For a discrete group $\Gamma$ the proalgebraic completion
$\Gamma^{\operatorname{alg}}$ over $k$ is a perfect group scheme over $k$.

  2. (ii)

Let $X$ be a connected topological space and $x_{0}\in X$ a point. Then the
proalgebraic fundamental group $\pi_{k}(X,x_{0})$ over $k$ is a perfect group
scheme over $k$.

######  Example 4.4.

For $\Gamma=\mathbb{Z}$ and $k$ algebraically closed of characteristic $p>0$
one may find the calculation of $\mathbb{Z}^{\operatorname{alg}}$ in [Sau02,
Annexe A]. There is a canonical isomorphism:

$\mathbb{Z}^{\operatorname{alg}}=\operatorname{Spec}k[k^{\times}]\times\mathbb{Z}_{p}\;.$

Here $\operatorname{Spec}k[k^{\times}]$ is the diagonalizable group scheme on
$k^{\times}$ over $k$ and $\mathbb{Z}_{p}$ denotes the unipotent, pro-étale
group scheme obtained as the projective limit of the constant group schemes
$\mathbb{Z}/p^{n}$. Since $k$ is perfect, it is clear that
$\mathbb{Z}^{\operatorname{alg}}$ is perfect as well in accordance with the
corollary.

###### Proof.

(i) By definition, $\Gamma^{\operatorname{alg}}$ is the Tannakian dual of the
category $\operatorname{Rep}(\Gamma)$ of representations of $\Gamma$ on finite
dimensional $k$-vector spaces $V$. The functor $\operatorname{Fr}_{+}$ sends
the $\Gamma$-representation $V$ to the induced $\Gamma$-representation on the
vector space $\operatorname{Fr}_{+}(V)$. The $k$-linear automorphism $t(V)$ in
Proposition 4.1 (i) is $\Gamma$-equivariant. It follows that the functor
$\operatorname{Fr}_{+}:\operatorname{Rep}(\Gamma)^{(p)}\to\operatorname{Rep}(\Gamma)$
is isomorphic to the functor $\\_\otimes_{k,(\,)^{p}}k$ which is an
equivalence of categories. Now we use Corollary 4.2 (ii).
(ii) By definition $\pi_{k}(X,x_{0})$ is the Tannakian dual of the category
$\mathbb{Loc}_{k}(X)$ of locally constant sheaves $\mathcal{E}$ of finite
dimensional $k$-vector spaces on $X$ together with the fibre functor
$\omega(\mathcal{E})=\mathcal{E}_{x_{0}}$. The functor $\operatorname{Fr}_{+}$
sends $\mathcal{E}$ to the locally constant sheaf
$\operatorname{Fr}_{+}(\mathcal{E})$. We define a map of sheaves

$t(\mathcal{E}):\mathcal{E}\otimes_{k,(\,)^{p}}k\longrightarrow\operatorname{Fr}_{+}(\mathcal{E})$

by sending $s\otimes\lambda$ for a local section $s$ of $\mathcal{E}$ to the
image in $\operatorname{Fr}_{+}(\mathcal{E})$ of
$\lambda[s\otimes\ldots\otimes s]$. For the stalk at a point $x\in X$ we have
$t(\mathcal{E})_{x}=t(\mathcal{E}_{x})$. Since the $t(\mathcal{E}_{x})$ are
isomorphisms of $k$-vector spaces by Proposition 4.1 (i), it follows that
$t(\mathcal{E})$ is a $k$-linear isomorphism of sheaves. Hence the functor
$\operatorname{Fr}_{+}:\mathbb{Loc}_{k}(X)^{(p)}\to\mathbb{Loc}_{k}(X)$ is
isomorphic to the functor $\\_\otimes_{k,(\,)^{p}}k$ which is an equivalence
of categories. Now we use Corollary 4.2 (ii). ∎

######  Remark 4.5.

In the remark after [Den, Theorem 2.5] we had previously shown that
$\pi_{k}(X,x_{0})$ was reduced.

##  5\. The proalgebraic fundamental group of the Kucharczyk-Scholze space

Given a field $F$ of characteristic zero containing all roots of unity, and an
injective homomorphism $\mu(F)\hookrightarrow\mathbb{C}^{\times}$, Kucharczyk
and Scholze have defined a scheme $\mathfrak{X}_{F}$ over $\mathbb{C}$ and a
canonical isomorphism [KS18, Corollary 4.15]

$\pi^{\operatorname{et}}_{1}(\mathfrak{X}_{F}(\mathbb{C}),x)=G_{F}\;.$ (19)

Here $\mathfrak{X}_{F}(\mathbb{C})$ carries the complex topology and
$x\in\mathfrak{X}_{F}(\mathbb{C})$ is a point. The étale fundamental group
$\pi^{\operatorname{et}}_{1}(Z,z)$ of a pointed connected topological space
$(Z,z)$ is defined in [KS18, Section 2.2] using the formalism of Grothendieck
categories. It is a profinite group which classifies the finite coverings of
$Z$. Finally $G_{F}$ is a version of the absolute Galois group of $F$ defined
via a fibre functor related to $(\mathfrak{X}_{F},x)$. The group $G_{F}$ is
isomorphic to the usual Galois group $\operatorname{Aut}_{F}(\overline{F})$
where $\overline{F}$ is an algebraic closure of $F$, the isomorphism being
unique up to an inner isomorphism. By [Den, Theorem 3.6], the étale
fundamental group $\pi^{\operatorname{et}}_{1}(Z,z)$ viewed as a group-scheme
over a field $k$ is canonically isomorphic to the maximal pro-étale quotient
of the proalgebraic group $\pi_{k}(Z,z)$ i.e.

$\pi_{k}(Z,z)^{\operatorname{et}}\xrightarrow{\sim}\pi^{\operatorname{et}}_{1}(Z,z)\;.$

In particular, we have

$\pi_{k}(\mathfrak{X}_{F}(\mathbb{C}),x)^{\operatorname{et}}\xrightarrow{\sim}\pi^{\operatorname{et}}_{1}(\mathfrak{X}_{F}(\mathbb{C}),x)=G_{F}\;.$

Thus the group of connected components of
$\pi_{k}(\mathfrak{X}(\mathbb{C}),x)$ may be identified with $G_{F}$. The
connected component $\pi_{k}(\mathfrak{X}_{F}(\mathbb{C}),x)^{0}$ is abelian
and we determine its structure. We also discuss relations with the motivic
Galois groups of $F$ and $\overline{F}$. For the calculation of
$\pi_{k}(\mathfrak{X}_{F}(\mathbb{C}),x)$ we use the same approach as in
[KS18] applied to $\pi_{k}$ instead of $\pi^{\operatorname{et}}_{1}$.

Given a connected topological space $Z$ and a continuous path $\gamma:[0,1]\to
Z$ with $\gamma(0)=z_{0}$ and $\gamma(1)=z_{1}$ we obtain a
$\otimes$-isomorphism $\omega_{\gamma}:\omega_{z_{0}}\to\omega_{z_{1}}$
between the fibre functors
$\omega_{z_{i}}:\mathbb{Loc}_{k}(Z)\to\operatorname{Vec}_{k}$ for $i=0,1$ as
follows: For $\mathcal{E}$ in $\mathbb{Loc}_{k}(Z)$ the locally constant sheaf
$\gamma^{-1}\mathcal{E}$ is constant on $[0,1]$. Evaluation therefore provides
isomorphisms

$\omega_{z_{0}}(\mathcal{E})=\mathcal{E}_{z_{0}}\underset{\operatorname{ev}_{z_{0}}}{\xleftarrow{\sim}}\Gamma([0,1],\gamma^{-1}\mathcal{E})\underset{\operatorname{ev}_{z_{1}}}{\xrightarrow{\sim}}\mathcal{E}_{z_{1}}\cong\omega_{z_{1}}(\mathcal{E})\;.$

The natural transformation
$\omega_{\gamma}=\operatorname{ev}_{z_{1}}\circ\operatorname{ev}^{-1}_{z_{0}}$
depends only on the homotopy class of $\gamma$. Let

$\operatorname{Aut}^{\otimes}(\omega_{z_{0}})\underset{L_{\omega_{\gamma}}}{\xrightarrow{\sim}}\operatorname{Iso}^{\otimes}(\omega_{z_{0}},\omega_{z_{1}})\underset{R_{\omega_{\gamma}}}{\xleftarrow{\sim}}\operatorname{Aut}^{\otimes}(\omega_{z_{1}})$

be defined by left resp. right translation of the point
$\omega_{\gamma}\in\operatorname{Iso}^{\otimes}(\omega_{z_{0}},\omega_{z_{1}})(k)$.
We obtain an isomorphism of $k$-groups

$\gamma_{k}=R^{-1}_{\omega_{\gamma}}\circ
L_{\omega_{\gamma}}:\pi_{k}(Z,z_{0})\xrightarrow{\sim}\pi_{k}(Z,z_{1})\;.$

######  Proposition 5.1.

Let $f,g:X\to Z$ be continuous maps between connected topological spaces. Let
$H:X\times[0,1]\to Z$ be a homotopy between $f$ and $g$. For a fixed point
$\tilde{x}\in X$ consider the path $\gamma:[0,1]\to Z$ from $f(\tilde{x})$ to
$g(\tilde{x})$ defined by $\gamma(t)=H(\tilde{x},t)$. Then the following
diagram of $k$-groups commutes

$\textstyle{\pi_{k}(Z,f(\tilde{x}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\wr\;\gamma_{k}}$$\textstyle{\pi_{k}(X,\tilde{x})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{\ast}}$$\scriptstyle{g_{\ast}}$$\textstyle{\pi_{k}(Z,g(\tilde{x}))\;.}$

For the proof we need a lemma on local systems. The formulation is a little
bit involved because apart from connectedness the topological spaces $X,Z$ in
Proposition 5.1 are arbitrary.

######  Lemma 5.2.

Consider continuous maps
$Y\overset{\displaystyle\xrightarrow{\pi}}{\underset{S}{\longleftarrow}}X$
between connected topological spaces such that $\pi\circ s=\operatorname{id}$.
Assume that for all $x\in X$ the following conditions hold:

  1. (i)

The fibre $\pi^{-1}(x)$ is a quasi-compact subspace of $Y$ and any two
different points of $\pi^{-1}(x)$ have disjoint neighborhoods in $Y$.

  2. (ii)

For any open subset $V$ of $Y$ with $\pi^{-1}(x)\subset V$ there is an open
neighborhood $x\in U\subset X$ with $\pi^{-1}(U)\subset V$.

  3. (iii)

All local systems in $\mathbb{Loc}_{k}(\pi^{-1}(x))$ are constant.

Then for any local system $\mathcal{F}$ in $\mathbb{Loc}_{k}(Y)$ the natural
map of sheaves $\pi_{\ast}\mathcal{F}\to s^{-1}\mathcal{F}$ is an isomorphism.

######  Example 5.3.

Let $K$ be a compact, path-connected, locally path-connected and simply
connected topological space. Let $X$ be a connected topological space. Then
conditions (i)-(iii) in Lemma 5.2 are satisfied for $Y=X\times
K\overset{\displaystyle\xrightarrow{\pi}}{\underset{s}{\longleftarrow}}X$,
where $s$ is any continuous section of the projection $\pi$.

Namely, (i) is clear and (iii) follows from the equivalences

$\mathbb{Loc}_{k}(\pi^{-1}(x))\cong\mathbb{Loc}_{k}(K)\cong\operatorname{Rep}(\pi_{1}(K,t))\;,\;t\in
K$

which are valid by our hypothesis, and the fact that $\pi_{1}(K,t)$ is
trivial. As for (ii), let $V\subset Y$ be open with
$V\supset\pi^{-1}(x)=\\{x\\}\times K$. For any point $t\in K$, there are open
neighborhoods $U_{t}$ of $x$ in $X$ and $W_{t}$ of $t$ in $K$ such that
$V\supset U_{t}\times W_{t}$. Since $K$ is compact, there are finitely many
points $t_{1},\ldots,t_{n}\in K$ such that $W_{t_{1}},\ldots,W_{t_{n}}$ cover
$K$. Then $U=U_{t_{1}}\cap\ldots\cap U_{t_{n}}$ is an open neighborhood of $x$
in $X$ with $V\supset U\times W_{t_{i}}$ for all $i$, and hence $V\supset
U\times K=\pi^{-1}(U)$.

Proof of Lemma 5.2. We recall the definition of the map
$\pi_{\ast}\mathcal{F}\to s^{-1}\mathcal{F}$. Let $s^{0}\mathcal{F}$ be the
presheaf on $X$ with sections

$(s^{0}\mathcal{F})(U)=\operatorname{colim}_{V\supset s(U)}\mathcal{F}(V)\;.$

The relation $\pi\circ s=\operatorname{id}$ implies that $\pi^{-1}(U)\supset
s(U)$. The resulting natural maps

$(\pi_{*}\mathcal{F})(U)=\mathcal{F}(\pi^{-1}(U))\longrightarrow(s^{0}\mathcal{F})(U)$

give a map of presheaves $\pi_{*}\mathcal{F}\to s^{0}\mathcal{F}$ and hence a
map of sheaves $\pi_{*}\mathcal{F}\to s^{-1}\mathcal{F}$. It suffices to check
that the induced maps on the stalks are isomorphisms

$(\pi_{*}\mathcal{F})_{x}=\operatorname{colim}_{U\ni
x}\mathcal{F}(\pi^{-1}(U))\longrightarrow(s^{-1}\mathcal{F})_{x}=\mathcal{F}_{s(x)}\;.$

For this, note that because of (ii) we have

$(\pi_{*}\mathcal{F})_{x}=\operatorname{colim}_{V\supset\pi^{-1}(x)}\mathcal{F}(V)\;.$

Condition (i) allows us to apply [Sta23, Tag 09V3] to deduce that

$\operatorname{colim}_{V\supset\pi^{-1}(x)}\mathcal{F}(V)=\Gamma(\pi^{-1}(x),\mathcal{F}\,|_{\pi^{-1}(x)})\;.$

Finally, since $\mathcal{F}\,|_{\pi^{-1}(x)}$ is a constant local system by
condition (iii), the evaluation map in the point $s(x)\in\pi^{-1}(x)$ gives an
isomorphism

$\Gamma(\pi^{-1}(x),\mathcal{F}\,|_{\pi^{-1}(x)})\underset{\operatorname{ev}_{s(x)}}{\xrightarrow{\sim}}\mathcal{F}_{s(x)}=(s^{-1}\mathcal{F})_{x}\;.$

The composition of these three isomorphisms is the map
$(\pi_{*}\mathcal{F})_{x}\to(s^{-1}\mathcal{F})_{x}$ above and we are done.
$\Box$

Now we can turn to the proof of Proposition 5.1. Using the homotopy $H$ we
first construct an isomorphism of $\otimes$-functors

$\underline{H}:f^{-1}\cong
g^{-1}:\mathbb{Loc}_{k}(Z)\longrightarrow\mathbb{Loc}_{k}(X)\;.$

For $\mathcal{E}$ in $\mathbb{Loc}_{k}(Z)$ the isomorphism

$\underline{H}(\mathcal{E}):f^{-1}(\mathcal{E})\xrightarrow{\sim}g^{-1}(\mathcal{E})$

is obtained as follows. Consider the diagram

$\textstyle{X\times[0,1]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s_{1}}$$\scriptstyle{s_{0}}$

where $\pi$ is the projection and $s_{0}(x)=(x,0),s_{1}(x)=(x,1)$. Setting
$Y=X\times[0,1]$ and $s=s_{0}$ or $s_{1}$, by Example 5.3 we may apply Lemma
5.2 to the sheaf $\mathcal{F}=H^{-1}(\mathcal{E})$. We obtain canonical
isomorphisms

$f^{-1}(\mathcal{E})=s^{-1}_{0}\mathcal{F}\xleftarrow{\sim}\pi_{*}\mathcal{F}\xrightarrow{\sim}s^{-1}_{1}\mathcal{F}=g^{-1}(\mathcal{E})\;.$

Using these we define the isomorphism $\underline{H}(\mathcal{E})$. Its fibre
in a point $x$ is a functorial isomorphism

$\underline{H}(\mathcal{E})_{x}:\mathcal{E}_{f(x)}=(f^{-1}\mathcal{E})_{x}\xrightarrow{\sim}(g^{-1}\mathcal{E})_{x}=\mathcal{E}_{g(x)}\;,$

so that
$\underline{H}(\;)_{x}\in\operatorname{Iso}^{\otimes}(\omega_{f(x)},\omega_{g(x)})$.
By construction, we have $\underline{H}(\;)_{\tilde{x}}=\omega_{\gamma}$ as
defined above, where $\gamma:[0,1]\to Z$ is the path
$\gamma(t)=H(\tilde{x},t)$. For the commutativity of the diagram we have to
show that $L_{\omega_{\gamma}}\circ f_{*}=R_{\omega_{\gamma}}\circ g_{*}$ i.e.
that for any $k$-algebra $R$ and any

$\alpha\in\pi_{k}(X,\tilde{x})(R)=\underline{\operatorname{Aut}}^{\otimes}(\omega_{\tilde{x},R})$

we have

$\omega_{\gamma}\circ
f_{*}(\alpha)=g_{*}(\alpha)\circ\omega_{\gamma}:\omega_{f(\tilde{x})}\longrightarrow\omega_{g(\tilde{x})}\;.$

In other words, we claim that for all $\mathcal{E}$ in $\mathbb{Loc}_{k}(Z)$:

$(\underline{H}(\mathcal{E})_{\tilde{x}}\otimes\operatorname{id}_{R})\circ\alpha(f^{-1}(\mathcal{E}))=\alpha(g^{-1}(\mathcal{E}))\circ(\underline{H}(\mathcal{E})_{\tilde{x}}\otimes\operatorname{id}_{R})\;.$

This holds since $\underline{H}(\mathcal{E}):f^{-1}(\mathcal{E})\to
g^{-1}(\mathcal{E})$ is a morphism in $\mathbb{Loc}_{k}(X)$ and since $\alpha$
is a natural transformation. $\Box$

As in [KS18, section 2.2] we obtain the following consequences:

######  Corollary 5.4.

Let $H$ be a pointed homotopy of the pointed connected topological spaces
$(X,\tilde{x})$ and $(Z,\tilde{z})$ and set $f=H(\;,0)$ and $g=H(\;,1)$. Then

$f_{*}=g_{*}:\pi_{k}(X,\tilde{x})\longrightarrow\pi_{k}(Z,\tilde{z})\;.$

###### Proof.

For the path $\gamma$ in Proposition 5.1 we have
$\gamma(t)=H(\tilde{x},t)=\tilde{z}$ since $H$ is a pointed homotopy. Hence
the corresponding isomorphism
$\omega_{\gamma}:\mathcal{E}_{\tilde{z}}\to\mathcal{E}_{\tilde{z}}$ is the
identity for all $\mathcal{E}$ in $\mathbb{Loc}_{k}(Z)$ and therefore the map
$\gamma_{k}$ in Proposition 5.1 is the identity as well. ∎

######  Corollary 5.5.

Let $(X,x)\overset{\xrightarrow{\;f\;}}{\underset{g}{\longleftarrow}}(Y,y)$ be
pointed maps of pointed topological spaces such that $f\circ g$ and $g\circ f$
are homotopic to the identities via pointed homotopies. Then $X$ is connected
if and only if $Y$ is connected. If this is the case, the functors
$\mathbb{Loc}_{k}(X)\overset{\xrightarrow{f^{-1}}}{\underset{g^{-1}}{\longleftarrow}}\mathbb{Loc}_{k}(Y)$
are quasi-inverse equivalences of categories and
$\pi_{k}(X,x)\overset{\xrightarrow{f_{*}}}{\underset{g_{*}}{\longleftarrow}}\pi_{k}(Y,y)$
are isomorphisms of $k$-groups which are inverse to each other.

###### Proof.

The assertion on connectedness is proved in the corresponding corollary [KS18,
Corollary 2.18]. The rest is a formal consequence of Corollary 5.4. ∎

Consider a field $F$ of characteristic zero containing all roots of unity in
an algebraic closure $\overline{F}$ of $F$ and fix an injective homomorphism
$\iota:\mu(F)=\mu(\overline{F})\hookrightarrow\mathbb{C}^{\times}$. Let
$W_{\operatorname{rat}}(F)$ denote the ring of rational Witt vectors of $F$,
c.f. [KS18, section 4.1]. The Teichmüller map $[\;]:F^{\times}\to
W_{\operatorname{rat}}(F)$ is multiplicative and we obtain ring homomorphisms

$\mathbb{Z}[\mu(F)]\xrightarrow{[\;]}W_{\operatorname{rat}}(F)\quad\text{and}\quad\mathbb{Z}[\mu(F)]\xrightarrow{\iota}\mathbb{C}\;,$

defined on the group ring of $\mu(F)$. Set

$\mathfrak{X}_{F}=\operatorname{Spec}(W_{\operatorname{rat}}(F)\otimes_{\mathbb{Z}[\mu(F)]}\mathbb{C})\;.$

The complex points of this scheme over $\operatorname{Spec}\mathbb{C}$ carry
the “complex topology” induced by the topology of $\mathbb{C}$, c.f. [KS18,
section 5.2]. By definition, we have

$\mathfrak{X}_{F}(\mathbb{C})=\\{\alpha\in\operatorname{Hom}(W_{\operatorname{rat}}(F),\mathbb{C})\mid\alpha\circ[\;]=\iota\\}\;.$

The topological space $\mathfrak{X}_{F}(\mathbb{C})$ is connected and it has
the following Galois-theoretic description. Let
$G_{F}=\operatorname{Aut}_{F}(\overline{F})$ be the absolute Galois group of
$F$. The Teichmüller map $\overline{F}^{\times}\to
W_{\operatorname{rat}}(\overline{F})$ induces an isomorphism
$\mathbb{Z}\overline{F}^{\times}\xrightarrow{\sim}W_{\operatorname{rat}}(\overline{F})$
and hence an isomorphism

$\mathbb{Z}[\overline{F}^{\times}]^{G}\xrightarrow{\sim}W_{\operatorname{rat}}(\overline{F})^{G}=W_{\operatorname{rat}}(F)\;.$

This implies that

$\mathfrak{X}_{\overline{F}}(\mathbb{C})=\\{\chi:\overline{F}^{\times}\longrightarrow\mathbb{C}^{\times}\mid\chi\,|_{\mu(\overline{F})}=\iota\\}\;.$

By [KS18, Lemma 4.9], the natural map

$\operatorname{Hom}(\mathbb{Z}[\overline{F}^{\times}],\mathbb{C})/G\xrightarrow{\sim}\operatorname{Hom}(\mathbb{Z}[\overline{F}^{\times}]^{G},\mathbb{C})$

is a bijection. This gives a bijection

$\mathfrak{X}_{F}(\mathbb{C})=\\{\chi:\overline{F}^{\times}\to\mathbb{C}^{\times}\mid\chi\,|_{\mu(\overline{F})}=\iota\\}/G\;.$

Note that since $\mu(\overline{F})=\mu(F)$, the Galois group $G$ fixes
$\iota$. This bijection is actually a homeomorphism if the set of characters
is given the topology of pointwise convergence and its quotient by $G$ the
quotient topology. Kucharczyk and Scholze prove that the categories of finite
étale coverings of $\mathfrak{X}_{F}$ and $\mathfrak{X}_{F}(\mathbb{C})$ are
naturally equivalent and that their isomorphic étale fundamental groups with
respect to a $\mathbb{C}$-valued point $x$ can be canonically identified with
the absolute Galois group of $F$ if the latter is defined using a certain
fibre functor defined by $x$. We will not consider the scheme
$\mathfrak{X}_{F}$ in the following but only the connected topological space
$\mathfrak{X}_{F}(\mathbb{C})$ with the complex topology. Let
$(\overline{F}^{\times})^{\vee}=\operatorname{Hom}(\overline{F}^{\times},S^{1})$
be the Pontrjagin dual of the discrete group $\overline{F}^{\times}$ and
consider the closed subspace

$X_{\overline{F}}=\\{\chi\in(\overline{F}^{\times})^{\vee}\mid\chi\,|_{\mu(F)}=\iota\\}\;.$

The group $G_{F}$ acts by homeomorphisms and we endow the quotient
$X_{F}=G\setminus X_{\overline{F}}$ with the quotient topology. By [KS18,
Proposition 5.1], the $G_{F}$ action on $X_{\overline{F}}$ is proper and free
and the space $X_{F}$ is a non-empty, connected, compact Hausdorff-space. By
[KS18, Theorem 5.7, ii)] the natural $G_{F}$-equivariant projection
$\mathfrak{X}_{\overline{F}}(\mathbb{C})\to
X_{\overline{F}},\chi\mapsto\chi/|\chi|$ and the induced projection
$\mathfrak{X}_{F}(\mathbb{C})\to X_{F}$ are (strong) deformation retractions.
Here we view $X_{\overline{F}}$ as a closed subspace of
$\mathfrak{X}_{\overline{F}}(\mathbb{C})$ via
$X_{\overline{F}}=X_{\overline{F}}\times\\{0\\}$ and similarly for $X_{F}$.
From Corollary 5.5 we thus obtain:

######  Corollary 5.6.

For any point $x\in X_{F}\subset\mathfrak{X}_{F}(\mathbb{C})$ the above map
$\mathfrak{X}_{F}(\mathbb{C})\to X_{F}$ induces an isomorphism of $k$-groups

$\pi_{k}(\mathfrak{X}_{F}(\mathbb{C}),x)\xrightarrow{\sim}\pi_{k}(X_{F},x)\;.$

It is sufficient therefore to determine $\pi_{k}(X_{F},x)$. For a finite
extension $F^{\prime}/F$ in $\overline{F}$ the natural map $X_{F^{\prime}}\to
X_{F}$ is a finite étale covering of compact connected Hausdorff spaces, c.f.
[KS18, Theorem 5.2]. Let $\overline{x}\in
X_{\overline{F}}\subset\mathfrak{X}_{\overline{F}}(\mathbb{C})$ be a point and
let $x_{F^{\prime}}$ be its image in $X_{F^{\prime}}$. By [Den, Theorem 3.6]
we have a natural isomorphism

$\pi_{k}(X_{F},x)^{0}=\varprojlim_{(Y,y)}\pi_{k}(Y,y)$

where $(Y,y)$ runs over the directed set of isomorphism classes of pointed
finite étale coverings of $(X_{F},x)$. According to [KS18, Theorem 5.2], the
above coverings $(X_{F^{\prime}},x_{F^{\prime}})\to(X_{F},x)$ form a cofinal
system, so that we obain an isomorphism

$\pi_{k}(X_{F},x)^{0}=\varprojlim_{F^{\prime}/F}\pi_{k}(X_{F^{\prime}},x_{F^{\prime}})\;.$
(20)

Since we are dealing with compact Hausdorff spaces, the natural map

$X_{\overline{F}}\longrightarrow\varprojlim_{F^{\prime}/F}X_{\overline{F}}/\operatorname{Gal}(\overline{F}/F^{\prime})=\varprojlim_{F^{\prime}/F}X_{F^{\prime}}$

is a homeomorphism. Applying [Den, Theorem 4.3] we get an isomorphism

$\pi_{k}(X_{\overline{F}},\overline{x})=\varprojlim_{F^{\prime}/F}\pi_{k}(X_{F^{\prime}},x_{F^{\prime}})\;.$
(21)

Combining (20) and (21) with [Den, Theorem 3.6] and the [KS18, Theorem 5.2]
identification of $\pi_{1}(X_{F},x)$ with $G_{F}$, we obtain the following
result:

######  Corollary 5.7.

For any point $\overline{x}\in X_{\overline{F}}$ mapping to $x\in X_{F}$ we
have a canonical short exact sequence of affine $k$-groups

$1\longrightarrow\pi_{k}(X_{\overline{F}},\overline{x})\longrightarrow\pi_{k}(X_{F},x)\longrightarrow(G_{F})_{/k}\longrightarrow
1\;.$

There are canonical isomorphisms

$\pi_{k}(X_{\overline{F}},\overline{x})=\pi_{k}(X_{F},x)^{0}\quad\text{and}\quad\pi_{k}(X_{F},x)^{\operatorname{et}}=\pi^{\operatorname{et}}_{1}(X_{F},x)_{/k}=(G_{F})_{/k}\;.$

Here, for a profinite group $G=\varprojlim_{i}G_{i}$ with $G_{i}$ finite,
$G_{/k}=\varprojlim_{i}(G_{i/k})$ denotes the pro-étale $k$-group obtained as
the limit of the finite constant $k$-groups $G_{i/k}$ attached to the
$G_{i}$’s. The global sections $\Gamma(G_{/k},\mathcal{O})$ of $G$ consists of
the continuous maps $G\to k$ where $k$ carries the discrete topology. Note
that for the $k$-rational points, we have

$\pi_{k}(X_{F},x)^{\operatorname{et}}(k)=\pi^{\operatorname{et}}_{1}(X_{F},x)=G_{F}\;.$

Next we show that the connected $k$-group
$\pi_{k}(X_{\overline{F}},\overline{x})$ is abelian. The inclusion
$\mu_{F}\hookrightarrow\overline{F}^{\times}$ induces a short exact sequence
of compact abelian topological group

$1\longrightarrow(\overline{F}^{\times}/\mu_{F})^{\vee}\longrightarrow(\overline{F}^{\times})^{\vee}\xrightarrow{\pi}\mu^{\vee}_{F}\longrightarrow
1\;.$

Here $\pi$ is the map which restricts a character
$\chi:\overline{F}^{\times}\to S^{1}$ to $\mu_{F}$. All fibres of $\pi$ are
homeomorphic and in particular $X_{\overline{F}}$, the fibre of our fixed
embedding $\iota:\mu_{F}\hookrightarrow S^{1}$ is homeomorphic to $V^{\vee}$,
the Pontrjagin dual of the $\mathbb{Q}$-vectorspace
$V=\overline{F}^{\times}/\mu_{F}$. Write $V=\varinjlim_{\Gamma}\Gamma$ as a
filtered union of finite free $\mathbb{Z}$-submodules. Then
$V^{\vee}=\varprojlim_{T}T$ where
$T=\Gamma^{\vee}\cong(S^{1})^{\operatorname{rk}\Gamma}$ is a torus. Let $y\in
V^{\vee}$ be the image of $\overline{x}$ under a homeomorphism
$X_{\overline{F}}\xrightarrow{\sim}V^{\vee}$ and let $y_{T}$ be the image of
$y$ in $T$. Using [Den, Theorems 4.1 and 4.3] we obtain isomorphisms of
$k$-groups

$\pi_{k}(X_{\overline{F}},\overline{x})\cong\pi_{k}(V^{\vee},y)=\varprojlim_{T}\pi_{k}(T,y_{T})=\varprojlim_{T}\pi_{1}(T,y_{T})^{\operatorname{alg}}\;.$

Since $\pi_{1}(T,y_{T})$ is commutative, it follows that
$\pi_{k}(X_{\overline{F}},\overline{x})$ is commutative as well. If $k$ is
perfect, then it follows from [Mil17, Theorem 16.13] that there is a canonical
isomorphism of $k$ groups

$\pi_{k}(X_{\overline{F}},\overline{x})\xrightarrow{\sim}\pi^{u}_{k}(X_{\overline{F}},\overline{x})\times\pi^{\operatorname{red}}_{k}(X_{\overline{F}},\overline{x})\;.$
(22)

Here $\pi^{u}_{k}(X_{\overline{F}},\overline{x})$ is the pro-unipotent radical
of $\pi_{k}(X_{\overline{F}},\overline{x})$ and
$\pi^{\operatorname{red}}_{k}(X_{\overline{F}},\overline{x})$, the maximal
pro-reductive quotient of $\pi_{k}(X_{\overline{F}},\overline{x})$ can be
identified with the greatest $k$-subgroup of
$\pi_{k}(X_{\overline{F}},\overline{x})$ of multiplicative type. Both
$\pi^{u}_{k}(X_{\overline{F}},\overline{x})$ and
$\pi^{\operatorname{red}}_{k}(X_{\overline{F}},\overline{x})$ are commutative
and connected. The cohomologies of $X_{\overline{F}}$ with coefficients in
$\mathbb{Z}/n$ and $\mathbb{Q}$ have been calculated in [KS18]. We recall the
argument for $H^{1}$ and coefficients in any abelian group $A$.

Recall the pro-torus $V^{\vee}=\varprojlim_{T}T$ above where
$V=\overline{F}^{\times}/\mu_{F}$. We have

$H^{1}(V^{\vee},\underline{A})=\varinjlim_{T}H^{1}(T,\underline{A})\;.$

It follows that translation by an element of $V^{\vee}$ induces the identity
on $H^{1}(V^{\vee},\underline{A})$ since this is well known for the finite
dimensional tori $T$. Hence the homeomorphisms
$V^{\vee}\xrightarrow{\sim}X_{\overline{F}},\alpha\mapsto\alpha\chi$ for
$\chi\in X_{\overline{F}}$ all induce the same isomorphism, which we will view
as an identification

$H^{1}(X_{\overline{F}},\underline{A})=H^{1}(V^{\vee},\underline{A})\;.$ (23)

We have

$\displaystyle H^{1}(T,\underline{A})$
$\displaystyle=H^{1}(\Gamma^{\vee},\underline{A})=\operatorname{Hom}(\pi_{1}(\Gamma^{\vee}),A)$
$\displaystyle=\operatorname{Hom}(\operatorname{Hom}(\Gamma,\mathbb{Z}),A)=\Gamma\otimes
A$

and hence

$H^{1}(X_{\overline{F}},\underline{A})=\varinjlim_{T}H^{1}(T,\underline{A})=\overline{F}^{\times}/\mu_{F}\otimes
A\;.$ (24)

The category of commutative unipotent groups $U$ over $k\supset\mathbb{Q}$ is
anti-equivalent to the category of $k$-vector spaces by sending $U$ to
$Y(U)=\operatorname{Hom}_{k}(U,\mathbb{G}_{a})$. Given $Y(U)$ we recover $U$
as

$U=\operatorname{Hom}_{k}(Y(U),\mathbb{G}_{a/k})\;,$

where by definition for all $k$-schemes $S$ we have

$\operatorname{Hom}_{k}(Y(U),\mathbb{G}_{a/k})(S):=\operatorname{Hom}_{k}(Y(U),\Gamma(S,\mathcal{O}))\;.$

Here, on the right $\operatorname{Hom}_{k}$ means $k$-linear maps. Again using
[Den, Proposition 4.4], we obtain

$Y(\pi^{u}_{k}(X_{\overline{F}},\overline{x}))=\operatorname{Hom}_{k}(\pi_{k}(X_{\overline{F}},\overline{x}),\mathbb{G}_{a/k})=H^{1}(X_{\overline{F}},\underline{k})\;,$
(25)

and then

$\pi^{u}_{k}(X_{\overline{F}},\overline{x})=\operatorname{Hom}_{k}(H^{1}(X_{\overline{F}},\underline{k}),\mathbb{G}_{a/k})\;.$
(26)

We now turn to the multiplicative part. Since we want to apply [Den,
Proposition 4.4] which gives a formula for
$\operatorname{Hom}_{k}(\pi_{k}(X_{\overline{F}},\overline{x}),\mathbb{G}_{m})$,
but not for
$\operatorname{Hom}_{k^{\operatorname{sep}}}(\pi_{k}(X_{\overline{F}},\overline{x})\otimes_{k}k^{\operatorname{sep}},\mathbb{G}_{m})$,
we restrict our discussion to the case where $k=\overline{k}$. Then the
category of multiplicative groups over $k$ is equivalent to the category of
abelian groups by sending $H$ to
$X^{*}(H)=\operatorname{Hom}_{k}(H,\mathbb{G}_{m})$. The group $H$ is
recovered from $X^{*}(H)$ as

$H=\operatorname{Hom}(X^{*}(H),\mathbb{G}_{m/k})\;.$

Here the right hand side is defined by its $S$-valued points, which are the
groups

$\operatorname{Hom}(X^{*}(H),\mathbb{G}_{m/k})(S):=\operatorname{Hom}(X^{*}(H),\Gamma(S,\mathcal{O}^{\times}))\;.$

Using [Den, Proposition 4.4], we find

$X^{*}(\pi^{\operatorname{red}}_{k}(X_{\overline{F}},\overline{x}))=\operatorname{Hom}_{k}(\pi_{k}(X_{\overline{F}},\overline{x}),\mathbb{G}_{m/k})=H^{1}(X_{\overline{F}},\underline{k}^{\times})\;$

and therefore

$\pi^{\operatorname{red}}_{k}(X_{\overline{F}},\overline{x})=\operatorname{Hom}(H^{1}(X_{\overline{F}},\underline{k}^{\times}),\mathbb{G}_{m/k})\;.$
(27)

Combining (25), (26) with (27) we get the following

######  Corollary 5.8.

In the situation of Corollary 5.7 we have canonical isomorphisms of
commutative $k$-groups:

$\pi_{k}(X_{\overline{F}},\overline{x})\xrightarrow{\sim}\pi^{u}_{k}(X_{\overline{F}},\overline{x})\times\pi^{\operatorname{red}}_{k}(X_{\overline{F}},\overline{x})\;,\quad\text{if
$k$ is perfect}$

with

$\pi^{u}_{k}(X_{\overline{F}},\overline{x})=\operatorname{Hom}_{k}(\overline{F}^{\times}/\mu_{F}\otimes_{\mathbb{Q}}k,\mathbb{G}_{a/k})\;,\quad\text{if}\;k\supset\mathbb{Q}$

and

$\pi^{\operatorname{red}}_{k}(X_{\overline{F}},\overline{x})=\operatorname{Hom}(\overline{F}^{\times}/\mu_{F}\otimes_{\mathbb{Q}}k^{\times}/\mu_{k},\mathbb{G}_{m/k})\;,\quad\text{if}\;k=\overline{k}\supset\mathbb{Q}\;.$

######  Remark 5.9.

Since the character group
$\overline{F}^{\times}/\mu_{F}\otimes_{\mathbb{Z}}k^{\times}=\overline{F}^{\times}/\mu_{F}\otimes_{\mathbb{Q}}k^{\times}/\mu_{F}$
of $\pi^{\operatorname{red}}_{k}(X_{\overline{F}},\overline{x})$ is a
$\mathbb{Q}$-vector space, it has no torsion and hence
$\pi^{\operatorname{red}}_{k}(X_{\overline{F}},\overline{x})$ is connected.
The unipotent part $\pi^{u}_{k}(X_{\overline{F}},\overline{x})$ being
connected, it follows that $\pi_{k}(X_{\overline{F}},\overline{x})$ is
connected. We know this of course, since
$\pi_{k}(X_{\overline{F}},\overline{x})=\pi_{k}(X_{F},x)^{0}$ by Corollary
5.7.

######  Remark 5.10.

Let $G\to G^{\prime\prime}$ be a surjective i.e. faithfully flat map of affine
$k$-groups. Equivalently, the corresponding map of Hopf-algebras
$A^{\prime\prime}\to A$ is injective. For algebraically closed fields $k$ the
induced map $G(k)\to G^{\prime\prime}(k)$ is surjective, c.f. [DG70, III, §3,
Collaire 7.6]. In particular, the map $G(k)\to\pi_{0}(G)(k)$ is surjective.

It follows from Corollary 5.7 that for $k=\overline{k}$ we have a short exact
sequence of pro-discrete groups

$1\longrightarrow\pi^{u}_{k}(X_{\overline{F}},\overline{x})(k)\times\pi^{\operatorname{red}}_{k}(X_{\overline{F}},\overline{x})(k)\longrightarrow\pi_{k}(X_{F},x)(k)\longrightarrow
G_{F}\longrightarrow 1\;.$ (28)

If additionally $\operatorname{char}k=0$, then using Corollary 5.8, we get

$\pi^{u}_{k}(X_{\overline{F}},\overline{x})(k)=\operatorname{Hom}_{k}(\overline{F}^{\times}/\mu_{F}\otimes_{\mathbb{Q}}k,k)=\operatorname{Hom}_{\mathbb{Q}}(\overline{F}^{\times}/\mu_{F},k)$

and

$\pi^{\operatorname{red}}_{k}(X_{\overline{F}},\overline{x})(k)=\operatorname{Hom}(\overline{F}^{\times}/\mu_{F}\otimes_{\mathbb{Q}}k^{\times}/\mu_{F},k^{\times})\;.$

Going through the constructions, one sees that the $G_{F}$-action on these
groups via (28) is the same as the one induced by the natural $G_{F}$-action
on $\overline{F}^{\times}/\mu_{F}$. The actions are continuous for the pro-
discrete topologies on $\pi^{u}_{k}(X_{\overline{F}},\overline{x})(k)$ and
$\pi^{\operatorname{red}}_{k}(X_{\overline{F}},\overline{x})(k)$ but not for
the discrete topologies (the $G_{F}$-orbits do not have to be finite). Thus,
although these groups are $\mathbb{Q}$-vector spaces, it is not clear whether
the extension (28) is trivial i.e. a semi-direct product or not. The
corresponding question is open for the extension of affine groups in Corollary
5.7 as well.

The étale fundamental group of $X_{F}$ is isomorphic to the Galois group of
$F$. We now discuss the question, whether the proalgebraic fundamental group
$\pi_{k}(X_{F},x)$ is related to the motivic Galois group
$G_{\mathcal{M}_{F}(k)}$ if $k\supset\mathbb{Q}$. We assume that our embedding
$\iota:\mu_{F}\hookrightarrow\mathbb{C}$ is the restriction of an embedding
$\iota:\overline{F}\hookrightarrow\mathbb{C}$, which then also provides a
point $\overline{x}$ of $X_{\overline{F}}$. Let
$\mathcal{M}_{F}(k),\mathcal{M}_{\overline{F}}(k)$ be the Tannakian categories
of mixed motives over $F$ with coefficients in $k$ neutralized by the Betti-
realization via $\iota$. Thus e.g. for $M=H^{w}(X)$, with $X$ a variety over
$\overline{F}$ the fibre functor sends $M$ to
$H^{w}_{B}(X\otimes_{\overline{F},\iota}\mathbb{C},\underline{k})$. We can
think of motives in terms of realizations as in [Jan90]. By definition,
$G_{\mathcal{M}_{F}(k)}$ resp. $G_{\mathcal{M}_{\overline{F}}(k)}$ are the
Tannakian duals of $\mathcal{M}_{F}(k)$ resp. $\mathcal{M}_{\overline{F}}(k)$
with respect to this fibre functor. There is a canonical short exact sequence
of $k$-groups

$1\longrightarrow G^{0}_{\mathcal{M}_{F}(k)}\longrightarrow
G_{\mathcal{M}_{F}(k)}\longrightarrow G_{F/k}\longrightarrow 1\;,$

where $G_{\mathcal{M}_{\overline{F}}(k)}=G^{0}_{\mathcal{M}_{F}(k)}$
canonically. As explained in the introduction of [KS18] there should be a
“better” space $\tilde{X}_{F}$ mapping to $X_{F}$ and we would expect
$\pi_{k}(\tilde{X}_{F},\tilde{x})$ to be closely related to
$G_{\mathcal{M}_{F}(k)}$. This suggests a map
$G_{\mathcal{M}_{F}(k)}\longrightarrow\pi_{k}(X_{F},x)$ of $k$-groups and
hence a map

$G_{\mathcal{M}_{\overline{F}}(k)}=G^{0}_{\mathcal{M}_{F}(k)}\longrightarrow\pi_{k}(X_{F},x)^{0}=\pi_{k}(X_{\overline{F}},\overline{x})\;.$

It would map unipotent radicals to unipotent radicals and induce map between
the maximal pro-reductive quotients. Since
$\pi_{k}(X_{\overline{F}},\overline{x})$ is commutative, we would therefore
get maps

$(G^{u}_{\mathcal{M}_{\overline{F}}(k)})^{\operatorname{ab}}\longrightarrow\pi^{u}_{k}(X_{\overline{F}},\overline{x})$
(29)

and

$(G^{\operatorname{red}}_{\mathcal{M}_{\overline{F}}(k)})^{\operatorname{ab}}\longrightarrow\pi^{\operatorname{red}}_{k}(X_{\overline{F}},\overline{x})\;.$
(30)

We will now construct a canonical non-trivial map (29) for any field
$k\supset\mathbb{Q}$. To do so is equivalent to giving a map of $k$-vector
spaces

$Y(\pi^{u}_{k}(X_{\overline{F}},\overline{x}))\longrightarrow
Y((G^{u}_{\mathcal{M}_{\overline{F}}(k)})^{\operatorname{ab}})=\operatorname{Hom}(G^{u}_{\mathcal{M}_{\overline{F}}(k)},\mathbb{G}_{a})\;.$
(31)

First note that by (25) and (24) we have

$Y(\pi^{u}_{k}(X_{\overline{F}},\overline{x}))=\overline{F}^{\times}/\mu_{F}\otimes
k\;.$ (32)

Next, any $a\in\overline{F}^{\times}$ defines a motive
$E_{a}=H^{1}(\mathbb{G}_{m},\\{1,a\\})(1)$ over $\overline{F}$. It can also be
described as $E_{a}=H^{1}(X_{a})(1)$, where $X_{a}$ is the curve obtained from
$\mathbb{G}_{m,\overline{F}}$ by identifying $1$ and $a$. In Deligne’s theory
of $1$-motives, we have $E_{a}=[\mathbb{Z}\xrightarrow{\phi}\mathbb{G}_{m}]$,
where $\phi(1)=a$, c.f. [Del74, § 10]. The motive $E_{a}$ fits into a
canonical extension

$0\longrightarrow\mathbb{Q}(0)\longrightarrow
E_{a}\longrightarrow\mathbb{Q}(1)\longrightarrow
0\quad\text{in}\;\mathcal{M}_{\overline{F}}(\mathbb{Q})\;.$

Extending coefficients to $k$ we get an injective homomorphism

$\overline{F}^{\times}/\mu\otimes
k\hookrightarrow\operatorname{Ext}^{1}_{\mathcal{M}_{\overline{F}}(k)}(k(0),k(1))\;,\;(a\;\operatorname{mod}\;\mu)\otimes
1\mapsto E_{a}\otimes k\;.$ (33)

The motives $k(0)$, resp. $k(1)$ correspond to the trivial representation of
$G_{\mathcal{M}_{\overline{F}}}(k)$ on $\mathbb{G}_{a}$ over $k$ resp. to the
representation denoted $\mathbb{G}_{a}(1)$ on $\mathbb{G}_{a}$ by the
character $\chi:G_{\mathcal{M}_{\overline{F}}}(k)\to\mathbb{G}_{m/k}$, dual to
the full embedding of $\langle k(1)\rangle^{\otimes}$ into
$\mathcal{M}_{\overline{F}}(k)$. Then we have

$\operatorname{Ext}^{1}_{\mathcal{M}_{\overline{F}}(k)}(k(0),k(1))=\operatorname{Ext}^{1}_{G_{\mathcal{M}_{\overline{F}}}(k)}(\mathbb{G}_{a},\mathbb{G}_{a}(1))=H^{1}(G_{\mathcal{M}_{\overline{F}}(k)},\mathbb{G}_{a}(1))\;.$
(34)

In the middle, extensions are taken within the category of finite dimensional
representations of $G_{\mathcal{M}_{\overline{F}}(k)}$ on $k$-vector spaces.
For the Hochschild cohomology on the right compare [Mil17, Ch. 15]. Consider
the canonical extension of $k$-groups

$1\longrightarrow G^{u}_{\mathcal{M}_{\overline{F}}(k)}\longrightarrow
G_{\mathcal{M}_{\overline{F}}(k)}\longrightarrow Q\longrightarrow 1\;.$

Here $Q=G^{\operatorname{red}}_{\mathcal{M}_{\overline{F}}(k)}$ can be
interpreted as the Tannakian dual of the category of (direct sums of) pure
motives over $\overline{F}$ with coefficients in $k$. We now follow the
argument in [Jan90, Appendix C4]. Since Hochschild cohomology of reductive
groups over fields of characteristic zero vanishes in positive degrees, the
Hochschild-Serre spectral sequence

$H^{i}(Q,H^{j}(G^{u}_{\mathcal{M}_{\overline{F}}(k)},\mathbb{G}_{a}(1))\Rightarrow
H^{i+j}(G_{\mathcal{M}_{\overline{F}}(k)},\mathbb{G}_{a}(1))$

degenerates into isomorphisms

$H^{j}(G^{u}_{\mathcal{M}_{\overline{F}}(k)},\mathbb{G}_{a}(1))^{Q}=H^{j}(G_{\mathcal{M}_{\overline{F}}(k)},\mathbb{G}_{a}(1))\;.$

In particular, using (34) we may identify:

$\operatorname{Ext}^{1}_{\mathcal{M}_{\overline{F}}(k)}(k(0),k(1))=H^{1}(G^{u}_{\mathcal{M}_{\overline{F}}(k)},\mathbb{G}_{a}(1))^{Q}\;.$
(35)

The character $\chi$ above factors over $Q$, the maximal pro-reductive
quotient of $G_{\mathcal{M}_{\overline{F}}(k)}$, and hence the restriction of
$\chi$ to $G^{u}_{\mathcal{M}_{\overline{F}}(k)}$ is trivial. Hence we have

$H^{1}(G^{u}_{\mathcal{M}_{\overline{F}}(k)},\mathbb{G}_{a}(1))^{Q}=H^{1}(G^{u}_{\mathcal{M}_{\overline{F}}(k)},\mathbb{G}_{a})(1)^{Q}=\operatorname{Hom}(G^{u}_{\mathcal{M}_{\overline{F}}(k)},\mathbb{G}_{a})(1)^{Q}\;.$
(36)

The group on the right is the $\chi^{-1}$ eigenspace of the $Q$-action on
$\operatorname{Hom}(G^{u}_{\mathcal{M}_{\overline{F}}(k)},\mathbb{G}_{a})$. In
any case, we get the following inclusion by combining (35) and (36)

$\operatorname{Ext}^{1}_{\mathcal{M}_{\overline{F}}(k)}(k(0),k(1))\subset\operatorname{Hom}(G^{u}_{\mathcal{M}_{\overline{F}}(k)},\mathbb{G}_{a})\;.$
(37)

The promised map (31) is obtained as the following composition

$Y(\pi^{u}_{k}(X_{\overline{F}},\overline{x}))\overset{\eqref{eq:5-21}}{=}\overline{F}^{\times}/\mu_{F}\otimes
k\overset{\eqref{eq:5-22}}{\hookrightarrow}\operatorname{Ext}^{1}_{\mathcal{M}_{\overline{F}}(k)}(k(0),k(1))\overset{\eqref{eq:5-26}}{\hookrightarrow}\operatorname{Hom}(G^{u}_{\mathcal{M}_{\overline{F}}(k)},\mathbb{G}_{a})\;.$

(The map (33) is conjectured to be an isomorphism.)

Contrary to (29), there does not seem to be a non-trivial map (30). The group
$G^{\operatorname{red}}_{\mathcal{M}_{\overline{F}}}(k)$ is the motivic Galois
group for pure motives over $\overline{F}$ with coefficients in $k$, and

$\operatorname{Hom}_{k}((G^{\operatorname{red}}_{\mathcal{M}_{\overline{F}}(k)})^{\operatorname{ab}},\mathbb{G}_{m})=\operatorname{Hom}_{k}(G^{\operatorname{red}}_{\mathcal{M}_{\overline{F}}(k)},\mathbb{G}_{m})$
(38)

is the group of isomorphism classes or rank $1$ motives over $\overline{F}$
with coefficients in $k$. It follows that for any
$k=\overline{k}\supset\mathbb{Q}$, (30) would induce a homomorphism from the
$\mathbb{Q}$-vector space

$\operatorname{Hom}(\pi^{\operatorname{red}}_{k}(X_{\overline{F}},\overline{x}),\mathbb{G}_{m})=\operatorname{Hom}(\pi_{k}(X_{\overline{F}},\overline{x}),\mathbb{G}_{m})=H^{1}(X_{\overline{F}},\underline{k}^{\times})=\overline{F}^{\times}/\mu_{F}\otimes
k^{\times}$

to the group (38), i.e. a map

$\overline{F}^{\times}/\mu_{F}\otimes\mathbb{Q}^{\times}\longrightarrow\operatorname{Hom}_{k}((G^{\operatorname{red}}_{\mathcal{M}_{\overline{F}}(k)})^{\operatorname{ab}},\mathbb{G}_{m})\;.$
(39)

For $k=\overline{\mathbb{Q}}$, the $\overline{\mathbb{Q}}$-group

$S=(G^{\operatorname{red}}_{\mathcal{M}_{\overline{F}}(\overline{\mathbb{Q}})})^{\operatorname{ab}}$

is the connected Serre group of $F$ over $\overline{\mathbb{Q}}$. Its
character group $\operatorname{Hom}_{\overline{\mathbb{Q}}}(S,\mathbb{G}_{m})$
is known [Ser94, § 7] and does not depend on $F$! It does not contain any
infinitely divisible elements except zero and hence any map (39) has to be
trivial. Note in this regard that as pointed out in [KS18] the Steinberg
relations do not hold in the rational cohomology of $X_{F}$ (hence the above
mentioned suggestion of a better space $\tilde{X}_{F}$ mapping to $X_{F}$).
Morover no information about weights is built into the category
$\mathbb{Loc}_{k}(X_{F})$ as we saw in our construction of the map (29). Both
deficits, and also that $\pi_{k}(X_{F},x)^{0}$ is commutative show that
$X_{F}$ is quite far from a topological space whose proalgebraic fundamental
group is more deeply related to the motivic Galois group of $F$.

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