. Springer, Berlin, 1986.
  * (3) Hyman Bass.  Traces and Euler characteristics.  In Homological group theory (Proc. Sympos., Durham, 1977), volume 36 of London Math. Soc. Lecture Note Ser., pages 1–26. Cambridge Univ. Press, Cambridge-New York, 1979.
  * (4) Tilman Bauer.  Bousfield localization and the Hasse square.  In Topological modular forms. Based on the Talbot workshop, North Conway, NH, USA, March 25–31, 2007, pages 79–88. Providence, RI: American Mathematical Society (AMS), 2014.
  * (5) G. Baumslag, E. Dyer, and A. Heller.  The topology of discrete groups.  J. Pure Appl. Algebra, 16:1–47, 1980.
  * (6) A. J. Berrick and J. A. Hillman.  Whitehead’s asphericity question and its relation to other open problems.  In Algebraic topology and related topics, Trends Math., pages 27–49. Birkhäuser/Springer, Singapore, 2019.
  * (7) Mladen Bestvina and Noel Brady.  Morse theory and finiteness properties of groups.  Invent. Math., 129(3):445–470, 1997.
  * (8) William A. Bogley.  J. H. C. Whitehead’s asphericity question.  In Two-dimensional homotopy and combinatorial group theory, volume 197 of London Math. Soc. Lecture Note Ser., pages 309–334. Cambridge Univ. Press, Cambridge, 1993.
  * (9) A. K. Bousfield and V. K. A. M. Gugenheim.  On ${\rm PL}$ de Rham theory and rational homotopy type.  Mem. Amer. Math. Soc., 8(179):ix+94, 1976.
  * (10) A. K. Bousfield and D. M. Kan.  Homotopy limits, completions and localizations.  Lecture Notes in Mathematics, Vol. 304. Springer-Verlag, Berlin-New York, 1972.
  * (11) R. Brown and J. Huebschmann.  Identities among relations.  In Low-dimensional topology (Bangor, 1979), volume 48 of London Math. Soc. Lecture Note Ser., pages 153–202. Cambridge Univ. Press, Cambridge-New York, 1982.
  * (12) Ronald Brown and Jean-Louis Loday.  Van Kampen theorems for diagrams of spaces.  Topology, 26(3):311–335, 1987.  With an appendix by M. Zisman.
  * (13) Pierre Cartier.  A primer of Hopf algebras.  In Frontiers in number theory, physics, and geometry. II, pages 537–615. Springer, Berlin, 2007.
  * (14) Ian M. Chiswell, Donald J. Collins, and Johannes Huebschmann.  Aspherical group presentations.  Math. Z., 178:1–36, 1981.
  * (15) Aristide Deleanu.  On the theorem of Baumslag, Dyer and Heller linking group theory and topology.  Cah. Topologie Géom. Différ. Catégoriques, 23:231–242, 1982\.
  * (16) J. Dieudonné.  Introduction to the theory of formal groups.  Marcel Dekker, Inc., New York, 1973.  Pure and Applied Mathematics, 20.
  * (17) Beno Eckmann.  Poincaré duality groups of dimension two are surface groups.  In Combinatorial group theory and topology (Alta, Utah, 1984), volume 111 of Ann. of Math. Stud., pages 35–51. Princeton Univ. Press, Princeton, NJ, 1987.
  * (18) Michael D. Fried and Moshe Jarden.  Field arithmetic, volume 11 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics].  Springer-Verlag, Berlin, third edition, 2008.  Revised by Jarden.
  * (19) P. Gabriel and M. Zisman.  Calculus of fractions and homotopy theory.  Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 35. Springer-Verlag New York, Inc., New York, 1967.
  * (20) Alexander Grothendieck.  Sur quelques points d’algèbre homologique.  Tohoku Math. J. (2), 9:119–221, 1957.
  * (21) Richard Hain and Makoto Matsumoto.  Weighted completion of Galois groups and Galois actions on the fundamental group of $\mathbb{P}^{1}-\\{0,1,\infty\\}$.  Compositio Math., 139(2):119–167, 2003.
  * (22) Richard M. Hain.  Algebraic cycles and extensions of variations of mixed Hodge structure.  In Complex geometry and Lie theory (Sundance, UT, 1989), volume 53 of Proc. Sympos. Pure Math., pages 175–221. Amer. Math. Soc., Providence, RI, 1991.
  * (23) Allen Hatcher.  Algebraic topology.  Cambridge University Press, Cambridge, 2002.
  * (24) G. Hochschild and J.-P. Serre.  Cohomology of group extensions.  Trans. Amer. Math. Soc., 74:110–134, 1953.
  * (25) Sze-tsen Hu.  Homotopy theory.  Pure and Applied Mathematics, Vol. VIII. Academic Press, New York-London, 1959.
  * (26) Andrei Jaikin-Zapirain.  Approximation by subgroups of finite index and the Hanna Neumann conjecture.  Duke Math. J., 166(10):1955–1987, 2017.
  * (27) Jens Carsten Jantzen.  Representations of algebraic groups, volume 107 of Mathematical Surveys and Monographs.  American Mathematical Society, Providence, RI, second edition, 2003.
  * (28) Daniel M. Kan.  A relation between ${\rm CW}$-complexes and free c.s.s. groups.  Amer. J. Math., 81:512–528, 1959.
  * (29) Kevin P. Knudson.  Relative completions and the cohomology of linear groups over local rings.  J. London Math. Soc. (2), 65(1):183–203, 2002.
  * (30) Helmut Koch.  Galois theory of $p$-extensions.  Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2002.  With a foreword by I. R. Shafarevich, Translated from the 1970 German original by Franz Lemmermeyer, With a postscript by the author and Lemmermeyer.
  * (31) Alexander Lubotzky.  Pro-finite presentations.  J. Algebra, 242(2):672–690, 2001.
  * (32) Alexander Lubotzky and Andy Magid.  Cohomology of unipotent and prounipotent groups.  J. Algebra, 74(1):76–95, 1982.
  * (33) Alexander Lubotzky and Andy R. Magid.  Cohomology, Poincaré series, and group algebras of unipotent groups.  Amer. J. Math., 107(3):531–553, 1985.
  * (34) Andy R. Magid.  Relation modules of prounipotent groups.  J. Algebra, 109(1):52–68, 1987.
  * (35) J. Peter May.  Simplicial objects in algebraic topology.  Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 1992.  Reprint of the 1967 original.
  * (36) A. M. Mikhovich.  Quasirationality and aspherical (pro-$p$-) presentations.  Mat. Zametki, 105(4):553–563, 2019.
  * (37) A. M. Mikhovich.  Quasirationality and prounipotent crossed modules.  J. Knot Theory Ramifications, 28(13):1940012, 23, 2019.
  * (38) Andrey Mikhovich.  Complete group rings as Hecke algebras.  Topology Appl., 275:107027, 9, 2020.
  * (39) Andrey M. Mikhovich.  Identity theorem for pro-$p$-groups.  In Knots, low-dimensional topology and applications, volume 284 of Springer Proc. Math. Stat., pages 363–387. Springer, Cham, 2019.
  * (40) John Milnor.  On spaces having the homotopy type of a ${\rm CW}$-complex.  Trans. Amer. Math. Soc., 90:272–280, 1959.
  * (41) Susan Montgomery.  Hopf algebras and their actions on rings, volume 82 of CBMS Regional Conference Series in Mathematics.  Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1993.
  * (42) A. Mutlu and T. Porter.  Free crossed resolutions from simplicial resolutions with given CW-basis.  Cahiers Topologie Géom. Différentielle Catég., 40(4):261–283, 1999.
  * (43) T. Porter.  Profinite algebraic homotopy.  247, 2009.
  * (44) Daniel Quillen.  Rational homotopy theory.  Ann. of Math. (2), 90:205–295, 1969.
  * (45) Daniel G. Quillen.  Homotopical algebra.  Lecture Notes in Mathematics, No. 43. Springer-Verlag, Berlin-New York, 1967.
  * (46) Daniel G. Quillen.  The geometric realization of a Kan fibration is a Serre fibration.  Proc. Amer. Math. Soc., 19:1499–1500, 1968.
  * (47) Luis Ribes and Pavel Zalesskii.  Profinite groups, volume 40 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics].  Springer-Verlag, Berlin, second edition, 2010.
  * (48) N. S. Romanovskiĭ.  On metabelian pro-$p$-groups with a single relation.  Algebra Logika, 59(1):116–122, 2020.
  * (49) Stephan Rosebrock.  The Whitehead conjecture—an overview.  Sib. Èlektron. Mat. Izv., 4:440–449, 2007.
  * (50) Walter Rudin.  Functional analysis.  International Series in Pure and Applied Mathematics. McGraw-Hill, Inc., New York, second edition, 1991.
  * (51) Jean-Pierre Serre.  Lie algebras and Lie groups. 1964 lectures, given at Harvard University. 2nd ed, volume 1500.  Berlin etc.: Springer-Verlag, 2nd ed. edition, 1992.
  * (52) Jean-Pierre Serre.  Galois cohomology.  Springer Monographs in Mathematics. Springer-Verlag, Berlin, english edition, 2002.  Translated from the French by Patrick Ion and revised by the author.
  * (53) Igor R. Shafarevich.  Basic notions of algebra, volume 11 of Encyclopaedia of Mathematical Sciences.  Springer-Verlag, Berlin, 2005.  Translated from the 1986 Russian original by Miles Reid, Reprint of the 1997 English translation [MR1634541], Algebra, I.
  * (54) Dennis Sullivan.  Genetics of homotopy theory and the Adams conjecture.  Ann. of Math. (2), 100:1–79, 1974.
  * (55) Dennis P. Sullivan.  Geometric topology: localization, periodicity and Galois symmetry, volume 8 of $K$-Monographs in Mathematics.  Springer, Dordrecht, 2005.  The 1970 MIT notes, Edited and with a preface by Andrew Ranicki.
  * (56) V. M. Tsvetkov.  On pro-$p$-groups of cohomology dimension two.  Dokl. Akad. Nauk SSSR, 271(6):1329–1332, 1983.
  * (57) A. Vezzani.  The pro-unipotent completion.  11, 2012.
  * (58) William C. Waterhouse.  Introduction to affine group schemes, volume 66 of Graduate Texts in Mathematics.  Springer-Verlag, New York-Berlin, 1979.
  * (59) J. H. C. Whitehead.  On adding relations to homotopy groups.  Ann. of Math. (2), 42:409–428, 1941.