###### Abstract

Composition principles for reduced moduli are extended to the case of domains
in the $n$-dimensional Euclidean space, $n>2$. As a consequence analogues of
extremal decomposition theorems of Kufarev, Dubinin and Kirillova in the
planer case are obtained.

Extremal decomposition problems in the Euclidean space

K.A. Gulyaeva111Far Eastern Federal University, Vladivostok, Russia,
<EMAIL_ADDRESS>S.I. Kalmykov222Far Eastern Federal University,
Vladivostok, Russia, Institute of Applied Mathematics, Far Eastern Branch of
the Russian Academy of Sciences, Vladivostok, Russia<EMAIL_ADDRESS>E.G. Prilepkina333Far Eastern Federal University, Vladivostok, Russia,
Institute of Applied Mathematics, Far Eastern Branch of the Russian Academy of
Sciences, Vladivostok, Russia<EMAIL_ADDRESS>
Subject Classification: 31B99

Keywords: reduced modulus, Robin function, Neumann function, nonoverlapping
domain, extremal decomposition problem.

##  1 Introduction and notations

Extremal decomposition problems have a rich history and go back to M.A.
Lavrentiev’s inequality for the product of conformal radii of non-overlapping
domains. There exist two methods of their study: the extremal-metric method
and the capacitive method. The first one has been systematically developed in
papers by G.V. Kuz’mina, E.G. Emel’yanov, A.Yu. Solynin, A. Vasil’ev, and Ch.
Pommerenke [9, 14, 6, 11]. The second approach is developed mainly in works of
V.N. Dubinin and his students [4, 5, 2, 3]. In particular, a series of well-
known results about extremal decomposition follows one way from composition
principles for generalized reduced moduli (see [1, p. 56] and [12]). In the
present paper we extend the mentioned composition principles to the case of
spatial domains. As a consequence we get theorem about extremal decomposition
for the harmonic radius [7] obtained earlier in [5].

Throughout the paper, $\mathbb{R}^{n}$ denotes the $n$-dimensional Euclidean
space consisting of points $x=(x_{1},\dots,\,x_{n})$, $n\geq 3$, and
$|x|=\sqrt{x_{1}^{2}+\dots+x_{n}^{2}}$ is the length of a vector
$x\in\mathbb{R}^{n}$. We introduce the following notations:

$B(a,r)=\\{x\in\mathbb{R}^{n}:\,|a-x|<r\\},$

$S(a,r)=\\{x\in\mathbb{R}^{n}:\,|a-x|=r\\}$, $a\in\mathbb{R}^{n}$;

$\omega_{n-1}=2\pi^{n/2}/\Gamma(n/2)$ is the area of the unit sphere $S(0,1)$;

$\lambda_{n}=((n-2)\omega_{n-1})^{-1}$.

$D$ is a bounded domain in $\mathbb{R}^{n}$, $\Gamma$ is a closed subset of
$\partial D$. The pair $(D,\Gamma)$ is admissible if there exists the Robin
function, $g_{\Gamma}(z,z_{0},D)$ harmonic in $D\setminus\\{z_{0}\\}$,
continuous in $\overline{D}\setminus\\{z_{0}\\}$ and

$\displaystyle\frac{\partial g_{\Gamma}}{\partial n}$
$\displaystyle=0\,\,\text{on}\,\,(\partial D)\setminus\Gamma,$ (1)
$\displaystyle g_{\Gamma}$ $\displaystyle=0\,\,\text{on\,\,$\Gamma$},$ (2)

and in a neighborhood of $z_{0}$ there is an expansion

$g_{\Gamma}(z,z_{0},D)=\lambda_{n}\left(|z-z_{0}|^{2-n}-r(D,z_{0},\Gamma)^{2-n}+o(1)\right),\,\,z\rightarrow
z_{0},$ (3)

where $\partial/\partial n$ means the inward normal derivative on the
boundary. In what follows all such pairs are assumed to be admissible.

In the case $\Gamma=\emptyset$ we change the condition (1) by the condition

$\frac{\partial g_{\Gamma}}{\partial n}=\frac{1}{\mu_{n-1}(\partial
D)}\,\,\text{on}\,\,\partial D,$

where $\mu_{n-1}(\partial D)$ is the area of boundary.

By analogy with the definition of the Robin radius for plain domains from the
paper [3] we will call the constant $r(D,z_{0},\Gamma)$ the Robin radius of
the domain $D$ and the set $\Gamma$. Note that in the case of $\Gamma=\partial
D$ we get the harmonic radius [7, 10, 5].

Let $\Delta=\\{\delta_{k}\\}_{k}^{m}$ be a collection of real numbers and
$Z=\\{z_{k}\\}_{k=1}^{m}$ be points of the domain $D$. For $\Gamma=\emptyset$
we additionally require

$\sum_{k=1}^{m}\delta_{k}=0.$

Define the potential function for the domain $D$, the set $\Gamma$, the
collection of points $Z$, and numbers $\Delta$:

$u(z)=u(z;Z,D,\Gamma,\Delta)=\sum_{k=1}^{m}\delta_{k}g_{\Gamma}(z,z_{k},D).$

Note that for $\Gamma=\emptyset$ the function
$g_{\Gamma}\left(z,z_{k},D\right)$ is defined up to an additive constant.
Nevertheless, the function $u(z)$ is defined uniquely and characterized by the
condition

$\frac{\partial u}{\partial n}=0\ \ \text{on}\ \ \partial D.$

It is clear from the definition of the potential function that in a
neighborhood of $z_{k}$ we have

$u(z)=\delta_{k}\lambda_{n}|z-z_{k}|^{2-n}+a_{k}+o(1),\,\,k=1,\dots,m,$

where

$a_{k}=-\delta_{k}\lambda_{n}r(D,z_{k},\Gamma)^{2-n}+\sum_{\begin{subarray}{c}l=1\\\
l\neq k\end{subarray}}^{m}\delta_{l}g_{\Gamma}(z_{l},z_{k},D).$

Now if we introduce the following notation

$g_{\Gamma}(z_{k},z_{k},D)=-\lambda_{n}r(D,z_{k},\Gamma)^{2-n},$

then the constant in the expansion of the potential function in a
neighbourhood of $z_{k}$ is

$a_{k}=\sum_{l=1}^{m}\delta_{l}g_{\Gamma}(z_{l},z_{k},D).$ (4)

A function $v(z)$ is admissible for $D$, $Z$, $\Delta$, and $\Gamma$ if
$v(z)\in\text{Lip}$ in a neighbourhood of each point of $D$ except maybe
finitely many such points, continuous in
$\overline{D}\setminus\bigcup_{k=1}^{m}\\{z_{k}\\}$, $v(z)=0$ on $\Gamma$, and
in neighbourhood of $z_{k}$ there is an expansion

$v(z)=\delta_{k}\lambda_{n}|z-z_{k}|^{2-n}+b_{k}+o(1),\,\,z\rightarrow z_{k.}$
(5)

The Dirichlet integral is the following

$I(f,D)=\int_{D}\left|\nabla f\right|^{2}d\mu,$

where $d\mu=dx_{1}\dots dx_{n}$.

##  2 Main results

######  Lemma 2.1

The asymptotic formula

$I(u,D_{r})=\left(\sum_{k=1}^{m}\delta_{k}^{2}\right)\lambda_{n}r^{2-n}+\sum_{k=1}^{m}\delta_{k}a_{k}+o(1),\,\,r\rightarrow
0,$

is true, where $u$ is the potential function and $a_{k}$, $k=1,\dots,m$ are
defined in $(\ref{eq:ak})$ and $D_{r}=D\setminus\bigcup_{k=1}^{m}B(z_{k},r)$.

Proof. The Green’s identity

$\int_{V}|\nabla u|^{2}d\mu=-\int_{\partial V}u\frac{\partial u}{\partial
n}ds$

gives

$I\left(u,D_{r}\right)=-\int_{\partial D_{r}}u\frac{\partial u}{\partial
n}ds=-\sum_{k=1}^{m}\int_{S(z_{k},r)}u\frac{\partial u}{\partial n}ds.$ (6)

The second equality in (6) holds because $u\dfrac{\partial u}{\partial n}=0$
on $\partial D$. Note that

$u=\delta_{k}\lambda_{n}r^{2-n}+a_{k}+o(1),\,\,z\rightarrow z_{k}$

in a neighbourhood of $z_{k}$.

We calculate the integral $\int_{S(z_{k},r)}u\frac{\partial u}{\partial n}ds.$
Let $u(z)=h(z)+g(z)$, where $h(z)=\lambda_{n}\delta_{k}|z-z_{k}|^{2-n}$ and
$g(z)$ is a harmonic function. Note that $g(z_{k})=a_{k}$. For $|z-z_{k}|=r$
we have the following correlations

$r^{n-1}u\frac{\partial u}{\partial n}=r^{n-1}\left(h\frac{\partial
h}{\partial n}+h\frac{\partial g}{\partial n}+g\frac{\partial h}{\partial
n}+g\frac{\partial g}{\partial n}\right)$
$=(2-n)\lambda_{n}^{2}\delta_{k}^{2}r^{2-n}+r\lambda_{n}\delta_{k}\frac{\partial
g}{\partial n}+(2-n)g\lambda_{n}\delta_{k}+g\frac{\partial g}{\partial
n}r^{n-1}$
$=-\frac{\lambda_{n}\delta_{k}^{2}}{\omega_{n-1}}r^{2-n}-\frac{g(z_{k})\delta_{k}}{\omega_{n-1}}+o(1),r\rightarrow
0.$

Therefore

$\int_{S(z_{k},r)}u\frac{\partial u}{\partial
n}ds=\int_{S(0,1)}u\frac{\partial u}{\partial n}r^{n-1}ds$
$=-\lambda_{n}\delta_{k}^{2}r^{2-n}-\delta_{k}a_{k}+o(1),\ \ r\rightarrow 0.$

Substituting it in (6) we get the lemma. $\square$

######  Lemma 2.2

For an admissible function $v$ and the potential function $u$ we have

$I(v-u,D_{r})=I(v,D_{r})-I(u,D_{r})-2\sum_{k=1}^{m}\delta_{k}(b_{k}-a_{k})+o(1),\,\,r\rightarrow
0.$

Proof. One may observe that

$I(v-u,D_{r})=\int_{D_{r}}\left(|\nabla v|^{2}+|\nabla u|^{2}-2\nabla u\nabla
v\right)d\mu\\\ =\int_{D_{r}}\left(|\nabla v|^{2}-|\nabla
u|^{2}\right)d\mu+2\int_{\partial D_{r}}(v-u)\frac{\partial u}{\partial
n}ds\\\
=I(v,D_{r})-I(u,D_{r})+2\sum_{k=1}^{m}\int_{S(z_{k},r)}(v-u)\frac{\partial
u}{\partial n}ds\\\
=I(v,D_{r})-I(u,D_{r})-2\sum_{k=1}^{m}\delta_{k}(b_{k}-a_{k})+o(1),\,\,r\rightarrow
0.$

Here we calculated the integral $\int_{S(z_{k},r)}(v-u)\frac{\partial
u}{\partial n}$ a similar way as in the proof of lemma 2.1 and used the
Green’s identity

$\int_{D_{r}}(\nabla u\cdot\nabla v)d\mu=-\int_{\partial D_{r}}v\frac{\partial
u}{\partial n}ds,$

where $n$ is the inner normal vector. $\square$

The quantity

$\sum_{k=1}^{m}\delta_{k}a_{k}=\sum_{k=1}^{m}\sum_{l=1}^{m}\delta_{k}\delta_{l}g_{\Gamma}(z_{l},z_{k},D)$

we call the reduced modulus and denote it by $M(D,\Gamma,Z,\Delta).$ According
to lemma 2.1

$M(D,\Gamma,Z,\Delta)=\lim_{r\rightarrow
0}\left(I(u,D_{r})-\left(\sum_{k=1}^{n}\delta_{k}^{2}\right)\lambda_{n}r^{2-n}\right).$

######  Theorem 2.3

Let sets $D$, $\Gamma$, collections $Z=\\{z_{k}\\}_{k=1}^{m}$,
$\Delta=\\{\delta_{k}\\}_{k=1}^{m}$, be as in the definition of the reduced
modulus $M=M(D,\Gamma,$ $Z,\Delta),$ $u(z)$ be the potential function for $D$,
$\Gamma$, $Z$, $\Delta$, and let $D_{i}\subset D$ be pairwise non-overlapping
subdomains of $D$, $\Gamma_{i}$, $Z_{i}=\\{z_{ij}\\}_{j=1}^{n_{i}}$,
$\Delta_{i}=\\{\delta_{ij}\\}_{j=1}^{n_{i}}$, be from the definition of the
reduced moduli $M_{i}=M(B_{i},\Gamma_{i},$ $Z_{i},$ $\Delta_{i}),$ $u_{i}(z)$
be the potential function for $D_{i},$ $\Gamma_{i}$, $Z_{i},$ $\Delta_{i},$
$i=1,...,p.$ Assume that the following conditions are fulfilled:

1)$(D\cap\partial D_{i})\subset\Gamma_{i},$ $i=1,...,p;$
---
2)
$\Gamma\subset\left(\bigcup_{i=1}^{p}\Gamma_{i}\right)\bigcup\left[\mathbb{R}^{n}\setminus\left(\bigcup_{i=1}^{p}\overline{D}_{i}\right)\right]$;
3)$Z=\bigcup_{i=1}^{p}Z_{i},$ _that is each point $z_{k}\in Z$ coincides with
some point_
$z_{ij}\in Z_{i}$ _for $k=k(i,j)$ and vice versa;_
4) $\delta_{k}=\delta_{ij}$ for $k=k(i,j).$

Then the inequality

$M\geq\sum_{i=1}^{p}M_{i}+\sum\limits_{i=1}^{p}I(u-u_{i},D_{i})\geq\sum_{i=1}^{p}M_{i}$

holds.

Proof. Consider the function

$v(z)=\left\\{\begin{array}[]{ll}u_{i}(z),&z\in D_{i},\\\ 0,&z\in
D\setminus\left(\bigcup_{i=1}^{p}D_{i}\right).\end{array}\right.$

The condition 1) guarantees that the function $v(z)$ is continuous in
$\overline{D}\backslash\bigcup_{k=1}^{m}\\{z_{k}\\}$. From the conditions 2)
and 3) it follows that $v(z)=0$ for $z\in\Gamma$ and in a neighbourhood of
$z_{k},$ $k=1,...,m$, there is the expansion (5). Applying lemma 2.2, we get

$I(v-u,D)=I(v,D_{r})-I(u,D_{r})-2\sum\limits_{k=1}^{p}\delta_{k}(b_{k}-a_{k})+o(1),\
r\rightarrow 0,$ (7)

here $a_{k}$ and $b_{k}$ from (4) and (5) respectively. By lemma 2.1

$I(v,D_{r})=\lambda_{n}r^{2-n}\sum\limits_{i=1}^{p}\sum\limits_{j=1}^{n_{i}}\delta_{ij}^{2}+\sum\limits_{i=1}^{p}M_{i}+o(1)=\lambda_{n}r^{2-n}\sum\limits_{k=1}^{m}\delta_{k}^{2}+\sum\limits_{i=1}^{p}M_{i}+o(1),$
$I(u,D_{r})=\lambda_{n}r^{2-n}\sum\limits_{k=1}^{m}\delta_{k}^{2}+M+o(1),\qquad
r\rightarrow 0,$

taking into account 3), we have

$\sum\limits_{k=1}^{m}\delta_{k}(b_{k}-a_{k})=\sum\limits_{i=1}^{p}M_{i}-M.$

Substituting the obtained correlations in (7), we see that the inequality

$\sum\limits_{i=1}^{p}I(u-u_{i},D_{i})\leq
I(v-u,D)=M-\sum\limits_{i=1}^{p}M_{i}+o(1),\qquad r\rightarrow 0,$

is true. Theorem is proved. $\square$

######  Theorem 2.4

Let sets $D$, $\Gamma$, collections $Z=\\{z_{k}\\}_{k=1}^{m}$,
$\Delta=\\{\delta_{k}\\}_{k=1}^{m}$, be as in the definition of the reduced
modulus $M:=M(D,\Gamma,Z,\Delta),$ $u(z)$ be the potential function for $D$,
$\Gamma$, $Z$, $\Delta$, and let $D_{i}\subset D$, $i=1,...,p,$ be pairwise
non-overlapping domains, $\Gamma_{i}$, $Z_{i}=\\{z_{ij}\\}_{j=1}^{n_{i}}$,
$\Delta_{i}=\\{\delta_{ij}\\}_{j=1}^{n_{i}}$, be from the definition of the
reduced moduli $M_{i}=M(D_{i},\Gamma_{i},Z_{i},\Delta_{i}),$ $u_{i}(z)$ be the
potential function for $D_{i},$ $\Gamma_{i}$, $Z_{i},$ $\Delta_{i},$
$i=1,...,p.$ Assume that $\Gamma_{i}\subset\Gamma,\ i=1,...,p,$
$Z=\bigcup_{i=1}^{m}Z_{i},$ (that is each point $z_{k}\in Z$ coincides with
some point $z_{ij}\in Z_{i}$ for $k=k(i,j)$ and vice versa),
$\delta_{k}=\delta_{ij}.$ Then the inequality

$\sum_{i=1}^{p}M_{i}\geq M+\sum\limits_{i=1}^{p}I(u-u_{i},D_{i})\geq M$

holds.

Proof. The function $u$ is admissible for $D_{i}$, $i=1,\dots,p$. Let $b_{k}$
be constants from the expansion of the function $u$ in a neighbourhood of
$z_{k},$ $b_{ij}=b_{k}$ if $k=k(i,j).$ Applying lemmata 2.1 and 2.2 with the
potential functions $u_{k}$ for $D_{k}$ we get

$\sum_{i=1}^{p}\sum_{j=1}^{n_{i}}(\delta_{ij})^{2}r^{2-n}\lambda_{n}+\sum_{i=1}^{p}\sum_{j=1}^{n_{i}}\delta_{ij}a_{ij}+o(1)=\sum_{i=1}^{p}I(u_{i},(D_{i})_{r})=\\\
=\sum_{i=1}^{p}\left(I(u,(D_{i})_{r})-2\sum_{j=1}^{n_{i}}\delta_{ij}(b_{ij}-a_{ij})-I(u-u_{i},(D_{i})_{r})\right)+o(1)\\\
\leq
I(u,D_{r})-\sum_{i=1}^{p}I(u-u_{i},(D_{i})_{r})-2\sum_{i=1}^{p}\sum_{j=1}^{n_{i}}\delta_{ij}(b_{ij}-a_{ij})+o(1)\\\
=\sum_{i=1}^{p}\sum_{j=1}^{n_{i}}(\delta_{ij})^{2}r^{2-n}\lambda_{n}+\sum_{i=1}^{p}\sum_{j=1}^{n_{i}}\delta_{ij}b_{ij}-2\sum_{i=1}^{p}\sum_{j=1}^{n_{i}}\delta_{ij}(b_{ij}-a_{ij})\\\
-\sum_{i=1}^{p}I(u-u_{i},(D_{i})_{r})+o(1),\,\,r\rightarrow 0.$

It implies that

$\sum_{i=1}^{p}\sum_{j=1}^{n_{i}}\delta_{ij}b_{ij}\leq\sum_{i=1}^{p}\sum_{j=1}^{n_{i}}\delta_{ij}a_{ij}-\sum_{i=1}^{p}I(u-u_{i},D_{i})$

or equivalently

$\sum_{i=1}^{p}I(u-u_{i},D_{i})+M(D,\Gamma,Z,\Delta)\leq\sum_{i=1}^{p}M(D_{i},\Gamma_{i},Z_{i},\Delta_{i}).$

Here we used the fact that the function $u-u_{i}$ has no singularity in
$D_{i}$. $\square$

Denote by $r(D_{l},x_{l})=r(D_{l},x_{l},\partial D)$ the harmonic radius.
Directly from theorem 2.3 we get theorem 2 of the paper [5]

######  Corollary 2.5

For any non-overlapping domains $D_{l}\subset\mathbb{R}^{n}$, $n\geq 3$,
points $x_{l}\in D_{l}$ and real numbers $\delta_{l}$, $l=1,\dots,m$ the
inequality

$-\sum_{l=1}^{m}\delta_{l}^{2}r\left(D_{l},x_{l}\right)^{2-n}\leq\sum_{l=1}^{m}\sum_{\begin{subarray}{c}p=1\\\
p\neq l\end{subarray}}^{m}\delta_{l}\delta_{p}|x_{l}-x_{p}|^{2-n}$

holds true.

Proof. The Green’s function of the ball $B(0,\rho)$ is

$\lambda_{n}\left(|x-x_{0}|^{2-n}-\left|\frac{|x_{0}|x}{\rho}-\frac{\rho
x_{0}}{|x_{0}|}\right|^{2-n}\right).$

Denote by $D_{l}(\rho)$ the intersection $D_{l}\cap B(0,\rho)$. By theorem 2.3

$M(\rho)\geq\sum_{l=1}^{m}M_{l}(\rho),$

where $M(\rho)$ is the modulus of the ball $B(0,\rho)$, the collections
$\left\\{x_{l}\right\\}_{l=1}^{m}$,
$\Delta=\left\\{\delta_{l}\right\\}_{l=1}^{m}$, and $\Gamma=\partial B$,

$M_{l}(\rho)=-\delta_{l}^{2}r\left(D_{l}(\rho),x_{l}\right)^{2-n}\lambda_{n}.$

It is sufficient to take a limit as $\rho\rightarrow\infty$. $\square$

Theorems 2.3 and 2.4 imply for $p=1$ monotonicity of the quadratic form

$\sum_{l=1}^{m}\sum_{p=1}^{m}\delta_{l}\delta_{p}g_{\Gamma}(z_{l},z_{p},D)$

under extension of a domain. Following [2] we will say that a domain
$\tilde{D}$ is obtained by extending a domain $D$ across a part of its
boundary $\gamma\subset\partial D$ if $D\subset\tilde{D}$ and $(\partial
D)\cap\tilde{D}$ lies in $\gamma$.

######  Corollary 2.6

If $\tilde{D}$ is obtained by extending $D$ across $\Gamma$,
$\tilde{\Gamma}\subset\left(\Gamma\bigcup(\mathbb{R}^{n}\setminus\overline{D})\right),$
then for any real numbers $\delta_{l}$ and points $z_{l}\in D$

$\sum_{l=1}^{m}\sum_{p=1}^{m}\delta_{l}\delta_{p}g_{\tilde{\Gamma}}\left(z_{l},z_{p},\tilde{D}\right)\geq\sum_{l=1}^{m}\sum_{p=1}^{m}\delta_{l}\delta_{p}g_{\Gamma}\left(z_{l},z_{p},D\right)+I(u-\tilde{u},D)$
$\geq\sum_{l=1}^{m}\sum_{p=1}^{m}\delta_{l}\delta_{p}g_{\Gamma}\left(z_{l},z_{p},D\right).$

If $\tilde{D}$ is obtained by extending $D$ across the part of $(\partial
D)\setminus\Gamma$, $\tilde{\Gamma}=\Gamma$, then

$\sum_{l=1}^{m}\sum_{p=1}^{m}\delta_{l}\delta_{p}g_{\tilde{\Gamma}}\left(z_{l},z_{p},\tilde{D}\right)\leq\sum_{l=1}^{m}\sum_{p=1}^{m}\delta_{l}\delta_{p}g_{\Gamma}\left(z_{l},z_{p},D\right)-I(u-\tilde{u},D)$
$\leq\sum_{l=1}^{m}\sum_{p=1}^{m}\delta_{l}\delta_{p}g_{\Gamma}\left(z_{l},z_{p},D\right),$

here $u$ and $\tilde{u}$ are the potential functions for $D$, $\Gamma$,
$Z=\left\\{z_{l}\right\\}_{l=1}^{m}$,
$\Delta=\left\\{\delta_{l}\right\\}_{l=1}^{m}$ and $\tilde{D}$,
$\tilde{\Gamma}$, $Z=\left\\{z_{l}\right\\}_{l=1}^{m}$,
$\Delta=\left\\{\delta_{l}\right\\}_{l=1}^{m}$, respectively.

In [3] the notion of the Robin radius

$r\left(D,z_{0},\Gamma\right)=\exp\lim_{z\rightarrow
z_{0}}\left(g_{D}\left(z,z_{0},\Gamma\right)+\log|z-z_{0}|\right)$

was introduced. This quantity generalized the notion of the conformal radius.
An analogue of Kufarev’s theorem (see [8]) for non-overlapping domains
$D_{1}$, $D_{2}$ lying in the unit disk $U$ under the condition
$\left(\left(\partial D_{k}\right)\cap
U\right)\subset\Gamma_{k}\subset\partial D_{k}$, $a_{k}\in D_{k}$, $k=1,2$ is
the inequality

$r\left(D_{1},a_{1},\Gamma_{1}\right)r\left(D_{2},a_{2},\Gamma_{2}\right)\leq|a_{2}-a_{1}|^{2}\left[1-\left|\frac{a_{2}-a_{1}}{1-\overline{a_{1}}a_{2}}\right|^{2}\right]^{-1}.$

By setting in theorem 2.3 $p=2$, $\Gamma=\emptyset$, we obtain in
$\mathbb{R}^{n}$ the following inequality.

######  Corollary 2.7

Let $D_{1}$ and $D_{2}$ be non-overlapping and lie in the ball $U=B(0,1)$,
$a_{k}\in D_{k}$, $\left(\partial D_{k}\right)\cap
U\subset\Gamma_{k}\subset\partial D_{k}$, $k=1,2$. Then

$-\lambda_{n}r\left(D_{1},a_{1},\Gamma_{1}\right)^{2-n}-\lambda_{n}r\left(D_{2},a_{2},\Gamma_{2}\right)^{2-n}\leq
M\left(U,\emptyset,\left\\{a_{1},a_{2}\right\\},\left\\{1,-1\right\\}\right).$
(8)

To calculate
$M\left(U,\emptyset,\left\\{a_{1},a_{2}\right\\},\left\\{1,-1\right\\}\right)$
we need to know the Neumann function of the unit ball. Note that it is a quite
comlicated problem in $\mathbb{R}^{n}$. In particular, for $n=3$ (see [13])

$g_{\emptyset}\left(x,y,U\right)=\frac{1}{4\pi}\left(\frac{1}{|x-y|}+\frac{|y|}{|x|y|^{2}-y|}-\log\left|1-(x,y)+\frac{\left|x|y|^{2}-y\right|}{\left|y\right|}\right|\right).$

In [13] there is an analytic view of $g_{\emptyset}(D,x,y)$ for $n=4,5$. So,
for $n=3$ the inequality (8) has the following form

$-r\left(D_{1},a_{1},\Gamma_{1}\right)^{-1}-r\left(D_{2},a_{2},\Gamma_{2}\right)^{-1}\leq-\frac{2}{|a_{1}-a_{2}|}-\frac{2|a_{2}|}{|a_{1}|a_{2}|^{2}-a_{2}|}\\\
+2\log\left|1-(a_{1},a_{2})+\frac{\left|a_{1}|a_{2}|^{2}-a_{2}\right|}{\left|a_{2}\right|}\right|+\frac{1}{1-|a_{1}|^{2}}+\frac{1}{1-|a_{2}|^{2}}\\\
-\log(4(1-|a_{1}|^{2})(1-|a_{2}|^{2})).$

Acknowledgements. This work was supported by the Russian Science Foundation
under grant 14-11-00022.

## References

  * [1] V.N. Dubinin, Condenser capacities and symmetrization in geometric function theory, Birkhäuser/Springer, Basel, 2014.
  * [2] V.N. Dubinin and N.V. Eyrikh, Some applications of generalized condensers in the theory of analytic functions, Journal of Mathematical Sciences, 133(6) (2006), 1634-1647.
  * [3] V.N. Dubinin and D.A. Kirillova, On extremal decomposition problems, Journal of Mathematical Sciences, 157(4) (2009), 573-583.
  * [4] V.N. Dubinin and L.V. Kovalev, The reduced module of the complex sphere, Journal of Mathematical Sciences, 105(4) (2001), 2165-2179.
  * [5] V.N. Dubinin and E.G. Prilepkina, Extremal decomposition of spatial domains, Journal of Mathematical Sciences, 105(4) (2001), 2180-2189.
  * [6] E.G. Emel’yanov, On problems on the extremal decomposition, Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta Imeni V. A. Steklova, 154 (1986), 76-89.
  * [7] J. Hersch, Transplantation harmonique, transplantation par modules, et théoremes isoperimetriques, Commentarii Mathematici Helvetici, 44 (1969), 354-366.
  * [8] P.P. Kufarev, On the question of conformal mappings of complementary domains, Dokl. Akad. Nauk SSSR, 73 (1950), 881-884.
  * [9] G.V. Kuz’mina, Methods of geometric function theory. I, II, St. Petersburg Mathematical Journal, 9(3) (1998), 455-507; 9(5) (1998), 889-930.
  * [10] B.E. Levitskii, The reduced p-module and the inner p-harmonic radius, Dokl. Akad. Nauk SSSR, 316 (1991), 812-815.
  * [11] Ch. Pommerenke and A. Vasil’ev, Angular derivatives of bounded univalent functions and extremal partions of the unit disk, Pacific J. Math., 206(2) (2002), 425-450.
  * [12] E.G. Prilepkina, On composition principles for reduced moduli, Siberian Mathematical Journal, 52(6) (2011), 1079-1091.
  * [13] M.A. Sadybekov and B.T. Torebek and B.Kh. Turmetov, Representation of Green’s function of the Neumann problem for a multi-dimensional ball, Complex Variables and Elliptic Equations, http://dx.doi.org/10.1080/17476933.2015.1064402
  * [14] A.Yu. Solynin, Modules and extremal metric problems, St. Petersburg Mathematical Journal, 11(1) (2000), 1–65.


# Further examples of non-geometric sections of arithmetic fundamental groups

Mohamed Saïdi

We show the existence of group-theoretic sections of certain geometrically
pro-nilpotent by abelian arithmetic fundamental groups of hyperbolic curves
over $p$-adic local fields which are non-geometric, i.e., which do not arise
from rational points. Among these quotients is the geometrically metabelian
arithmetic fundamental group.

## §0. Introduction/Statement of the Main Result

Grothendieck’s anabelian section conjecture predicts that sections of
arithmetic fundamental groups of hyperbolic curves over finitely generated
fields over $\mathbb{Q}$ arise from rational points (cf. [Saïdi] for a more
precise formulation of the conjecture). Accordingly, sections of arithmetic
fundamental groups of hyperbolic curves over $p$-adic local fields; which are
defined over number fields, and which arise from global sections, should arise
from rational points. In this context it is tempting to predict a $p$-adic
analog of Grothendieck’s anabelian section conjecture. In [Saïdi1] we
investigated such analog, and exhibited two necessary and sufficient
conditions for a section of the arithmetic fundamental group of a hyperbolic
curve over a $p$-adic local field to be geometric, i.e., to arise from a
rational point (cf. loc. cit. Theorem 4.5).

For the time being there are no examples of sections of (the full) arithmetic
fundamental groups of hyperbolic curves over $p$-adic local fields which are
non-geometric, and one can still hope the validity of a $p$-adic analog of the
section conjecture. On the other hand, recent examples were found of group-
theoretic sections of certain (geometrically characteristic) quotients of
arithmetic fundamental groups of curves over $p$-adic local fields which are
non-geometric. Hoshi constructed examples of sections of the geometrically
pro-$p$ quotient of arithmetic fundamental groups of curves over $p$-adic
local fields which are non-geometric (cf. [Hoshi]). (Actually, Hoshi’s example
arises from group-theoretic sections of geometrically pro-$p$ fundamental
groups of hyperbolic curves over number fields (cf. loc. cit.).) In [Saïdi] we
constructed examples of group-theoretic sections of geometrically prime-to-$p$
fundamental groups of hyperbolic curves over $p$-adic local fields which are
non-geometric (cf. loc. cit. $\S 3$). Further, in [Saïdi2] we provided
examples of group-theoretic sections of the ”étale by geometrically abelian”
fundamental group of hyperbolic curves over $p$-adic local fields which are
non-geometric. The existence of these examples is crucial for our
understanding of the $p$-adic section conjecture. Indeed, if the $p$-adic
version of the section conjecture holds true then it may possibly hold true
even for smaller quotients of the arithmetic fundamental group, and one would
like to know these quotients in this case. On the other hand, more elaborate
examples of non-geometric sections as above may lead to a counterexample for
the $p$-adic version of the section conjecture.

In this note we provide further examples of sections of certain quotients of
arithmetic fundamental groups of curves over $p$-adic local fields which are
non-geometric. These quotients include the geometrically metabelian and
certain geometrically pro-nilpotent by abelian quotients.

Next, we fix notations and state our main results.

$\bullet$ Let

$1\to H^{\prime}\to H@>{\operatorname{pr}}>>G\to 1$

be an exact sequence of profinite groups. We will refer to a continuous
homomorphism $s:G\to H$ satisfying $\operatorname{pr}\circ
s=\operatorname{id}_{G}$ as a (group-theoretic) section, or splitting, of the
above sequence, or simply a section of the projection
$\operatorname{pr}:H\twoheadrightarrow G$. We denote by
$\operatorname{Sect}\left(H\twoheadrightarrow G\right)$ the set of sections of
the projection $H\twoheadrightarrow G$.

$\bullet$ Given a profinite group $H$, and a prime integer $\ell$, we will
denote by $H^{\ell}$ the maximal pro-$\ell$ quotient of $H$,
$H^{\operatorname{ab}}$ the maximal abelian quotient of $H$, and
$H^{\operatorname{ab},\ell}$ its maximal abelian pro-$\ell$ quotient. Thus
$H^{\operatorname{ab},\ell}=(H^{\ell})^{\operatorname{ab}}$.

Let $p\geq 2$ be a prime integer, and $k$ a $p$-adic local field; meaning
$k/\mathbb{Q}_{p}$ is a finite extension, with ring of integers $\Cal{O}_{k}$,
and residue field $F$. Thus $F$ is a finite field of characteristic $p$. Let
$X\to\operatorname{Spec}k$ be a proper, smooth, and geometrically connected
hyperbolic (i.e., $\operatorname{genus}(X)\geq 2$) curve over $k$. Let $\eta$
be a geometric point of $X$ above its generic point, which determines an
algebraic closure $\overline{k}$ of $k$, and a geometric point $\bar{\eta}$ of
$\overline{X}\overset\text{def}\to{=}X\times_{k}\overline{k}$. There exists a
canonical exact sequence of profinite groups (cf. [Grothendieck], Exposé IX,
Théorème 6.1)

$1\to\pi_{1}(\overline{X},\bar{\eta})\to\pi_{1}(X,\eta)\to G_{k}\to 1.$

Here, $\pi_{1}(X,\eta)$ denotes the arithmetic étale fundamental group of $X$
with base point $\eta$, $\pi_{1}(\overline{X},\bar{\eta})$ the étale
fundamental group of
$\overline{X}\overset\text{def}\to{=}X\times_{k}\overline{k}$ with base point
$\bar{\eta}$, and
$G_{k}\overset\text{def}\to{=}\operatorname{Gal}(\overline{k}/k)$ the absolute
Galois group of $k$.

$\bullet$ Let $\Pi$ be a quotient of $\pi_{1}(X,\eta)$ such that the
projection $\pi_{1}(X,\eta)\twoheadrightarrow G_{k}$ factors as
$\pi_{1}(X,\eta)\twoheadrightarrow\Pi\twoheadrightarrow G_{k}$, and which is
geometrically non-trivial; meaning $\operatorname{Ker}(\Pi\twoheadrightarrow
G_{k})$ is non-trivial. Given a section $s:G_{k}\to\Pi$ of the projection
$\Pi\twoheadrightarrow G_{k}$, we say that $s$ is geometric if $s(G_{k})$ is
contained in (hence equal to) the decomposition group $D_{x}\subset\Pi$
associated to a rational point $x\in X(k)$. In this case we say $s$ arises
from the rational point $x$. We say that the section $s$ is non-geometric if
$s$ is not geometric in the above sense, i.e., $s(G_{k})$ is not contained in
the decomposition group associated to a rational point $x\in X(k)$. (Note that
in the above discussion the decomposition group $D_{x}$ is only defined up to
conjugation by elements of $\operatorname{Ker}(\Pi\twoheadrightarrow G_{k})$.)

$\bullet$ We assume $X(k)\neq\emptyset$. We fix a $k$-rational point $x\in
X(k)$, and $s\overset\text{def}\to{=}s_{x}:G_{k}\to\pi_{1}(X,\eta)$ a section
of the projection $\pi_{1}(X,\eta)\twoheadrightarrow G_{k}$ associated to $x$.
Thus $s$ is defined only up to conjugation by
$\pi_{1}(\overline{X},\bar{\eta})$. Note that the section $s$ induces a
structure of $G_{k}$-group on any characteristic quotient of
$\pi_{1}(\overline{X},\bar{\eta})$.

$\bullet$ Let $\Delta$ be a quotient of $\pi_{1}(\overline{X},\bar{\eta})$
which fits in an exact sequence

$1\to\widetilde{H}\to\pi_{1}(\overline{X},\bar{\eta})\to\Delta\to 1,$

where
$\widetilde{H}\overset\text{def}\to{=}\operatorname{Ker}(\pi_{1}(\overline{X},\bar{\eta})\twoheadrightarrow\Delta)$.
We consider the following condition $(\star)$ on $\Delta$.

###### Condition ($\star$)

(i) $\Delta$ is pro-nilpotent, and is a characteristic quotient of
$\pi_{1}(\overline{X},\bar{\eta})$.

(ii) $H^{0}(U,\Delta)=0$ for every open subgroup $U$ of $G_{k}$.

(iii) The quotient
$\pi_{1}(\overline{X},\bar{\eta})\twoheadrightarrow\pi_{1}(\overline{X},\bar{\eta})^{\operatorname{ab}}$
factors as

$\pi_{1}(\overline{X},\bar{\eta})\twoheadrightarrow\Delta\twoheadrightarrow\pi_{1}(\overline{X},\bar{\eta})^{\operatorname{ab}}.$

(iv) Let $H\overset\text{def}\to{=}\widetilde{H}^{\operatorname{ab}}$, and
$\Gamma\overset\text{def}\to{=}\pi_{1}(\overline{X},\bar{\eta})/\operatorname{Ker}(\widetilde{H}\twoheadrightarrow
H)$. We have a push-out diagram

$\begin{CD}1@>{}>{}>\widetilde{H}@>{}>{}>\pi_{1}(\overline{X},\bar{\eta})@>{}>{}>\Delta
@>{}>{}>1\\\ @V{}V{}V@V{}V{}V\Big{\|}\\\ 1@>{}>{}>H@>{}>{}>\Gamma
@>{}>{}>\Delta @>{}>{}>1\\\ \end{CD}$

where the middle and left vertical maps are surjective. There exists a prime
integer $\ell\neq p$, such that the natural surjective map

$\Gamma^{\ell}\twoheadrightarrow\Delta^{\ell}$

is not an isomorphism.

Let
$\Pi\overset\text{def}\to{=}\pi_{1}(X,\eta)/\operatorname{Ker}(\pi_{1}(\overline{X},\bar{\eta})\twoheadrightarrow\Gamma)$,
and
$\widetilde{\Pi}\overset\text{def}\to{=}\pi_{1}(X,\eta)/\operatorname{Ker}(\pi_{1}(\overline{X},\bar{\eta})\twoheadrightarrow\Delta)$.
We have the following push-out diagrams

$\begin{CD}1@>{}>{}>\pi_{1}(\overline{X},\bar{\eta})@>{}>{}>\pi_{1}(X,\eta)@>{}>{}>G_{k}@>{}>{}>1\\\
@V{}V{}V@V{}V{}V\Big{\|}\\\ 1@>{}>{}>\Gamma @>{}>{}>\Pi
@>{}>{}>G_{k}@>{}>{}>1\\\ @V{}V{}V@V{}V{}V\Big{\|}\\\ 1@>{}>{}>\Delta
@>{}>{}>\widetilde{\Pi}@>{}>{}>G_{k}@>{}>{}>1\\\ \end{CD}$

where the vertical maps are surjective. Thus
$\operatorname{Ker}(\Pi\twoheadrightarrow\widetilde{\Pi})=\operatorname{Ker}(\Gamma\twoheadrightarrow\Delta)=H$.

$\bullet$ Examples of quotients $\Delta$ satisfying the condition $(\star)$
are: $\Delta=\pi_{1}(\overline{X},\bar{\eta})^{\operatorname{ab}}$ is the
maximal abelian quotient of $\pi_{1}(\overline{X},\bar{\eta})$ (cf. [Saïdi3],
Lemma 1.3, for the condition $(\star)$(ii)); in this case
$\widetilde{\Pi}\overset\text{def}\to{=}\pi_{1}(X,\eta)^{(\operatorname{ab})}$
is the geometrically abelian quotient of $\pi_{1}(X,\eta)$, $\Gamma$ is the
maximal metabelian quotient of $\pi_{1}(\overline{X},\bar{\eta})$, and $\Pi$
is the geometrically metabelian quotient of $\pi_{1}(X,\eta)$. More generally,
any pro-nilpotent characteristic quotient $\Delta$ of
$\pi_{1}(\overline{X},\bar{\eta})$ which satisfies conditions $(\star)$(ii),
$(\star)$(iii), and for which there exists a prime integer $\ell\neq p$ such
that the natural projection
$\pi_{1}(\overline{X},\bar{\eta})^{\ell}\twoheadrightarrow\Delta^{\ell}$ is
not an isomorphism, satisfies the condition $(\star)$.

Given a finite extension $k^{\prime}/k$ (all finite extensions of $k$ we
consider are contained in $\overline{k}$), and the corresponding open subgroup
$G_{k^{\prime}}\subseteq G_{k}$, we will denote by $\Pi_{k^{\prime}}$ the
pullback of the group extension $\Pi$ by $G_{k^{\prime}}\hookrightarrow
G_{k}$. Thus we have a commutative diagram of exact sequences

$\begin{CD}1@>{}>{}>\Gamma
@>{}>{}>\Pi_{k^{\prime}}@>{}>{}>G_{k^{\prime}}@>{}>{}>1\\\
\Big{\|}@V{}V{}V@V{}V{}V\\\ 1@>{}>{}>\Gamma @>{}>{}>\Pi
@>{}>{}>G_{k}@>{}>{}>1\\\ \end{CD}$

where the right square is cartesian. Likewise we write
$\widetilde{\Pi}_{k^{\prime}}\overset\text{def}\to{=}\widetilde{\Pi}\times_{G_{k}}G_{k^{\prime}}$.
Note that $\Pi_{k^{\prime}}$ is a quotient of $\pi_{1}(X_{k^{\prime}},\eta)$,
where
$X_{k^{\prime}}\overset\text{def}\to{=}X\times_{\operatorname{Spec}k}\operatorname{Spec}k^{\prime}$
and $\eta$ is naturally induced by the above geometric point $\eta$. Our first
main result in this paper is the following.

###### Theorem A

We use notations as above. Let $X$ be a proper, smooth, and geometrically
connected hyperbolic curve over the $p$-adic local field $k$. Assume that
$X(k)\neq\emptyset$, and $X$ has potentially good reduction. Let $\Delta$ be a
quotient of $\pi_{1}(\overline{X},\bar{\eta})$ satisfying the condition
$(\star)$, and $\Pi$ the corresponding quotient of $\pi_{1}(X,\eta)$ as in the
above discussion which fits in the exact sequence $1\to\Gamma\to\Pi\to
G_{k}\to 1$. Then there exists a finite extension $\tilde{k}/k$ such that the
following holds. For every finite extension $k^{\prime}/\tilde{k}$, there
exists a section $s:G_{k^{\prime}}\to\Pi_{k^{\prime}}$ of the projection
$\Pi_{k^{\prime}}\twoheadrightarrow G_{k^{\prime}}$ which is non-geometric.

As a corollary of Theorem A we obtain the following (cf. examples discussed
after introducing the condition $(\star)$).

###### Corollary B

There exist non-geometric sections of geometrically metabelian arithmetic
fundamental groups of hyperbolic curves over $p$-adic local fields.

Let $m\geq 1$ be an integer. With the notations above, let
$\Delta_{m}\overset\text{def}\to{=}\Delta_{m,X}$ be the maximal $m$-step
solvable pro-$p$ quotient of $\pi_{1}(\overline{X},\bar{\eta})$, and
$\Pi_{m}\overset\text{def}\to{=}\Pi_{m,X}$ the geometrically $m$-step solvable
pro-$p$ quotient of $\pi_{1}(X,\eta)$ which sits in the exact sequence

$1\to\Delta_{m}\to\Pi_{m}\to G_{k}\to 1$

(cf. [Saïdi3], $\S 1$). Note that $\Delta_{m}$ doesn’t satisfy condition
$(\star)$(iii). It is plausible, in light of Hoshi’s example in [Hoshi] (cf.
above discussion), that there exist non-geometric sections of $\Pi_{m}$ for a
suitable $X/k$ as above (this is easily seen if $m=1$, using the Kummer exact
sequence associated to $\operatorname{Pic}^{0}_{X}$). In this context we prove
the following.

###### Theorem C

We use notations as above. There exists an integer $N\geq 2$, such that the
following holds. For every prime integer $p\geq N$ there exists a proper,
smooth, and geometrically connected hyperbolic curve $X$ over a $p$-adic local
field $k$, an integer $m\geq 2$, and a section $s:G_{k}\to\Pi_{m,X}$ of the
projection $\Pi_{m,X}\twoheadrightarrow G_{k}$ which is non-geometric.

## Acknowledgment

I would like to thank Akio Tamagawa for very fruitful discussions during which
the proofs of Theorem A and Theorem C were established, while the author was
visiting the Research Institute for Mathematical Sciences at Kyoto university
(RIMS), the idea of proof of Theorem C is due to him. I would also like to
thank the referee for his/her very helpful comments.

## §1. Proof of Theorem A

We use the notations introduced in $\S 0$, as well as the notations and
assumptions in Theorem A. Thus $X$ is a proper, smooth, and geometrically
connected hyperbolic curve over the $p$-adic local field $k$,
$X(k)\neq\emptyset$, and we assume (without loss of generality) that $X$ has
good reduction over $\Cal{O}_{k}$. Further, $\Delta$ is a characteristic
quotient of $\pi_{1}(\overline{X},\bar{\eta})$ satisfying the condition
$(\star)$, and $\Pi$ is the corresponding quotient of $\pi_{1}(X,\eta)$ as
above which fits in the exact sequence $1\to\Gamma\to\Pi\to G_{k}\to 1$ (cf.
$\S 0$). We have the following commutative diagram of exact sequences.

$\begin{CD}11\\\ @V{}V{}V@V{}V{}V\\\ H@=H\\\ @V{}V{}V@V{}V{}V\\\
1@>{}>{}>\Gamma @>{}>{}>\Pi @>{}>{}>G_{k}@>{}>{}>1\\\
@V{}V{}V@V{}V{}V\Big{\|}|\\\ 1@>{}>{}>\Delta
@>{}>{}>\widetilde{\Pi}@>{}>{}>G_{k}@>{}>{}>1\\\ @V{}V{}V@V{}V{}V\\\ 11\\\
\end{CD}$ $None$

Recall $x\in X(k)$ is a $k$-rational point, and
$s\overset\text{def}\to{=}s_{x}:G_{k}\to\pi_{1}(X,\eta)$ a section of the
projection $\pi_{1}(X,\eta)\twoheadrightarrow G_{k}$ associated to $x$.
Further, $s$ induces sections:
$s_{1}\overset\text{def}\to{=}s_{1,x}:G_{k}\to\widetilde{\Pi}$ of the
projection $\widetilde{\Pi}\twoheadrightarrow G_{k}$, and
$s_{2}\overset\text{def}\to{=}s_{2,x}:G_{k}\to\Pi$ of the projection
$\Pi\twoheadrightarrow G_{k}$, which fit in a commutative diagram

$\begin{CD}G_{k}@>{s_{2}}>{}>\Pi\\\ \Big{\|}@V{}V{}V\\\
G_{k}@>{s_{1}}>{}>\widetilde{\Pi}\\\ \end{CD}$

where the right vertical map is the one in diagram (1.1).

The profinite group $\Delta$ is finitely generated, as follows from the well-
known finite generation of the profinite group
$\pi_{1}(\overline{X},\overline{\eta})$ which projects onto $\Delta$. Let
$\\{\Delta^{i}\\}_{i\geq 1}$ be a countable system of characteristic open
subgroups of $\Delta$ such that

$\Delta^{i+1}\subseteq\Delta^{i},\ \ \ \ \Delta^{1}=\Delta,\ \ \ \ \text{and}\
\ \bigcap_{i\geq 1}\Delta^{i}=\\{1\\}.$

Write $\Delta_{i}\overset\text{def}\to{=}\Delta/\Delta^{i}$. Thus $\Delta_{i}$
is a finite characteristic quotient of $\Delta$, and we have a push-out
diagram of exact sequences

$\begin{CD}1@>{}>{}>\Delta @>{}>{}>\widetilde{\Pi}@>{}>{}>G_{k}@>{}>{}>1\\\
@V{}V{}V@V{}V{}V\Big{\|}\\\
1@>{}>{}>\Delta_{i}@>{}>{}>\Pi_{i}@>{}>{}>G_{k}@>{}>{}>1\\\ \end{CD}$ $None$

which defines a (geometrically finite) quotient $\Pi_{i}$ of
$\widetilde{\Pi}$. The section $s_{1}$ induces a section

$\rho_{i}:G_{k}\to\Pi_{i}$

of the projection $\Pi_{i}\twoheadrightarrow G_{k}$, $\forall i\geq 1$. Write

$\widetilde{\Pi}^{i}\overset\text{def}\to{=}\widetilde{\Pi}^{i}[s_{1}]\overset\text{def}\to{=}\Delta^{i}.s_{1}(G_{k}).$

Note that $\widetilde{\Pi}^{i}\subseteq\widetilde{\Pi}$ is an open subgroup
which contains the image $s_{1}(G_{k})$ of $s_{1}$. Write $\Pi^{i}$ for the
inverse image of $\widetilde{\Pi}^{i}$ in $\pi_{1}(X,\eta)$. Thus
$\Pi^{i}\subseteq\pi_{1}(X,\eta)$ is an open subgroup corresponding to an
étale cover

$X_{i}\to X_{1}\overset\text{def}\to{=}X$

defined over $k$ (since $\Pi^{i}$ maps onto $G_{k}$ via the natural projection
$\pi_{1}(X,\eta)\twoheadrightarrow G_{k}$, by the very definition of
$\Pi^{i}$).

Note that the étale cover
$\overline{X}_{i}\overset\text{def}\to{=}X_{i}\times_{\operatorname{Spec}k}\operatorname{Spec}{\overline{k}}\to\overline{X}$
is Galois with Galois group $\Delta_{i}$, and we have a commutative diagram of
étale covers

$\begin{CD}\overline{X}_{i}@>{}>{}>\overline{X}\\\ @V{}V{}V@V{}V{}V\\\
X_{i}@>{}>{}>X\\\ \end{CD}$

where $\overline{X}_{i}\to X$ is Galois with Galois group $\Pi_{i}$, and
$\overline{X}_{i}\to X_{i}$ is Galois with Galois group $\rho_{i}(G_{k})$. We
have a commutative diagram of exact sequences

$\begin{CD}11\\\ @V{}V{}V@V{}V{}V\\\
1@>{}>{}>\widetilde{\Delta}^{i}=\pi_{1}(\overline{X_{i}},\bar{\eta})@>{}>{}>\Pi^{i}=\pi_{1}(X_{i},\eta)@>{}>{}>G_{k}@>{}>{}>1\\\
@V{}V{}V@V{}V{}V\Big{\|}\\\
1@>{}>{}>\pi_{1}(\overline{X},\bar{\eta})@>{}>{}>\pi_{1}(X,\eta)@>{}>{}>G_{k}@>{}>{}>1\\\
\end{CD}$

where $\widetilde{\Delta}^{i}$ is the inverse image of $\Delta^{i}$ in
$\pi_{1}(\overline{X},\bar{\eta})$, and the equalities
$\widetilde{\Delta}^{i}=\pi_{1}(\overline{X_{i}},\bar{\eta})$,
$\Pi^{i}=\pi_{1}(X_{i},\eta)$, are natural identifications; the base points
$\eta$ (resp. $\overline{\eta}$) of $X_{i}$ (resp. $\overline{X}_{i}$) are
those induced by the base points $\eta$ (resp. $\overline{\eta}$) of $X$
(resp. $\overline{X}$). Note that $\Pi^{i+1}\subseteq\Pi^{i}$, and
$\widetilde{\Delta}^{i+1}\subseteq\widetilde{\Delta}^{i}$, as follows from the
various definitions.

###### Lemma 1.1

With the above notations and those in $\S 0$, the following holds:

$\widetilde{H}=\bigcap_{i\geq 1}\widetilde{\Delta}^{i}.$

###### Demonstration Proof

Follows from the various definitions. ∎

We take this opportunity to correct a mistake that occurred in [Saïdi2], Lemma
1.1. The claim there that $\Cal{I}_{X}=\bigcap_{i\geq 1}\Pi_{i}$ is false,
however this doesn’t affect the validity of the results or other assertions
made in loc. cit..

For each integer $i\geq 1$, consider the push-out diagram

$\begin{CD}1@>{}>{}>\widetilde{\Delta}^{i}=\pi_{1}(\overline{X_{i}},\bar{\eta})@>{}>{}>\Pi^{i}=\pi_{1}(X_{i},\eta)@>{}>{}>G_{k}@>{}>{}>1\\\
@V{}V{}V@V{}V{}V\Big{\|}\\\
1@>{}>{}>\widetilde{\Delta}^{i,\operatorname{ab}}@>{}>{}>\Pi^{(i,\operatorname{ab})}@>{}>{}>G_{k}@>{}>{}>1\\\
\end{CD}$

where $\widetilde{\Delta}^{i,\operatorname{ab}}$ is the maximal abelian
quotient of $\widetilde{\Delta}^{i}$, and $\Pi^{(i,\operatorname{ab})}$ is the
geometrically abelian fundamental group of $X_{i}$. Consider the commutative
diagram

$\begin{CD}1@>{}>{}>H@>{}>{}>\Cal{H}\overset\text{def}\to{=}\Cal{H}[s_{1}]@>{}>{}>G_{k}@>{}>{}>1\\\
@V{}V{}V@V{}V{}V@V{s_{1}}V{}V\\\ 1@>{}>{}>H@>{}>{}>\Pi
@>{}>{}>\widetilde{\Pi}@>{}>{}>1\\\ \end{CD}$ $None$

where the right square is cartesian. Thus (the group extension) $\Cal{H}$ is
the pull-back of (the group extension) $\Pi$ via the section
$s_{1}:G_{k}\to\widetilde{\Pi}$.

###### Lemma 1.2

We have natural identifications $H\overset\sim\to{\rightarrow}\underset{i\geq
1}\to{\varprojlim}\ \widetilde{\Delta}^{i,\operatorname{ab}}$, and
$\Cal{H}\overset\sim\to{\rightarrow}\underset{i\geq 1}\to{\varprojlim}\
\Pi^{(i,\operatorname{ab})}$.

###### Demonstration Proof

Similar to the proof of Lemma 1.3 in [Saïdi2]. ∎

The section $s_{2}:G_{k}\to\Pi$, which lifts the section $s_{1}$, induces a
section $s_{2}:G_{k}\to\Cal{H}$ of the projection $\Cal{H}\twoheadrightarrow
G_{k}$ (since $s_{2}(G_{k})\subset\Cal{H}$). We fix the section
$s_{2}:G_{k}\to\Cal{H}$ as a base point of the torsor of splittings of the
upper sequence in diagram (1.3). Thus the set of splittings of the group
extension $\Cal{H}$, modulo conjugation by elements of $H$, is a torsor under
$H^{1}(G_{k},H)$; the $G_{k}$-module structure of $H$ being deduced from
diagram (1.3). The splitting $s_{2}:G_{k}\to\Cal{H}$ thus corresponds to $0\in
H^{1}(G_{k},H)$. Note that the set of splittings $\tilde{s}:G_{k}\to\Cal{H}$
of the group extension $\Cal{H}$ is in one-to-one correspondence with the set
of sections $\tilde{s}:G_{k}\to\Pi$ of the projection $\Pi\twoheadrightarrow
G_{k}$ which lift the section $s_{1}$.

Let $\tilde{s}:G_{k}\to\Cal{H}$ be a section of the group extension $\Cal{H}$,
which induces a section $\tilde{s}:G_{k}\to\Pi$ of the projection
$\Pi\twoheadrightarrow G_{k}$ which lift the section $s_{1}$. Let
$[\tilde{s}]$ be the class of $\tilde{s}$ in $H^{1}(G_{k},H)$ (cf. above
discussion).

###### Fact 1.3

The section $\tilde{s}:G_{k}\to\Pi$ is geometric if and only if
$[\tilde{s}]=0$. In this case the section $\tilde{s}$ is associated to the
rational point $x\in X(k)$.

###### Demonstration Proof

First, assume that the section $\tilde{s}:G_{k}\to\Pi$ is geometric and arises
from a rational point $\tilde{x}\in X(k)$. Both sections
$\tilde{s}:G_{k}\to\Pi$, and $s_{2}:G_{k}\to\Pi$, induce splittings
$\tilde{s}^{\operatorname{ab}}:G_{k}\to\pi_{1}(X,\eta)^{(\operatorname{ab})}$,
and $s_{2}^{\operatorname{ab}}:G_{k}\to\pi_{1}(X,\eta)^{(\operatorname{ab})}$
of the group extension
$1\to\pi_{1}(\overline{X},\bar{\eta})^{\operatorname{ab}}\to\pi_{1}(X,\eta)^{(\operatorname{ab})}\to
G_{k}\to 1$, where $\pi_{1}(X,\eta)^{(\operatorname{ab})}$ is the
geometrically abelian quotient of $\pi_{1}(X,\eta)$. Further one has
$\tilde{s}^{\operatorname{ab}}=s_{2}^{\operatorname{ab}}$ (cf. condition
$(\star)$(iii), and the fact that both $\tilde{s}$ and $s_{2}$ lift the
section $s_{1}$). A standard argument, resorting to the Kummer exact sequence
associated to the jacobian $\operatorname{Pic}^{0}_{X}$ of $X$, shows that
$\tilde{x}=x$ (cf. [Tamagawa], Proposition 2.8).

Next, we claim $[\tilde{s}]=0$. Indeed, the classes of $\tilde{s}$ and $s_{2}$
in $H^{1}(G_{k},\Gamma)$ coincide as both sections are geometric and
associated to the same rational point $x$, hence $\tilde{s}$ and $s_{2}$ are
conjugate by an element of $\Gamma$. Here we view the set of splittings of the
group extension $\Pi$ (of $\Gamma$ by $G_{k}$) as a torsor under
$H^{1}(G_{k},\Gamma)$, with base point the class of the section $s_{2}$.
Further the natural map $H^{1}(G_{k},H)\to H^{1}(G_{k},\Gamma)$ of pointed
cohomology sets is injective as follows from the condition $(\star)$(ii) (cf.
[Serre], I.$\S 5$, Proposition 38, and diagram (1.1)). (Here the
$G_{k}$-module structure on $H$ (resp. $G_{k}$-group structure on $\Gamma$) is
induced by the section $s_{1}$ (resp. $s_{2}$) (cf. diagram (1.1)).) Thus
$[\tilde{s}]=0$.

Conversely, if $[\tilde{s}]=0$, then $\tilde{s}$ is conjugate to $s_{2}$ by an
element of $H$ hence is geometric and associated to the rational point $x$. ∎

As a consequence we obtain the following.

###### Lemma 1.4

Let $k^{\prime}/k$ be a finite extension. There exists a section
$\tilde{s}:G_{k^{\prime}}\to\Pi_{k^{\prime}}$ of the projection
$\Pi_{k^{\prime}}\twoheadrightarrow G_{k^{\prime}}$, which lifts the section
$s_{1,k^{\prime}}:G_{k^{\prime}}\to\widetilde{\Pi}_{k^{\prime}}$ of the
projection $\widetilde{\Pi}_{k^{\prime}}\twoheadrightarrow G_{k}$ induced by
$s_{1}$, and which is non-geometric, if and only if
$H^{1}(G_{k^{\prime}},H)\neq 0$.

Thus proving Theorem A reduces to proving the following.

###### Proposition 1.5

There exists a finite extension $\tilde{k}/k$ such that
$H^{1}(G_{k^{\prime}},H)\neq 0$ for every finite extension
$k^{\prime}/\tilde{k}$.

The rest of this section is devoted to proving Proposition 1.5. Let $\ell\neq
p$ be a prime integer such that the map
$\Gamma^{\ell}\twoheadrightarrow\Delta^{\ell}$ is not an isomorphism (cf.
condition $(\star)$(iv)). Write $\pi_{1}(\overline{X},\bar{\eta})^{\ell}$ for
the maximal pro-$\ell$ quotient of $\pi_{1}(\overline{X},\bar{\eta})$, and

$\pi_{1}(X,\eta)^{(\ell)}\overset\text{def}\to{=}\pi_{1}(X,\eta)/\operatorname{Ker}(\pi_{1}(\overline{X},\bar{\eta})\twoheadrightarrow\pi_{1}(\overline{X},\bar{\eta})^{\ell})$

for the geometrically pro-$\ell$ quotient of $\pi_{1}(X,\eta)$, which fits in
the exact sequence

$1\to\pi_{1}(\overline{X},\bar{\eta})^{\ell}\to\pi_{1}(X,\eta)^{(\ell)}\to
G_{k}\to 1.$

Let $s^{\ell}=s^{\ell}_{x}:G_{k}\to\pi_{1}(X,\eta)^{(\ell)}$ be the section of
the projection $\pi_{1}(X,\eta)^{(\ell)}\twoheadrightarrow G_{k}$ induced by
the section $s=s_{x}$. This section induces a representation

$\rho^{\ell}:G_{k}\to\operatorname{Aut}(\pi_{1}(\overline{X},\bar{\eta})^{\ell})$

which factors as $G_{k}\twoheadrightarrow
G_{F}\to\operatorname{Aut}(\pi_{1}(\overline{X},\bar{\eta})^{\ell})$, where
$G_{F}$ is the quotient of $G_{k}$ by its inertia subgroup, since $X$ has good
reduction over $\Cal{O}_{k}$. Further the image of the representation
$\rho^{\ell}$ is almost pro-$\ell$, i.e., $\rho^{\ell}(G_{k})$ possesses an
open subgroup which is pro-$\ell$. In particular, there exists a finite
extension $\tilde{k}/k$ such that the restriction
$\rho^{\ell}_{\tilde{k}}:G_{\tilde{k}}\to\operatorname{Aut}(\pi_{1}(\overline{X},\bar{\eta})^{\ell})$
of $\rho^{\ell}$ to $G_{\tilde{k}}$ has a pro-$\ell$ image. In order to prove
Proposition 1.5 we will, without loss of generality, assume that the image of
$\rho^{\ell}$ is pro-$\ell$, and will show $H^{1}(G_{k},H)\neq 0$.

Let $\Delta^{\ell}$ be the maximal pro-$\ell$ quotient of $\Delta$, and
$\widetilde{\Pi}^{(\ell)}\overset\text{def}\to{=}\widetilde{\Pi}/\operatorname{Ker}(\Delta\twoheadrightarrow\Delta^{\ell})$
the geometrically pro-$\ell$ quotient of $\widetilde{\Pi}$. We have the
following commutative diagram of exact sequences

$\begin{CD}1@>{}>{}>\Delta @>{}>{}>\widetilde{\Pi}@>{}>{}>G_{k}@>{}>{}>1\\\
@V{}V{}V@V{}V{}V\Big{\|}\\\
1@>{}>{}>\Delta^{\ell}@>{}>{}>\widetilde{\Pi}^{(\ell)}@>{}>{}>G_{k}@>{}>{}>1\\\
\end{CD}$

where the left and middle vertical maps are surjective. For $i\geq 1$, let
$N^{i}$ be the image of $\Delta^{i}$ in $\Delta^{\ell}$, and
$\widetilde{N}^{i}$, $\widehat{N}^{i}$, the pre-images of $N^{i}$ in
$\pi_{1}(\overline{X},\bar{\eta})$, and
$\pi_{1}(\overline{X},\bar{\eta})^{\ell}$; respectively. Note that
$\widetilde{N}^{i}$ is a characteristic subgroup of
$\pi_{1}(\overline{X},\bar{\eta})$, and $\widehat{N}^{i}$ is stable by the
action of $s^{\ell}(G_{k})$.

Let $\widetilde{U}^{i}\overset\text{def}\to{=}\widetilde{N}^{i}.s(G_{k})$, and
$\widehat{U}^{i}\overset\text{def}\to{=}\widehat{N}^{i}.s^{\ell}(G_{k})$, for
$i\geq 1$. Thus $\widetilde{U}^{i}$ is an open subgroup of $\pi_{1}({X},\eta)$
corresponding to an étale cover $Y_{i}\to X$, and the étale cover $X_{i}\to X$
factorises as

$X_{i}\to Y_{i}\to X$

[$\pi_{1}(X_{i},\eta)\subset\pi_{1}(Y_{i},\eta)$ as follows from the various
definitions], where $X_{i}\to Y_{i}$ is an étale cover of degree prime-
to-$\ell$, since $\Delta$ (hence also $\Delta_{i}$, for $i\geq 1$) is pro-
nilpotent (see condition $(\star)$(i)). Further $\widetilde{U}^{i}$, and
$\widehat{U}^{i}$, are naturally identified with $\pi_{1}(Y_{i},\eta)$, and
$\pi_{1}(Y_{i},\eta)^{(\ell)}$; respectively, $\forall i\geq 1$. Here
$\pi_{1}(Y_{i},\eta)^{(\ell)}$ is the geometrically pro-$\ell$ quotient of
$\pi_{1}(Y_{i},\eta)$, and sits in an exact sequence

$1\to\pi_{1}(\overline{Y}_{i},\bar{\eta})^{\ell}\to\pi_{1}(Y_{i},\eta)^{(\ell)}\to
G_{k}\to 1,$

where $\overline{Y}_{i}\overset\text{def}\to{=}Y\times_{k}\overline{k}$.

The natural action of $G_{k}$ on $\pi_{1}(\overline{X},\bar{\eta})^{\ell}$,
and which factorises through $G_{F}^{\ell}$ by our assumption on the
representation $\rho^{\ell}$, is compatible with its action on the open
subgroup $\pi_{1}(\overline{Y}_{i},\bar{\eta})^{\ell}$, hence this latter
action also factorises through $G_{F}^{\ell}$.

There is a surjective homomorphism (recall Lemma 1.2)

$H^{1}(G_{k},H)=\varprojlim_{i\geq
1}H^{1}(G_{k},\widetilde{\Delta}^{i,\operatorname{ab}})\twoheadrightarrow\varprojlim_{i\geq
1}H^{1}(G_{k},\widetilde{\Delta}^{i,\operatorname{ab},\ell}).$

(Indeed, $\varprojlim_{i\geq
1}H^{1}(G_{k},\widetilde{\Delta}^{i,\operatorname{ab}})\overset\sim\to{\rightarrow}\prod_{l\in\mathfrak{Primes}}\varprojlim_{i\geq
1}H^{1}(G_{k},\widetilde{\Delta}^{i,\operatorname{ab},l})$, where the product
is over all prime integers $l$ and the above homomorphism is the projection
onto the $\ell$-th factor.) Further the étale covers $\\{X_{i}\to
Y_{i}\\}_{i\geq 1}$ induce a homomorphism (recall
$\widetilde{\Delta}^{i}=\pi_{1}(\overline{X}_{i},\bar{\eta})$)

$\varprojlim_{i\geq
1}H^{1}(G_{k},\widetilde{\Delta}^{i,\operatorname{ab},\ell})\to\varprojlim_{i\geq
1}H^{1}(G_{k},\pi_{1}(\overline{Y}_{i},\bar{\eta})^{\operatorname{ab},\ell}),$

which is surjective. More precisely, the map
$H^{1}(G_{k},\widetilde{\Delta}^{i,\operatorname{ab},\ell})\to
H^{1}(G_{k},\pi_{1}(\overline{Y}_{i},\bar{\eta})^{\operatorname{ab},\ell})$ is
surjective, $\forall i\geq 1$, as follows easily from a restriction-
corestriction argument using the fact that the degree of the cover $X_{i}\to
Y_{i}$ is prime-to-$\ell$ (observe the maps on cohomology induced by the
natural maps
$\pi_{1}(\overline{Y}_{i},\bar{\eta})^{\operatorname{ab},\ell}@>\operatorname{res}>>\widetilde{\Delta}^{i,\operatorname{ab},\ell}@>\operatorname{cor}>>\pi_{1}(\overline{Y}_{i},\bar{\eta})^{\operatorname{ab},\ell}$
arising from the morphisms
$\operatorname{Pic}^{0}(Y_{i})\to\operatorname{Pic}^{0}(X_{i})\to\operatorname{Pic}^{0}(Y_{i})$,
where the first one is the pull-back map of line bundles and the second is the
norm map). Thus in order to prove Proposition 1.5 it suffices to show the
following.

###### Proposition 1.6

With the above notations, it holds that

$\varprojlim_{i\geq
1}H^{1}(G_{k},\pi_{1}(\overline{Y}_{i},\bar{\eta})^{\operatorname{ab},\ell})\neq
0.$

###### Demonstration Proof

As discussed above the natural action of $G_{k}$ on
$\pi_{1}(\overline{Y}_{i},\bar{\eta})^{\operatorname{ab},\ell}$ factors
through $G_{F}^{\ell}$ (which is isomorphic to $\mathbb{Z}_{\ell}$). There is
an injective inflation map

$\inf:\varprojlim_{i\geq
1}H^{1}(G_{F}^{\ell},\pi_{1}(\overline{Y}_{i},\bar{\eta})^{\operatorname{ab},\ell})\hookrightarrow\varprojlim_{i\geq
1}H^{1}(G_{k},\pi_{1}(\overline{Y}_{i},\bar{\eta})^{\operatorname{ab},\ell}).$

Further

$(\varprojlim_{i\geq
1}\pi_{1}(\overline{Y}_{i},\bar{\eta})^{\operatorname{ab},\ell})_{G_{F}^{\ell}}=H^{1}(G_{F}^{\ell},\varprojlim_{i\geq
1}\pi_{1}(\overline{Y}_{i},\bar{\eta})^{\operatorname{ab},\ell})=\varprojlim_{i\geq
1}H^{1}(G_{F}^{\ell},\pi_{1}(\overline{Y}_{i},\bar{\eta})^{\operatorname{ab},\ell})$

where the notation $(\ \ )_{G_{F}^{\ell}}$ stands for the co-invariant module,
the first equality follows from the fact that $G_{F}^{\ell}$ is procyclic, and
the second follows from [Neukirch-Schmidt-Winberg] (2.3.5) Corollary.

There is a natural isomorphism

$\left(\operatorname{Ker}\left(\Gamma^{\ell}\to\Delta^{\ell}\right)\right)\overset\sim\to{\rightarrow}\varprojlim_{i\geq
1}\pi_{1}(\overline{Y}_{i},\bar{\eta})^{\operatorname{ab},\ell}.$

(Proof similar to the proof of Lemma 1.2.) Thus $\varprojlim_{i\geq
1}\pi_{1}(\overline{Y}_{i},\eta)^{\operatorname{ab},\ell}\neq 0$ (cf.
condition $(\star)$(iii), and our choice of $\ell$). The proof of Proposition
1.6 follows from the following.

###### Lemma 1.7

Let $T$ be an abelian pro-$\ell$ group, and $P$ an infinite pro-$\ell$ cyclic
group. Assume $T$ is a continuous $P$-module. Then the co-invariant module
$(T)_{P}=\\{0\\}$ is trivial if and only if $T=\\{0\\}$ itself is trivial.

###### Demonstration Proof

Let $T^{\wedge}$ be the Pontryagin dual of $T$ which is an $\ell$-primary
torsion group. The dual of $(T)_{P}$ is the invariant group
$(T^{\wedge})^{P}$. It suffices to show that $(T^{\wedge})^{P}$ is trivial if
and only if $(T^{\wedge})$ is trivial. The action of $P$ on $T^{\wedge}$ is
discrete, in particular $T^{\wedge}$ is the union of finite $\ell$-groups
which are stable $P$-submodules. We can thus reduce to the case where
$T^{\wedge}$ and $P$ are finite. Suppose $(T^{\wedge})$ is finite, and non-
trivial, then $(T^{\wedge})^{P}$ is non-trivial since its order is divisible
by $\ell$, and $(T^{\wedge})^{P}$ contains $0$. ∎

This finishes the proof of Proposition 1.6, hence the proof of Proposition
1.5, and the proof of Theorem A. ∎

## §2. Proof of Theorem C

The rest of this paper is devoted to proving Theorem C. We use the notations
as introduced in $\S 0$, and the statement of Theorem C.

Let $K$ be a number field (finite extension of $\mathbb{Q}$), and
$\overline{K}$ an algebraic closure of $K$. Let $X$ be a proper, smooth, and
geometrically connected hyperbolic curve over $K$. Write
$J\overset\text{def}\to{=}\operatorname{Pic}^{0}_{X}$ for the jacobian of $X$.
Assume $X(K)\neq\emptyset$. Fix a rational point $x\in X(K)$, and consider the
embedding $\iota:X\hookrightarrow J$ defined by $\iota(x)=0_{J}$. For any
field extension $\overline{K}\subset L$, with $L$ algebraically closed, let
$J^{\operatorname{tor}}\overset\text{def}\to{=}J(L)^{\operatorname{tor}}=J(\overline{K})^{\operatorname{tor}}$
be the torsion subgroup of $J$. The intersection $X\cap
J^{\operatorname{tor}}$ is finite by [Raynaud]. Let $M$ be the cardinality of
the subgroup of $J^{\operatorname{tor}}$ generated by $X\cap
J^{\operatorname{tor}}$. We assume $M\geq 2$ (this $M$ will be the integer $N$
required in theorem C).

Let $p>M$ be a prime integer, $k$ a $p$-adic completion of $K$, $\overline{k}$
an algebraic closure of $k$, $X_{k}\overset\text{def}\to{=}X\times_{K}k$, and
$X_{\overline{k}}\overset\text{def}\to{=}X\times_{K}\overline{k}$. Recall the
exact sequence of fundamental groups (cf. $\S 0$)

$1\to\pi_{1}(X_{\overline{k}},\bar{\eta})\to\pi_{1}(X_{k},\eta)\to G_{k}\to
1.$

Let $\Delta$ be the maximal pro-$p$ quotient of
$\pi_{1}(X_{\overline{k}},\bar{\eta})$, and

$\Pi\overset\text{def}\to{=}\pi_{1}(X_{k},\eta)/\operatorname{Ker}(\pi_{1}(X_{\overline{k}},\bar{\eta})\twoheadrightarrow\Delta)$

the geometrically pro-$p$ arithmetic fundamental group of $X_{k}$.

For an integer $m\geq 1$, let $\Delta_{m}$ be the maximal $m$-step solvable
pro-$p$ quotient of $\pi_{1}(X_{\overline{k}},\bar{\eta})$, and

$\Pi_{m}\overset\text{def}\to{=}\pi_{1}(X_{k},\eta)/\operatorname{Ker}(\pi_{1}(X_{\overline{k}},\bar{\eta})\twoheadrightarrow\Delta_{m})$

the geometrically $m$-step solvable pro-$p$ arithmetic fundamental group of
$X_{k}$. We have a commutative diagram of exact sequences

$\begin{CD}11\\\ @V{}V{}V@V{}V{}V\\\ \Delta[m]@=\Delta[m]\\\
@V{}V{}V@V{}V{}V\\\
1@>{}>{}>\Delta_{m+1}@>{}>{}>\Pi_{m+1}@>{}>{}>G_{k}@>{}>{}>1\\\
@V{}V{}V@V{}V{}V\Big{\|}\\\
1@>{}>{}>\Delta_{m}@>{}>{}>\Pi_{m}@>{}>{}>G_{k}@>{}>{}>1\\\
@V{}V{}V@V{}V{}V\\\ 11\\\ \end{CD}$ $None$

where
$\Delta[m]\overset\text{def}\to{=}\operatorname{Ker}(\Delta_{m+1}\twoheadrightarrow\Delta_{m})=\operatorname{Ker}(\Pi_{m+1}\twoheadrightarrow\Pi_{m})$
(cf. [Saïdi3], $\S 1$, for more details). Further we have natural
identifications

$\Delta=\varprojlim_{m\geq 1}\Delta_{m},\ \ \ \text{and}\ \ \ \
\Pi=\varprojlim_{m\geq 1}\Pi_{m}.$

Let $s_{x}:G_{k}\to\Pi$ be a section of the projection $\Pi\twoheadrightarrow
G_{k}$ associated to the $k$-rational point (image in $X_{k}$ of) $x$, which
induces sections $s_{x,m}:G_{k}\to\Pi_{m}$ of the projections
$\Pi_{m}\twoheadrightarrow G_{k}$, $\forall m\geq 1$. (Thus $s_{x}$ is defined
up to conjugation by $\Delta$.) We fix the section $s_{x,1}$ as a base point
of the torsor of splittings of the exact sequence $1\to\Delta_{1}\to\Pi_{1}\to
G_{k}\to 1$ ($\Pi_{1}$ is the geometrically abelian pro-$p$ arithmetic
fundamental group of $X_{k}$), which is a torsor under
$H^{1}(G_{k},\Delta_{1})$.

Let $y\in(X\cap J^{\operatorname{tor}})(K)\setminus\\{0_{J}\\}$ (the existence
of $y$ follows from our assumption $M\geq 2$), $s_{y}:G_{k}\to\Pi$ a section
of the projection $\Pi\twoheadrightarrow G_{k}$ associated to the $k$-rational
point (image in $X_{k}$ of) $y$, which induces sections
$s_{y,m}:G_{k}\to\Pi_{m}$ of the projections $\Pi_{m}\twoheadrightarrow
G_{k}$, $\forall m\geq 1$. The classes $[s_{x,1}]=0$, and $[s_{y,1}]$, of the
sections $s_{x,1}$, and $s_{y,1}$; respectively, in $H^{1}(G_{k},\Delta_{1})$
coincide. Indeed this follows easily from the (pro-$p$) Kummer exact sequence
associated to $J$, and the fact that $\iota(y)$ is a torsion point of order
prime-to-$p$ (recall $p>M$).

More generally, for $m\geq 1$, consider the following commutative diagram

$\begin{CD}1@>{}>{}>\Delta[m+1]@>{}>{}>E[m+1]@>{}>{}>G_{k}@>{}>{}>1\\\
\Big{\|}@V{}V{}V@V{s_{x,m}}V{}V\\\
1@>{}>{}>\Delta[m+1]@>{}>{}>\Pi_{m+1}@>{}>{}>\Pi_{m}@>{}>{}>1\\\ \end{CD}$
$None$

where the right square is cartesian. Thus the group extension $E[m+1]$ is the
pull-back of the group extension $\Pi_{m+1}$ via the section $s_{x,m}$.

The upper exact sequence in diagram (2.2) splits. Indeed this follows from the
existence of the section $s_{x,m+1}:G_{k}\to\Pi_{m+1}$ which lifts the section
$s_{x,m}$, and induces a splitting $s_{x,m+1}:G_{k}\to E[m+1]$ of the group
extension $E[m+1]$. We fix the section $s_{x,m+1}$ as a base point for the
torsor of splittings of the group extension $E[m+1]$, which is a torsor under
$H^{1}(G_{k},\Delta[m+1])$; the $G_{k}$-module structure of $\Delta[m+1]$ is
deduced from diagram (2.2). If $z\in X(k)$, and
$s_{z,m}=s_{x,m}:G_{k}\to\Pi_{m}$, then the section $s_{z,m+1}$ gives rise to
a splitting $s_{z,m+1}:G_{k}\to E[m+1]$ of the upper exact sequence in diagram
(2.2), hence to a class $[s_{z,m+1}]\in H^{1}(G_{k},\Delta[m+1])$.

Define $\Cal{S}_{m}$ to be the set of rational points $z\in X(k)$ such that
$s_{x,m}(G_{k})$ coincide with a decomposition group of $\Pi_{m}$ associated
to $z$. We have the following inclusions

$\cdots\subseteq\Cal{S}_{m+1}\subseteq\Cal{S}_{m}\subseteq\cdots\subseteq\Cal{S}_{2}\subseteq\Cal{S}_{1}=X(k)\cap
J^{\operatorname{tor},p^{\prime}}\subseteq X\cap J^{\operatorname{tor}}.$

The equality $\Cal{S}_{1}=X(k)\cap J^{\operatorname{tor},p^{\prime}}$ follows
from the (pro-$p$) Kummer exact sequence associated to $J$, and the well-known
structure of $J(k)$.

###### Lemma 2.1

The equality $\bigcap_{m\geq 1}\Cal{S}_{m}=\\{x\\}$ holds.

###### Demonstration Proof

Follows from [Mochizuki], Theorem C, and a limit argument using the fact that
$\Pi=\varprojlim_{m\geq 1}\Pi_{m}$. ∎

It follows from Lemma 2.1, and the above discussion, that there exists $m\geq
1$ such that

$\\{x\\}\subsetneq\Cal{S}_{m},\ \ \ \text{and}\ \ \ \\{x\\}=\Cal{S}_{m+1}.$

Let

$A\overset\text{def}\to{=}\\{[s_{z,m+1}]:z\in\Cal{S}_{m}\\}\subset
H^{1}(G_{k},\Delta[m+1]).$

Note that $\\{0\\}\subsetneq A$; which follows from the facts that
$\\{x\\}\subsetneq\Cal{S}_{m}$ and $\\{x\\}=\Cal{S}_{m+1}$. Further
$\operatorname{Card}(A)\leq\operatorname{Card}(\Cal{S}_{m})\leq M<p$. In
particular,

$\exists\alpha\in H^{1}(G_{k},\Delta[m+1])\setminus A,$

since $H^{1}(G_{k},\Delta[m+1])$ is $p$-primary. Thus $\alpha$ corresponds to
a section $\alpha:G_{k}\to\Pi_{m+1}$ of the projection
$\Pi_{m+1}\twoheadrightarrow G_{k}$, which lifts the section $s_{x,m}$.

###### Lemma 2.3

The section $\alpha:G_{k}\to\Pi_{m+1}$ is non-geometric.

###### Demonstration Proof

Follows from the various definitions, and the fact that $\alpha\notin A$. ∎

This finishes the proof of Theorem C. ∎

References.

[Grothendieck] Grothendieck, A., Revêtements étales et groupe fondamental,
Lecture Notes in Math. 224, Springer, Heidelberg, 1971.

[Hoshi] Hoshi, Y., Existence of nongeometric pro-$p$ Galois sections of
hyperbolic curves, Publ. Res. Inst. Math. Sci. 46 (2010), no. 4, 829-848.

[Mochizuki] Mochizuki, S., The Local Pro-$p$ Anabelian Geometry of Curves,
Invent. Math. 138 (1999), 319-423.

[Neukirch-Schmidt-Winberg] Neukirch, J., Schmidt, A., Winberg, K., Cohomology
of Number Fields, Grundlehren der mathematischen Wissenschaften, 323,
Springer, 2000.

[Raynaud] Raynaud, M., Courbes sur une variété abélienne et points de torsion.
Invent. Math. 71, 207-233, (1983).

[Saïdi] Saïdi, M., The cuspidalisation of sections of arithmetic fundamental
groups, Advances in Mathematics 230 (2012) 1931-1954.

[Saïdi1] Saïdi, M., The cuspidalisation of sections of arithmetic fundamental
groups II, Advances in Mathematics, 354 (2019),
https://doi.org/10.1016/j.aim.2019.106737354.

[Saïdi2] Saïdi, M., On the existence of non-geometric sections of arithmetic
fundamental groups, Mathematische Zeitschrift 277, no. 1-2 (2014), 361-372.

[Saïdi3] Saïdi, M., A local-global principle for torsors under geometric
prosolvable fundamental groups, Manuscripta Mathematica 145, no. 1-2 (2014),
163-174.

[Serre] Serre, J.-P., Cohomologie galoisienne, Seconde édition, Lecture Notes
in Mathematics, 5, Springer-Verlag, Berlin-Heidelberg-New York, 1962/1963.

[Tamagawa] Tamagawa, A., The Grothendieck conjecture for affine curves.
Compositio Mathematica, 109(2) (1997), 135-194.

Mohamed Saïdi

College of Engineering, Mathematics, and Physical Sciences

University of Exeter

Harrison Building

North Park Road

EXETER EX4 4QF

United Kingdom

M.Saidiexeter.ac.uk


# Every finite-dimensional analytic space is $\sigma$-homogeneous

Claudio Agostini  Institut für Diskrete Mathematik und Geometrie
Technische Universität Wien
Wiedner Hauptstraße 8–-10/104
1040 Vienna, Austria<EMAIL_ADDRESS>http://www.dmg.tuwien.ac.at/agostini/ and  Andrea Medini  Institut für
Diskrete Mathematik und Geometrie
Technische Universität Wien
Wiedner Hauptstraße 8–-10/104
1040 Vienna, Austria<EMAIL_ADDRESS>http://www.dmg.tuwien.ac.at/medini/

(Date: March 19, 2024)

###### Abstract.

All spaces are assumed to be separable and metrizable. Building on work of van
Engelen, Harrington, Michalewski and Ostrovsky, we obtain the following
results:

  * •

Every finite-dimensional analytic space is $\sigma$-homogeneous with analytic
witnesses,

  * •

Every finite-dimensional analytic space is $\sigma$-homogeneous with pairwise
disjoint $\mathbf{\Delta}^{1}_{2}$ witnesses.

Furthermore, the complexity of the witnesses is optimal in both of the above
results. This completes the picture regarding $\sigma$-homogeneity in the
finite-dimensional realm. It is an open problem whether every analytic space
is $\sigma$-homogeneous. We also investigate finite unions of homogeneous
spaces.

###### Key words and phrases:

Homogeneous, zero-dimensional, finite-dimensional, analytic.

###### 2020 Mathematics Subject Classification:

54H05, 03E15.

The first-listed author was supported by the FWF grant P 35655. The second-
listed author was supported by the FWF grant P 35588. The authors are grateful
to Zoltán Vidnyánszky for useful discussions, in which he “planted the seeds”
for the results of Section 3.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries, terminology and notation
  3. 3 Every zero-dimensional analytic space is $\sigma$-homogeneous
  4. 4 Pairwise disjoint witnesses
  5. 5 Optimality
  6. 6 Higher dimensions
  7. 7 Finite unions
  8. 8 Open questions

##  1\. Introduction

Throughout this article, unless we specify otherwise, we will be working in
the theory $\mathsf{ZF}+\mathsf{DC}$, that is, the usual axioms of Zermelo-
Fraenkel (without the Axiom of Choice) plus the principle of Dependent Choices
(see [CMM2, Section 2] for more details and references). By _space_ we will
always mean separable metrizable topological space.

A space $X$ is _homogeneous_ if for every $(x,y)\in X\times X$ there exists a
homeomorphism $h:X\longrightarrow X$ such that $h(x)=y$. For example, using
translations, it is easy to see that every topological group is homogeneous
(as [vE1, Corollary 3.6.6] shows, the converse is not true, not even for zero-
dimensional Borel spaces). Homogeneity is a classical notion in topology,
which has been studied in depth (see for example the survey [AvM1]).

Here, we will focus on a much less studied notion.111 See however [AvM2] and
[AvM3], [vEvM] and [vM1], where somewhat related questions are investigated.
We will say that a space $X$ is _$\sigma$ -homogeneous_ if there exist
homogeneous subspaces $X_{n}$ of $X$ for $n\in\omega$ such that
$X=\bigcup_{n\in\omega}X_{n}$. When each $X_{n}$ is closed in $X$
(respectively $X_{n}\in\mathbf{\Sigma}^{1}_{1}(X)$ or
$X_{n}\in\mathbf{\Pi}^{1}_{1}(X)$), we will say that $X$ is _$\sigma$
-homogeneous with closed witnesses_ (respectively _with analytic witnesses_ or
_with coanalytic witnesses_). Notice that the complexity of the witnesses is
always relative to $X$, as opposed to absolute (see Section 2 for the
definitions). Further instances of $\sigma$-homogeneity with special witnesses
will appear throughout the paper, but their intended meaning seems clear
enough not to warrant explicit definitions.

The following is the result that sparked our interest in this topic (see
[Os2]), while the subsequent three theorems are the main results of [MV].
Notice that the complexity “closed” is optimal, as $\omega+1$ gives a trivial
example of a space that is not $\sigma$-homogeneous with open witnesses.

######  Theorem 1.1 (Ostrovsky).

Every zero-dimensional Borel space is $\sigma$-homogeneous with pairwise
disjoint closed witnesses.

######  Theorem 1.2 (Medini, Vidnyánszky).

Assume $\mathsf{AD}$. Then every zero-dimensional space is
$\sigma$-homogeneous with pairwise disjoint closed witnesses.

######  Theorem 1.3 (Medini, Vidnyánszky).

In $\mathsf{ZFC}$, there exists a zero-dimensional space that is not
$\sigma$-homogeneous.

######  Theorem 1.4 (Medini, Vidnyánszky).

Assume $\mathsf{V=L}$. Then there exists a coanalytic zero-dimensional space
that is not $\sigma$-homogeneous.

It is clear from the above results that, in order to have a complete picture
of $\sigma$-homogeneity in the zero-dimensional realm, only one piece is
missing: is every zero-dimensional analytic space $\sigma$-homogeneous? This
is [MV, Question 13.2], and our main result gives an affirmative answer (see
Theorem 3.4). The witnesses can be chosen analytic in general, or
$\mathbf{\Delta}^{1}_{2}$ if one desires them to be pairwise disjoint (see
Theorem 4.2). In Section 5, we will see that these are the lowest-possible
complexities. Furthermore, as Ostrovsky already noticed in [Os2], results of
this kind can easily be extended to all finite dimensions (see Section 6).
Finally, in Section 7, we will examine the finite analogues of
$\sigma$-homogeneity.

##  2\. Preliminaries, terminology and notation

Our reference for general set theory is [Je]. Given a function
$f:Z\longrightarrow W$, $A\subseteq Z$ and $B\subseteq W$, we will use the
notation $f[A]=\\{f(x):x\in A\\}$ and $f^{-1}[B]=\\{x\in Z:f(x)\in B\\}$.

Our reference for topology is [En]. We will write $X\approx Y$ to mean that
the spaces $X$ and $Y$ are homeomorphic. A subset $M$ of a space $X$ is
_meager_ if there exist nowhere dense subsets $M_{n}$ of $X$ for $n\in\omega$
such that $M\subseteq\bigcup_{n\in\omega}M_{n}$. A space $X$ is _meager_ if
$X$ is a meager subset of $X$. A space is _Baire_ if no non-empty open subset
of $X$ is meager. Given spaces $X$ and $Z$, an _embedding_ of $X$ in $Z$ is a
function $j:X\longrightarrow Z$ such that $j:X\longrightarrow j[X]$ is a
homeomorphism.

A zero-dimensional space $X$ is _strongly homogeneous_ (or _h-homogeneous_) if
every non-empty clopen subspace of $X$ is homeomorphic to $X$. This notion has
been studied by several authors, both “instrumentally” and for its own sake
(see the list of references in [Me1]). It is well-known that every strongly
homogeneous space is homogeneous (see for example [vE1, 1.9.1] or [Me2,
Proposition 3.32]). On the other hand, the reverse implication depends on set-
theoretic assumptions (see [CMM1, Theorems 1.1 and 1.2]).

Given a space $X$, recall that the _Cantor-Bendixson derivative_ of $X$ is
defined as follows for every ordinal $\xi$:

  * •

$X^{(0)}=X$,

  * •

$X^{(\xi+1)}=X^{(\xi)}\setminus\\{x\in X^{(\xi)}:x\text{ is isolated in
}X^{(\xi)}\\}$,

  * •

$X^{(\xi)}=\bigcap_{\xi^{\prime}<\xi}X^{(\xi^{\prime})}$, if $\xi$ is a limit
ordinal.

A space $X$ is _scattered_ if $X^{(\xi)}=\varnothing$ for some ordinal $\xi$.
In this case, the minimal such $\xi$ is the _Cantor-Bendixson rank_ of $X$.

Our reference for descriptive set theory is [Ke]. We will assume familiarity
with the basic theory of Borel, analytic and coanalytic sets in Polish spaces.
We will denote by $\mathsf{Borel}(X)$ the collection of all Borel subsets of a
space $X$. Recall that, following [Ke, page 315], one can define the classes
$\mathbf{\Sigma}^{1}_{n}(X)$, $\mathbf{\Pi}^{1}_{n}(X)$ and
$\mathbf{\Delta}^{1}_{n}(X)=\mathbf{\Sigma}^{1}_{n}(X)\cap\mathbf{\Pi}^{1}_{n}(X)$
for an arbitrary (that is, not necessarily Polish) space $X$, where $1\leq
n<\omega$. Given $A\subseteq X$, simply declare
$A\in\mathbf{\Sigma}^{1}_{n}(X)$ (respectively $A\in\mathbf{\Pi}^{1}_{n}(X)$)
if there exist a Polish space $Z$, an embedding $j:X\longrightarrow Z$, and
$\widetilde{A}\in\mathbf{\Sigma}^{1}_{n}(Z)$ (respectively
$\widetilde{A}\in\mathbf{\Pi}^{1}_{n}(X)$) such that $j[A]=\widetilde{A}\cap
j[X]$.

A space $X$ is a _Borel space_ (respectively _analytic_ or _coanalytic_) if
there exist a Polish space $Z$ and an embedding $j:X\longrightarrow Z$ such
that $j[X]\in\mathsf{Borel}(Z)$ (respectively
$j[X]\in\mathbf{\Sigma}^{1}_{1}(Z)$ or $j[X]\in\mathbf{\Pi}^{1}_{1}(Z)$). It
is well-known that a space $X$ is Borel (respectively analytic or coanalytic)
iff $j[X]\in\mathsf{Borel}(Z)$ (respectively
$j[X]\in\mathbf{\Sigma}^{1}_{1}(Z)$ or $j[X]\in\mathbf{\Pi}^{1}_{1}(Z)$) for
every Polish space $Z$ and every embedding $j:X\longrightarrow Z$ (see [MZ,
Proposition 4.2]).

Next, we recall the fundamental definitions of Wadge theory, since it will be
convenient to use this language in Section 3 (see [CMM2] for more on this
topic). Given a space $Z$ and $A,B\subseteq Z$, we will write
$A\leq_{\mathsf{W}}B$ if there exists a continuous function
$f:Z\longrightarrow Z$ such that $A=f^{-1}[B]$. Define
$A\\!\\!\downarrow\,=\\{B\subseteq Z:B\leq_{\mathsf{W}}A\\}$. We will say that
$\mathbf{\Gamma}\subseteq\mathcal{P}(Z)$ is a _Wadge class_ in $Z$ if there
exists $A\subseteq Z$ such that $\mathbf{\Gamma}=A\\!\\!\downarrow\,$. We will
denote by $\widecheck{\mathbf{\Gamma}}=\\{Z\setminus A:A\in\mathbf{\Gamma}\\}$
the _dual_ class of $\mathbf{\Gamma}$.

We conclude this section with a simple proposition, which can be safely
assumed to be folklore. It shows that Baire spaces of sufficiently low
complexity are Baire for a very strong reason.

######  Lemma 2.1.

Let $X$ be a Baire space. Assume that $X$ is analytic or coanalytic. Then $X$
has a Polish dense subspace.

###### Proof.

Assume without loss of generality that $X$ is a dense subspace of a Polish
space $Z$. Since $X$ has the Baire property in $Z$, we can write $X=G\cup M$
by [Ke, Proposition 8.23.ii], where $G\in\mathbf{\Pi}^{0}_{2}(Z)$ and $M$ is
meager in $Z$. By [Ke, Theorem 3.11], it will be enough to show that $G$ is
dense in $Z$. Assume, in order to get a contradiction, that $U$ is a non-empty
open subset of $Z$ such that $U\cap G=\varnothing$. Observe that $U\cap X$ is
a non-empty open subset of $X$ because $X$ is dense in $Z$. Furthermore, using
the density of $X$ again, it is easy to see that $M=M\cap X$ is meager in $X$
(see [vM2, Exercise A.13.7]). Since $U\cap X\subseteq M$, this contradicts the
assumption that $X$ is a Baire space. ∎

##  3\. Every zero-dimensional analytic space is $\sigma$-homogeneous

The following theorem combines [vE2, Theorems 4.1(a) and 4.3(a)]. Similar (if
not identical) results were independently obtained by Ostrovsky (see [Os1])
and Medvedev (see [Mv1] and [Mv2]).

######  Theorem 3.1 (van Engelen).

Let $X$ be a zero-dimensional space that satisfies the following conditions:

  * •

Every non-empty clopen subspace of $X$ contains a closed subspace homeomorphic
to $X$,

  * •

$X$ is either a meager space or it has a Polish dense subspace.

Then $X$ is strongly homogeneous.

Another result in the same family was obtained by Steel (see [St, Theorem 2]),
by making use of an ingenious lemma due to Harrington. We will not apply
Steel’s theorem, but instead we will combine Theorem 3.1 with Harrington’s
lemma. This will make it easier to see that, for our purposes, no determinacy
assumptions are necessary. This strategy was first employed by Michalewski in
[Mi]. Before stating this lemma, we will need some preliminaries.

Given $i\in 2$, set

$Q_{i}=\\{x\in 2^{\omega}:x(n)=i\text{ for all but finitely many
}n\in\omega\\}.$

Notice that every element of $2^{\omega}\setminus(Q_{0}\cup Q_{1})$ is
obtained by alternating finite blocks of zeros and finite blocks of ones.
Define the function $\phi:2^{\omega}\setminus(Q_{0}\cup Q_{1})\longrightarrow
2^{\omega}$ by setting

$\phi(x)(n)=\left\\{\begin{array}[]{ll}0&\textrm{if the $n^{\text{th}}$ block
of zeros of $x$ has even length},\\\ 1&\textrm{otherwise,}\end{array}\right.$

where we start counting with the $0^{\text{th}}$ block of zeros. Notice that
$\phi$ is continuous. A Wadge class $\mathbf{\Gamma}$ in $2^{\omega}$ is
_reasonably closed_ if $\phi^{-1}[A]\cup Q_{0}\in\mathbf{\Gamma}$ for every
$A\in\mathbf{\Gamma}$. For the purposes of this article, the only fact
concerning these classes that we will need is that
$\mathbf{\Sigma}^{1}_{1}(2^{\omega})$ and $\mathbf{\Pi}^{1}_{1}(2^{\omega})$
are reasonably closed Wadge classes.

The original statement of Lemma 3.2 (see [St, Lemma 3]) asks that
$B\in\mathbf{\Gamma}\setminus\widecheck{\mathbf{\Gamma}}$ and that suitable
determinacy assumptions hold, from which $\mathbf{\Gamma}=B\\!\\!\downarrow\,$
follows by Wadge Lemma (see [CMM2, Lemma 4.4]). However, it is clear from the
proof that this is the only place where the determinacy assumptions are used.
Therefore, it is possible to give the following determinacy-free version of
this result.

######  Lemma 3.2 (Harrington).

Let $\mathbf{\Gamma}$ be a reasonably closed Wadge class in $2^{\omega}$, and
let $A,B\subseteq 2^{\omega}$. Assume that $A\leq_{\mathsf{W}}B$ in
$2^{\omega}$ and $\mathbf{\Gamma}=B\\!\\!\downarrow\,$. Then there exists an
injective continuous function $f:2^{\omega}\longrightarrow 2^{\omega}$ such
that $A=f^{-1}[B]$.

Now we have all the tools needed to prove the zero-dimensional version of our
main result (see Theorem 3.4). In fact, the core of our argument is contained
in the proof of the following lemma. As in [vE1], we will say that a space $X$
is _nowhere countable_ if $X$ is non-empty and no non-empty open subset of $X$
is countable.

######  Lemma 3.3.

Let $X$ be a zero-dimensional analytic space. Assume that $X$ is nowhere
countable, and that $X$ is either a meager space or a Baire space. Then $X$ is
$\sigma$-homogeneous with analytic witnesses.

###### Proof.

Let $\\{U_{n}:n\in\omega\\}$ be a base for $X$ consisting of non-empty clopen
sets. Using the fact that analytic sets have the perfect set property,
together with the assumption that $X$ is nowhere countable, it is a simple
exercise to obtain $K_{n,i}$ for $(n,i)\in\omega\times 2$ satisfying the
following conditions:

  * •

Each $K_{n,i}\approx 2^{\omega}$,

  * •

Each $K_{n,i}$ is nowhere dense in $X$,

  * •

Each $K_{n,i}\subseteq U_{n}$,

  * •

$K_{n,i}\cap K_{m,j}=\varnothing$ whenever $(n,i)\neq(m,j)$.

By [Ke, Theorem 26.1], we can fix $A\subseteq 2^{\omega}$ such that
$A\\!\\!\downarrow\,=\mathbf{\Sigma}^{1}_{1}(2^{\omega})$. Fix homeomorphisms
$h_{n,i}:2^{\omega}\longrightarrow K_{n,i}$ for $(n,i)\in\omega\times 2$, then
set $A_{n,i}=h_{n,i}[A]$. Define

$X_{i}=\left(X\setminus\bigcup_{n\in\omega}K_{n,i}\right)\cup\left(\bigcup_{n\in\omega}A_{n,i}\right)$

for $i\in 2$. It is clear that each $X_{i}$ is analytic, and that $X=X_{0}\cup
X_{1}$. Therefore, it remains to show that each $X_{i}$ is homogeneous.

So fix $i\in 2$. First observe that $X_{i}$ is dense in $X$. In particular, if
$X$ is a meager space, then $X_{i}$ is also a meager space (see [vM2, Exercise
A.13.7]). On the other hand, if $X$ is a Baire space then $X$ has a dense
Polish subspace by Lemma 2.1, hence the same is true of $X_{i}$, because the
$K_{n,i}$ are closed nowhere dense in $X$ (see [Ke, Theorem 3.11]).

By Theorem 3.1 plus the previous paragraph, it will be enough to show that
each $U_{n}\cap X_{i}$ contains a closed subspace homeomorphic to $X_{i}$. Let
$X_{i}^{\prime}$ be a subspace of $2^{\omega}$ such that
$X_{i}^{\prime}\approx X_{i}$, and observe that
$X_{i}^{\prime}\in\mathbf{\Sigma}^{1}_{1}(2^{\omega})$. It follows from our
choice of $A$ that $X_{i}^{\prime}\leq_{\mathsf{W}}A$. Therefore, by Lemma
3.2, this is witnessed by a continuous injection $f:2^{\omega}\longrightarrow
2^{\omega}$. Using the compactness of $2^{\omega}$, one sees that
$f[X_{i}^{\prime}]$ is a closed subspace of $A$ homeomorphic to $X_{i}$. So
$h_{n,i}[f[X_{i}^{\prime}]]$ is a closed subspace of $A_{n,i}$ homeomorphic to
$X_{i}$ for each $n$. The observation that each $A_{n,i}=K_{n,i}\cap X_{i}$ is
closed in $X_{i}$ concludes the proof. ∎

######  Theorem 3.4.

Let $X$ be a zero-dimensional analytic space. Then $X$ is $\sigma$-homogeneous
with analytic witnesses.

###### Proof.

Set

$\mathcal{U}=\\{U:U\text{ is a countable open subset of }X\\},$

and observe that $C=\bigcup\mathcal{U}$ is a countable open subset of $X$. Set
$D=X\setminus C$. It is easy to check that either $D$ is empty or it is
nowhere countable. Therefore, we will assume without loss of generality that
$X$ is nowhere countable.

Now set

$\mathcal{V}=\\{V:V\text{ is an open meager subset of }X\\},$

and observe that $M=\bigcup\mathcal{V}$ is an open meager subset of $X$. It
follows that $M$ is a meager space, and that $M$ is either empty or nowhere
countable. Set $B=X\setminus M$. It is easy to check that $B$ is a Baire
space, and that $B$ is either empty or nowhere countable. At this point, the
desired result clearly follows from Lemma 3.3. ∎

By carefully considering the above proofs, one sees that every nowhere
countable zero-dimensional analytic space can be written as the union of four
homogeneous spaces (two for the meager part, and two for the Baire part).
Therefore, one might wonder whether ever zero-dimensional analytic space can
be written as the union of finitely many homogeneous spaces. In Section 7,
among other things, we will show that this is not the case (see Corollary
7.2).

##  4\. Pairwise disjoint witnesses

The purpose of this section is to observe that a minor modification of the
arguments given in Section 3 yields $\sigma$-homogeneity with pairwise
disjoint witnesses. The price to pay for this improvement is an increase in
the complexity of the witnesses.

######  Lemma 4.1.

Let $X$ be a zero-dimensional analytic space. Assume that $X$ is nowhere
countable and that $X$ is either a meager space or a Baire space. Then $X$ is
$\sigma$-homogeneous with pairwise disjoint $\mathbf{\Delta}^{1}_{2}$
witnesses.

###### Proof.

We will use the same notation as in the proof of Lemma 3.3. In addition, set
$B=2^{\omega}\setminus A$ and $B_{n,0}=h_{n,0}[B]$ for $n\in\omega$. Observe
that

$X\setminus X_{0}=\bigcup_{n\in\omega}B_{n,0}$

is a coanalytic space. Therefore, it will be enough to show that $X\setminus
X_{0}$ is homogeneous. This can be achieved as in the proof of Lemma 3.3, by
applying Lemma 3.2 to the reasonably closed Wadge class
$\widecheck{\mathbf{\Gamma}}=\mathbf{\Pi}^{1}_{1}(2^{\omega})$. ∎

######  Theorem 4.2.

Let $X$ be a zero-dimensional analytic space. Then $X$ is $\sigma$-homogeneous
with pairwise disjoint $\mathbf{\Delta}^{1}_{2}$ witnesses.

###### Proof.

Proceed as in the proof of Theorem 3.4, but apply Lemma 4.1 instead of Lemma
3.3. ∎

##  5\. Optimality

In this section, we will point out that the complexity of the witnesses to
$\sigma$-homogeneity obtained in Section 3 is as low as possible. In the case
of Theorem 3.4, this follows from Theorem 5.1. In the case of Theorem 4.2,
this follows from Corollary 5.2.

A weaker version of the following result is stated as part of [MV, Theorem
12.2]. However, it is easy to realize that the exact same proof (see [MV,
Lemma 12.1]) actually yields the stronger result stated here.

######  Theorem 5.1 (Medini, Vidnyánszky).

Assume $\mathsf{V=L}$. Then there exists a zero-dimensional analytic space
that is not $\sigma$-homogeneous with coanalytic witnesses.

Given a space $X=\bigcup_{n\in\omega}X_{n}$, where the $X_{n}$ are pairwise
disjoint, it is clear that $X_{n}\in\mathbf{\Sigma}^{1}_{1}(X)$ for each $n$
iff $X_{n}\in\mathbf{\Pi}^{1}_{1}(X)$ for each $n$. Combining this observation
with Theorem 5.1 immediately yields the following result.

######  Corollary 5.2.

Assume $\mathsf{V=L}$. Then there exists a zero-dimensional analytic space $X$
with the following properties:

  * •

$X$ is not $\sigma$-homogeneous with pairwise disjoint analytic witnesses,

  * •

$X$ is not $\sigma$-homogeneous with pairwise disjoint coanalytic witnesses.

##  6\. Higher dimensions

In this section, we will see that the results obtained so far, together with
those of [MV], yield a complete picture of $\sigma$-homogeneity in the finite-
dimensional realm. We begin by extending Theorems 3.4 and 4.2. The proof is
essentially due to Ostrovsky, who applied the same argument in the Borel
context (see [Os2, page 663]). However, we do not know whether the assumption
“finite-dimensional” can be dropped in the following theorem (see Question
8.1).

######  Theorem 6.1.

Let $X$ be a finite-dimensional analytic space. Then:

  * •

$X$ is $\sigma$-homogeneous with analytic witnesses,

  * •

$X$ is $\sigma$-homogeneous with pairwise disjoint $\mathbf{\Delta}^{1}_{2}$
witnesses.

###### Proof.

It is a fundamental result of dimension theory that every finite-dimensional
space is homeomorphic to a subspace of $\mathbb{R}^{n}$ for some $n\in\omega$
(see for example [vM2, Theorem 3.3.5]). Since $\mathbb{Q}$ and
$\mathbb{R}\setminus\mathbb{Q}$ are both zero-dimensional and analytic, it
follows that $X$ can be written as $X=\bigcup_{k\in n}X_{k}$ for some
$n\in\omega$, where the $X_{k}$ are zero-dimensional, analytic, and pairwise
disjoint. The desired results now follow from Theorems 3.4 and 4.2. ∎

Similarly, one can extend Theorems 1.1 and 1.2 as follows.

######  Theorem 6.2 (Ostrovsky).

Every finite-dimensional Borel space is $\sigma$-homogeneous with pairwise
disjoint $\mathsf{G}_{\delta}$ witnesses.

######  Theorem 6.3.

Assume $\mathsf{AD}$. Then every finite-dimensional space is
$\sigma$-homogeneous with pairwise disjoint $\mathsf{G}_{\delta}$ witnesses.

Given that Theorems 1.1 and 1.2 give closed witnesses, it is natural to wonder
whether “$\mathsf{G}_{\delta}$” can be improved to “closed” in the above two
theorems. The following proposition shows that this is not the case, and that
$\mathsf{G}_{\delta}$ is the optimal complexity, even dropping the requirement
that the witnesses are pairwise disjoint.

######  Proposition 6.4.

There exists a one-dimensional compact space that is not $\sigma$-homogeneous
with $\mathsf{F}_{\sigma}$ witnesses.

###### Proof.

Let $P$ be the “propeller space” constructed by de Groot and Wille in [dGW,
Section 2].222 For our purposes, we could have chosen every propeller to be a
two-bladed propeller. Assume that $P=\bigcup_{n\in\omega}P_{n}$, where each
$P_{n}\in\mathbf{\Sigma}^{0}_{2}(P)$. We will show that some $P_{n}$ is not
homogeneous. Pick closed subsets $P_{n,k}$ of $P$ for
$(n,k)\in\omega\times\omega$ such that each
$P_{n}=\bigcup_{k\in\omega}P_{n,k}$. Since $P$ is a Baire space, we can fix
$(n,k)\in\omega\times\omega$ and a non-empty open subset $U$ of $P$ such that
$U\subseteq P_{n,k}$. Using the same notation as [dGW], one key property of
$P$ is that if $p\in P\setminus\\{a^{\prime}_{i}:1\leq i<\omega\\}$, then for
every neighborhood $V$ of $p$ in $P$ there exists a neighborhood
$V^{\prime}\subseteq V$ of $p$ in $P$ such that $V^{\prime}\setminus\\{p\\}$
is connected. On the other hand, no $a^{\prime}_{i}$ has this property. Now
fix $p\in U\setminus\\{a^{\prime}_{i}:1\leq i<\omega\\}$ and
$a^{\prime}_{j}\in U$. Since $U$ is also open in $P_{n}$, it is clear that
there can be no homeomorphism $h:P_{n}\longrightarrow P_{n}$ such that
$h(p)=a^{\prime}_{j}$. This shows that $P_{n}$ is not homogeneous, as desired.
∎

##  7\. Finite unions

The purpose of this section is to distinguish $\sigma$-homogeneity from its
finite analogues, and distinguish these finite analogues among themselves (see
Corollary 7.2). Given $n$ such that $1\leq n\leq\omega$, we will say that a
space $X$ is _$n$ -homogeneous_ if $X=\bigcup_{k\in n}X_{k}$, where each
$X_{k}$ is a homogeneous subspace of $X$. Similarly, we will say that a space
$X$ is _$n$ -discrete_ if $X=\bigcup_{k\in n}X_{k}$, where each $X_{k}$ is a
discrete subspace of $X$.

######  Theorem 7.1.

Let $1\leq n\leq\omega$, and let $X$ be a scattered space of Cantor-Bendixson
rank $n$. Then $X$ is $n$-discrete but not $\ell$-discrete for any $\ell<n$.

###### Proof.

Set $X_{k}=X^{(k)}\setminus X^{(k+1)}$ for $k<n$, and observe that
$\\{X_{k}:k\in n\\}$ is a partition of $X$ consisting of discrete spaces. So
it remains to show that $X$ is not $\ell$-discrete for any $\ell<n$. First
assume that $n<\omega$.

Given $k\in\omega$, we will denote by $\omega^{<k\uparrow}$ (respectively
$\omega^{k\uparrow}$) the collection of all strictly increasing sequences of
elements of $\omega$ of length less than $k$ (respectively of length exactly
$k$). When $k=0$, we declare $\langle
s_{0},\ldots,s_{k-1}\rangle=\\{s_{0},\ldots,s_{k-1}\\}=\varnothing$ and
$s_{-1}=-1$. By starting with an arbitrary $f(\varnothing)\in X_{n-1}$ and
continuing in the obvious way, one can construct a function
$f:\omega^{<n\uparrow}\longrightarrow X$ satisfying the following conditions:

  1. (1)

$f[\omega^{k\uparrow}]\subseteq X_{n-k-1}$ for every $k<n$,

  2. (2)

$\langle f(s^{\frown}i):s_{k-1}<i<\omega\rangle$ converges to $f(s)$ for all
$s=\langle s_{0},\ldots,s_{k-1}\rangle\in\omega^{<n\uparrow}$.

Now assume, in order to get a contradiction, that $X$ is $\ell$-discrete for
some $\ell<n$. Write $X=\bigcup_{i\in\ell}D_{i}$, where the $D_{i}$ are
pairwise disjoint discrete subspaces of $X$. Define the coloring
$c:[\omega]^{<n}\longrightarrow\ell$ by setting

$c(\\{s_{0},\ldots,s_{k-1}\\})=i\text{ iff }f(\langle
s_{0},\ldots,s_{k-1}\rangle)\in D_{i},$

where we are assuming that $s_{0}<\cdots<s_{k-1}$. By repeatedly applying the
classical Ramsey Theorem (see for example [Je, Theorem 9.1]), one obtains sets
$H_{k}$ for $k<n$ satisfying the following conditions:

  * •

$\omega=H_{0}\supseteq\cdots\supseteq H_{n-1}$,

  * •

$|c[[H_{k}]^{k}]|=1$ for each $k$.

Denote by $i_{k}$ the unique element of $c[[H_{k}]^{k}]$ for $k<n$. By the
Pigeonhole Principle, we can fix $i<\ell$ and $j,k<n$ such that $j<k$ and
$i_{j}=i_{k}=i$. Also fix $s_{0},\ldots,s_{j-1}\in H_{k}\subseteq H_{j}$ such
that $s_{0}<\cdots<s_{j-1}$, and set $s=\langle s_{0},\ldots,s_{j-1}\rangle$.
It is clear that $f(s)\in D_{i}$. Now set

$S=\\{f(s^{\frown}\langle s_{j},\ldots,s_{k-1}\rangle):s_{j},\ldots,s_{k-1}\in
H_{k}\text{ and }s_{j-1}<s_{j}<\cdots<s_{k-1}\\},$

and observe that $S\subseteq D_{i}$. Using condition (2), it is easy to
realize that $f(s)$ belongs to the closure of $S$ in $X$. Furthermore,
condition (1) guarantees that $f(s)\notin S$. This contradicts the assumption
that $D_{i}$ is discrete.

Finally, assume that $n=\omega$. If $X$ were $\ell$-discrete for some
$\ell<n$, then so would be $X\setminus X^{(\ell+1)}$. But this would
contradict the first part of this proof, since $X\setminus X^{(\ell+1)}$ is a
scattered space of Cantor-Bendixson rank $\ell+1$. ∎

######  Corollary 7.2.

Let $1\leq n\leq\omega$. Then there exists a zero-dimensional scattered space
that is $n$-homogeneous but not $\ell$-homogeneous for any $\ell<n$.

###### Proof.

Consider the ordinal $\omega^{n}$ with the order topology. It is not hard to
check that this space is scattered of Cantor-Bendixson rank $n$. The desired
result then follows from Theorem 7.1 plus the observation that a scattered
space is homogeneous iff it is discrete. ∎

##  8\. Open questions

The most pressing open question is the following. We remark that we would not
know the answer even if we substituted “analytic” with much more restrictive
conditions, such as “compact”.

######  Question 8.1.

Is every analytic space $\sigma$-homogeneous?

Recall that a space is _countable-dimensional_ if it can be written as a
countable union of zero-dimensional spaces. Notice that, by the
$\mathsf{G}_{\delta}$-Enlargement Theorem of dimension theory (see for example
[vM2, Corollary 3.3.12]), these zero-dimensional subspaces can be chosen to be
$\mathsf{G}_{\delta}$. Using this observation, it is straightforward to see
that Theorem 6.1 holds for all countable-dimensional spaces. The same is true
of Theorems 6.2 and 6.3, except that we do not know whether one can still
obtain pairwise disjoint witnesses (without increasing their complexity beyond
$\mathsf{G}_{\delta}$).

######  Question 8.2.

Do Theorems 6.2 and 6.3 hold for all countable-dimensional spaces?

Given the results of Section 4, it seems fitting to mention that the following
question is still open (see [MV, Question 13.5]).

######  Question 8.3 (Medini, Vidnyánszky).

In $\mathsf{ZFC}$, is there a zero-dimensional $\sigma$-homogeneous space that
is not $\sigma$-homogeneous with pairwise disjoint witnesses? At least under
additional set-theoretic assumptions?

Finally, we remark that the $\sigma$-homogeneity of the zero-dimensional
spaces considered in this article (as well as in [MV] and [Os2]) can always be
witnessed by _strongly_ homogeneous subspaces. This observation suggests the
following question.

######  Question 8.4.

In $\mathsf{ZFC}$, is there a zero-dimensional $\sigma$-homogeneous (or even
homogeneous) space that is not $\sigma$-homogeneous with strongly homogeneous
witnesses? At least under additional set-theoretic assumptions?

## References

  * [AvM1] A. V. Arhangel’skii, J. van Mill. Topological homogeneity. In: _Recent Progress in General Topology III._ Atlantis Press, 2014. 1–68.
  * [AvM2] A. V. Arhangel’skii, J. van Mill. Splitting Tychonoff cubes into homeomorphic and homogeneous parts. _Topology Appl._ 275 (2020), 107018.
  * [AvM3] A. V. Arhangel’skii, J. van Mill. Covering Tychonoff cubes by topological groups. _Topology Appl._ 281 (2020), 107189.
  * [CMM1] R. Carroy, A. Medini, S. Müller. Every zero-dimensional homogeneous space is strongly homogeneous under determinacy. _J. Math. Log._ 20:3 (2020), 2050015.
  * [CMM2] R. Carroy, A. Medini, S. Müller. Constructing Wadge classes. _Bull. Symb. Log._ 28:2 (2022), 207–-257.
  * [vE1] A. J. M. van Engelen. _Homogeneous zero-dimensional absolute Borel sets._ CWI Tract, 27. Stichting Mathematisch Centrum, Centrum voor Wiskunde en Informatica, Amsterdam, 1986. Available at `http://repository.cwi.nl/`.
  * [vE2] F. van Engelen. On the homogeneity of infinite products. _Topology Proc._ 17 (1992), 303–315.
  * [vEvM] F. van Engelen, J. van Mill. Decompositions of rigid spaces. _Proc. Am. Math. Soc._ 89:3 (1983), 533–536.
  * [En] R. Engelking. _General topology._ Revised and completed edition. Sigma Series in Pure Mathematics, vol. 6. Heldermann Verlag, Berlin, 1989.
  * [dGW] J. de Groot, R. J. Wille. Rigid continua and topological group-pictures. _Arch. Math._ 9 (1958), 441–446.
  * [Je] T. Jech. _Set theory._ The third millennium edition, revised and expanded. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2003.
  * [Ke] A. S. Kechris. _Classical descriptive set theory._ Graduate Texts in Mathematics, 156. Springer-Verlag, New York, 1995.
  * [Me1] A. Medini. Products and h-homogeneity. _Topology Appl._ 158:18 (2011), 2520–2527.
  * [Me2] A. Medini. _The topology of ultrafilters as subspaces of the Cantor set and other topics._ Ph.D. Thesis. University of Wisconsin - Madison. ProQuest LLC, Ann Arbor, MI, 2013. Available at `http://www.math.wisc.edu/~lempp/theses/medini.pdf`.
  * [MV] A. Medini, Z. Vidnyánszky. Zero-dimensional $\sigma$-homogeneous spaces. _Ann. Pure Appl. Logic._ 175:1 (2024), 103331.
  * [MZ] A. Medini, L. Zdomskyy. Between Polish and completely Baire. _Arch. Math. Logic._ 54:1-2 (2015), 231–245.
  * [Mv1] S. V. Medvedev. On properties of h-homogeneous spaces of first category. _Topology Appl._ 157:18 (2010), 2819–2828.
  * [Mv2] S. V. Medvedev. On properties of h-homogeneous spaces with the Baire property. _Topology Appl._ 159:3 (2012), 679–694.
  * [Mi] H. Michalewski. Homogeneity of $\mathcal{K}(\mathbf{Q})$. _Tsukuba J. Math._ 24:2 (2000), 297–302.
  * [vM1] J. van Mill. Homogeneous subsets of the real line. _Compositio Math._ 46:1 (1982), 3–13.
  * [vM2] J. van Mill. _The infinite-dimensional topology of function spaces._ North-Holland Mathematical Library, 64. North-Holland Publishing Co., Amsterdam, 2001.
  * [Os1] A. V. Ostrovskiĭ. On a question of L. V. Keldysh on the structure of Borel sets. _Mat. Sb. (N.S.)_ 131(173):3 (1986), 323–346, 414 (in Russian); English translation in: _Math. USSR-Sb._ 59:2 (1988), 317–337.
  * [Os2] A. Ostrovsky. $\sigma$-homogeneity of Borel sets. _Arch. Math. Logic._ 50:5-6 (2011), 661–664.
  * [St] J. R. Steel. Analytic sets and Borel isomorphisms. _Fund. Math._ 108:2 (1980), 83–88.