this section, the morphism $q_{E}\colon G/E\to G/K$ is equivalent to the
quotient morphism $G/E\to(G/E)/(K/E)$ and its kernel
$\ker(q_{E})=p_{E}(K)\cong K/E$ is therefore totally disconnected.

First assume that $p_{E}\mathcal{C}\in\mathop{\rm{\bf B}_{\mathcal{F}}}(G/E)$
and denote by $\eta$ the transfer function of $p_{E}\mathcal{C}$ with
$\eta(z)=e$. Then $\mathop{\rm
co}(\xi)=p_{K}\mathcal{C}=q_{E}p_{E}\mathcal{C}=\mathop{\rm co}(q_{E}\eta)$.
Since the base flow $\mathcal{F}$ is minimal and $q_{E}\eta(z)=e=\xi(z)$, it
follows that $q_{E}\eta=\xi$ on the whole $X$. This shows that (a) implies
(b).

Now assume that $\eta\colon X\to G/E$ is continuous and satisfies
$q_{E}\eta=\xi$. Then $q_{E}\mathop{\rm co}(\eta)=\mathop{\rm
co}(\xi)=p_{K}\mathcal{C}=q_{E}(p_{E}\mathcal{C})$ and hence
$(p_{E}\mathcal{C})^{-1}\mathop{\rm co}(\eta)\in\mathop{\rm{\bf
Z}_{\mathcal{F}}}(G/E)$ takes its values in a totally disconnected subgroup
$\ker(q_{E})$ of $G/E$. Now let $\chi\in(\ker(q_{E}))^{*}$. Since
$\ker(q_{E})$ is (compact and) totally disconnected, $\chi$ is a torsion
element of $(\ker(q_{E}))^{*}$ and hence
$\chi((p_{E}\mathcal{C})^{-1}\mathop{\rm co}(\eta))$ is a torsion element of
$\mathop{\rm{\bf Z}_{\mathcal{F}}}$. By Theorem 4.10 it follows that
$\chi((p_{E}\mathcal{C})^{-1}\mathop{\rm co}(\eta))=1$. As the latter identity
holds for every character $\chi$ of $\ker(q_{E})$, we conclude with
$p_{E}\mathcal{C}=\mathop{\rm co}(\eta)\in\mathop{\rm{\bf
B}_{\mathcal{F}}}(G/E)$. This shows that (b) implies (a). ∎

######  Proposition 5.3.

Let $\mathop{\rm\mathcal{F}\colon\Gamma\curvearrowright X}$ be a minimal flow,
$G\in\mathsf{CAGp}$ and $\mathcal{C}\in\mathop{\rm{\bf Z}_{\mathcal{F}}}(G)$.
Fix $z\in X$. Assume that $K\mathrel{\lhd}G$ is totally disconnected with
$p_{K}\mathcal{C}\in\mathop{\rm{\bf B}_{\mathcal{F}}}(G/K)$ (that is, with
$F(\mathcal{C})\subseteq K$), and denote by $\xi$ the transfer function of
$p_{K}\mathcal{C}$ with $\xi(z)=e$. If $X$ is connected and $\Gamma$ has no
non-trivial finite abelian quotient groups then the following statements hold:

  1. (i)

$F(\mathcal{C})$ equals the smallest of all the closed subgroups $E$ of $K$,
for which $\xi$ lifts across $q_{E}$ to a continuous map $\eta\colon X\to
G/E$,

  2. (ii)

$F(\mathcal{C})$ equals the smallest of all the closed subgroups $E$ of $K$,
for which the morphism $\xi^{*}\colon(G/K)^{*}\to C_{z}(X,\mathbb{T}^{1})$
extends through $(q_{E})^{*}\colon(G/K)^{*}\to(G/E)^{*}$ to a morphism
$\sigma\colon(G/E)^{*}\to C_{z}(X,\mathbb{T}^{1})$.

######  Remark 5.4.

We shall have an occasion to use the proposition also in a special case when
$K$ is finite. In such case every subgroup $E$ of $K$ is finite and hence
closed, and so the word “closed” becomes redundant in the reading of both (i)
and (ii).

###### Proof of Proposition 5.3.

We verify statement (i) by showing that the following statements are
equivalent for every closed subgroup $E$ of $K$:

  1. ($\alpha$)

$F(\mathcal{C})\subseteq E$,

  2. ($\beta$)

$\xi$ lifts across $q_{E}$ to a continuous map $\eta\colon X\to G/E$.

So fix a closed subgroup $E$ of $K$ and assume that $F(\mathcal{C})\subseteq
E$. Then, since $p_{F(\mathcal{C})}\mathcal{C}\in\mathop{\rm{\bf
B}_{\mathcal{F}}}(G/F(\mathcal{C}))$, it follows that
$p_{E}\mathcal{C}\in\mathop{\rm{\bf B}_{\mathcal{F}}}(G/E)$ and the existence
of a map $\eta$ as in ($\beta$) follows from Lemma 5.1. This verifies that
($\beta$) follows from ($\alpha$). Conversely, assume that $\xi$ lifts across
$q_{E}$ to a continuous map $\eta\colon X\to G/E$. Then, by Lemma 5.1,
$p_{E}\mathcal{C}\in\mathop{\rm{\bf B}_{\mathcal{F}}}(G/E)$ and so
$e=F(p_{E}\mathcal{C})=p_{E}F(\mathcal{C})$. Thus, $F(\mathcal{C})\subseteq
E$, which shows that ($\alpha$) follows from ($\beta$).

Statement (ii) follows from statement (i) by virtue of the following
observations:

  * •

if $\xi$ lifts across $q_{E}$ to a continuous map $X\to G/E$ then it lifts
also to a continuous base point preserving map $X\to G/E$; this has already
been observed in Remark 5.2,

  * •

the continuous base point preserving maps $X\to G/E$ are in a one-to-one
correspondence with the morphisms $(G/E)^{*}\to C_{z}(X,\mathbb{T}^{1})$, the
correspondence assigning the morphism $\eta^{*}$ to every $\eta\in
C_{z}(X,G/E)$; this follows from our discussion in Subsection 2.4.7, see
formula (2.14),

  * •

a map $\eta\in C_{z}(X,G/E)$ satisfies $q_{E}\eta=\xi$ if and only if the
morphism $\eta^{*}$ satisfies $\eta^{*}(q_{E})^{*}=\xi^{*}$; this follows from
the fact that the characters of $G/K$ separate the points of $G/K$.

∎

###  5.2 Relation to the monodromy action

Our aim in this section is to describe a relationship between the functor $F$
and the functor $\pi_{1}$. The situation that we shall consider is as follows.
Assume that $\mathop{\rm\mathcal{F}\colon\Gamma\curvearrowright X}$ is a
minimal flow and $\mathcal{C}\in\mathop{\rm\mathsf{CAGpZ}_{\mathcal{F}}}$ is
an extension of $\mathcal{F}$ taking its values in a group
$G\in\mathsf{CAGp}$. Suppose that $K$ is a closed totally disconnected
subgroup of $G$ with $F(\mathcal{C})\subseteq K$ or, equivalently, with
$p_{K}\mathcal{C}\in\mathop{\rm{\bf B}_{\mathcal{F}}}(G/K)$. Let $\xi$ be the
base point preserving transfer function for $p_{K}\mathcal{C}$. Is it possible
to express the group $F(\mathcal{C})$ in terms of the map $\xi$ or in terms of
the morphism $\xi^{\sharp}\colon\pi_{1}(X)\to\pi_{1}(G/K)$ induced by $\xi$?
We show that the answer to the proposed question is yes, provided $\Gamma$ is
a connected Lie group and $X$ is a compact (connected) manifold; in Theorem
5.14 we show that the group $F(\mathcal{C})$ equals the orbit closure of the
identity $e$ under the monodromy action of $\xi^{\sharp}\pi_{1}(X)$ on $K$.
Before proving this theorem, we study epimorphisms in $\mathsf{CAGp}$ with
totally disconnected kernels. We show that they have properties analogous to
those of the covering morphisms; in particular, a monodromy action of
$\pi_{1}(G/K)$ on $K$ is defined in the situation described above. After
gaining more information about the monodromy action in this setting, we prove
the mentioned Theorem 5.14, which is our main result in this section.

######  Lemma 5.5.

Let $G\in\mathsf{CAGp}$, $K$ be a closed totally disconnected subgroup of $G$
and $p_{K}\colon G\to G/K$ be the quotient morphism. Let $X\in\mathsf{CLAC}$
be a pointed space with the base point $z$ and $\xi\colon X\to G/K$ be a
continuous base point preserving map. Then the following conditions are
equivalent:

  1. (a)

$\xi$ lifts across $p_{K}$ to a continuous base point preserving map
$\zeta\colon X\to G$,

  2. (b)

$\xi^{\sharp}\pi_{1}(X)\subseteq p_{K}^{\sharp}\pi_{1}(G)$.

######  Remark 5.6.

Notice the following facts.

  * •

The map $\zeta$ from statement (a) is obviously unique, for the space $X$ is
connected and the kernel $K$ of $p_{K}$ is totally disconnected by the
assumptions of the lemma.

  * •

The statement of the lemma in case of a finite group $K$ follows at once from
the well known and standard results from algebraic topology, for in such case
the morphism $p_{K}$ is a covering map (see, e.g., [36, Proposition 1.33, p.
61]). In the situation when $K$ is infinite the morphism $p_{K}$ is no longer
a covering map and so the usual arguments from algebraic topology do not
apply.

  * •

From our proof of the lemma it will follow that the paths in $G/K$ do lift
across $p_{K}$ as well as do the homotopies of paths. Thus, if two loops
$f_{1},f_{2}$ in $G/K$ based at $e$ are path-homotopic, then their lifts
$\widetilde{f}_{1},\widetilde{f}_{2}$ across $p_{K}$ starting at $e$ satisfy
$\widetilde{f}_{1}(1)=\widetilde{f}_{2}(1)$. Consequently, there is a
monodromy action of the group $\pi_{1}(G/K)$ on the $p_{K}$-fibre $K$ of $e$,
just as in the case when $p_{K}$ is a covering map. The action is given by the
rule $\pi_{1}(G/K)\times K\ni(f,k)\mapsto k\widetilde{f}(1)\in K$, where
$\widetilde{f}$ is the lift of $f$ across $p_{K}$ starting at $e$.

###### Proof of Lemma 5.5.

We divide the proof into four steps. Before turning into it we remind the
reader that the base point for the fundamental group of a topological group
$G$ is always the identity $e$ of $G$. We also recall that the operation of
composition of paths is denoted by $*$, the reverse of a path $f$ by
$\overline{f}$ and the relation of homotopy of paths by $\simeq_{p}$.

_1st step._ We fix some notation that shall be used in the rest of the proof.

Since $K$ is a totally disconnected compact abelian group, it is a projective
limit of finite abelian groups. Hence there is a filter base $(H_{j})_{j\in
J}$ in $K$ formed by closed subgroups of $K$ with $\bigcap_{j\in J}H_{j}=e$
and with $K/H_{j}$ finite for every $j$. For $j,l\in J$ write $j\leq l$ when
(and only when) $H_{l}\subseteq H_{j}$. The quotient morphism $G\to G/H_{j}$
will be denoted by $p_{j}$ for every $j\in J$. Given $j,l\in J$ with $j\leq
l$, the inclusion $H_{l}\subseteq H_{j}$ yields the existence of a (unique)
quotient morphism $p_{jl}\colon G/H_{l}\to G/H_{j}$ with $p_{jl}p_{l}=p_{j}$.
The epimorphisms $p_{jl}$ ($j\leq l$) constitute an inverse system in
$\mathsf{CAGp}$ indexed by the directed set $J$. As follows from the equality
$\bigcap_{j\in J}H_{j}=e$, the inverse limit of this system is the group $G$
along with the morphisms $p_{j}$ ($j\in J$) as the limit projections. Finally,
for every $j\in J$, the inclusion $H_{j}\subseteq K$ yields the existence of a
(unique) quotient morphism $r_{j}\colon G/H_{j}\to G/K$ with
$r_{j}p_{j}=p_{K}$. An elementary argument shows that $r_{j}p_{jl}=r_{l}$ for
all $j\leq l$.

_2nd step._ Let $f\colon[0,1]\to G/K$ be a path starting at the identity $e$
of $G/K$. We show that $f$ lifts uniquely across $p_{K}$ to a path
$\widetilde{f}\colon[0,1]\to G$ starting at the identity $e$ of $G$.

The uniqueness part follows at once from the connectedness of $[0,1]$ and from
the fact that the kernel $K$ of $p_{K}$ is totally disconnected. To prove the
existence part we use an inverse limit argument. Given $j\in J$, the morphism
$r_{j}\colon G/H_{j}\to G/K$ is a covering morphism, for its kernel
$\ker(r_{j})=p_{j}(K)\cong K/H_{j}$ is finite. Consequently, $f$ lifts across
$r_{j}$ to a path $f_{j}$ in $G/H_{j}$ starting at $e$. If $j\leq l$ then
$p_{jl}f_{l}$ is a path in $G/H_{j}$ starting at $e$ and projecting by $r_{j}$
to $f$, and hence $p_{jl}f_{l}=f_{j}$ by uniqueness of $f_{j}$. Thus, the
paths $f_{j}$ ($j\in J$) form a system of maps compatible with the inverse
system of groups $G/H_{j}\stackrel{{\scriptstyle
p_{jl}}}{{\longleftarrow}}G/H_{l}$ and hence they give rise to a continuous
map $\widetilde{f}\colon[0,1]\to G$ with $p_{j}\widetilde{f}=f_{j}$ for every
$j\in J$. Clearly, $\widetilde{f}(0)=e$ and so it remains to show that
$\widetilde{f}$ is a lift of $f$ across $p_{K}$. This is immediate, for if
$j\in J$ then $p_{K}\widetilde{f}=r_{j}p_{j}\widetilde{f}=r_{j}f_{j}=f$.

_3rd step._ Let $F\colon[0,1]^{2}\to G/K$ be a homotopy of paths which starts
at the identity $e$ of $G/K$. We claim that $F$ lifts uniquely across $p_{K}$
to a homotopy of paths $\widetilde{F}\colon[0,1]^{2}\to G$ starting at the
identity $e$ of $G$.

The proof follows the line of the argument used in the second step of the
proof and so we omit the details. Let us only mention that in proving the
claim one uses an inverse limit argument along with the following two facts:

  * •

homotopies of paths do lift to homotopies of paths across the quotient
morphisms in $\mathsf{CAGp}$ with finite kernels,

  * •

the inverse limit of a system of homotopies of paths is again a homotopy of
paths.

_4th step._ We finish the proof of the lemma by verifying the equivalence of
(a) and (b). Implication (a)$\Rightarrow$(b) is clear and so we verify only
the converse implication (b)$\Rightarrow$(a). Our construction of the map
$\zeta$ as in (a) will proceed as in the case of $K$ finite, by using the
method of lifting appropriate paths across $p_{K}$. To verify the continuity
of $\zeta$ we shall use an inverse limit argument.

We start by verifying the following auxiliary claim:

  1. ($*$)

Let $f_{1},f_{2}$ be paths in $X$ with $f_{1}(0)=f_{2}(0)=z$ and
$f_{1}(1)=f_{2}(1)$, and let $\widetilde{f}_{1},\widetilde{f}_{2}$ be the
lifts of the paths $\xi f_{1},\xi f_{2}$ across $p_{K}$ with
$\widetilde{f}_{1}(0)=\widetilde{f}_{2}(0)=e$. Then
$\widetilde{f}_{1}(1)=\widetilde{f}_{2}(1)$.

By virtue of (b), there is $\widetilde{l}\in\pi_{1}(G)$ with
$\xi(f_{1}*\overline{f_{2}})\simeq_{p}p_{K}\widetilde{l}$. Then

$\xi
f_{1}\simeq_{p}\xi((f_{1}*\overline{f_{2}})*f_{2})=\xi(f_{1}*\overline{f_{2}})*\xi
f_{2}\simeq_{p}p_{K}\widetilde{l}*\xi f_{2}$

and hence, by the third step of the proof, the lifts $\widetilde{f}_{1}$ and
$\widetilde{l}*\widetilde{f}_{2}$ of $\xi f_{1}$ and $p_{K}\widetilde{l}*\xi
f_{2}$, respectively, are homotopic paths. Thus,
$\widetilde{f}_{1}(1)=(\widetilde{l}*\widetilde{f}_{2})(1)=\widetilde{f}_{2}(1)$,
as was to be shown.

Now, given $x\in X$, let $f$ be a path in $X$ from $z$ to $x$. Then $\xi f$ is
a path in $G/K$ from $e$ to $\xi(x)$ and hence, by the second step of the
proof, it lifts across $p_{K}$ to a (unique) path $\widetilde{f}$ in $G$ with
$\widetilde{f}(0)=e$. We set $\zeta(x)=\widetilde{f}(1)$. By virtue of ($*$),
$\zeta$ is a well defined map $X\to G$ with $p_{K}\zeta=\xi$ and $\zeta(z)=e$.
Thus, to finish the proof it remains to show that $\zeta$ is continuous. Given
$j\in J$, we have $\xi^{\sharp}\pi_{1}(X)\subseteq
p_{K}^{\sharp}\pi_{1}(G)=r_{j}^{\sharp}p_{j}^{\sharp}\pi_{1}(G)\subseteq
r_{j}^{\sharp}\pi_{1}(G/H_{j})$. Since, in addition, $r_{j}\colon G/H_{j}\to
G/K$ is a covering morphism, it follows that the map $\xi$ lifts uniquely
across $r_{j}$ to a continuous base point preserving map $\zeta_{j}\colon X\to
G/H_{j}$. If $x\in X$, $f$ is a path in $X$ from $z$ to $x$ and
$\widetilde{f}$ is the continuous lift of $\xi f$ across $p_{K}$ starting at
$e$, then both $p_{j}\widetilde{f}$ and $\zeta_{j}f$ are continuous lifts of
$\xi f$ across $r_{j}$ starting at $e$. Thus $p_{j}\widetilde{f}=\zeta_{j}f$
and hence $p_{j}\zeta(x)=p_{j}\widetilde{f}(1)=\zeta_{j}f(1)=\zeta_{j}(x)$.
This shows that $p_{j}\zeta=\zeta_{j}$ is continuous for every $j\in J$ and
hence the map $\zeta$ itself is continuous. ∎

######  Corollary 5.7.

Let $\mathop{\rm\mathcal{F}\colon\Gamma\curvearrowright X}$ be a minimal flow
with $\Gamma\in\mathsf{LieGp}$ connected and with $X$ a compact connected
manifold. Let $G\in\mathsf{CAGp}$, $K\mathrel{\lhd}G$ be totally disconnected
and $p_{K}\colon G\to G/K$ be the quotient morphism. Given a continuous base
point preserving map $\xi\colon X\to G/K$, the following conditions are
equivalent:

  1. (a)

$\mathop{\rm co}(\xi)$ lifts across $p_{K}$ to an extension
$\mathcal{C}\in\mathop{\rm{\bf Z}_{\mathcal{F}}}(G)$,

  2. (b)

$\xi^{\sharp}\mathcal{F}_{z}^{\sharp}\pi_{1}(\Gamma)\subseteq
p_{K}^{\sharp}\pi_{1}(G)$,

  3. (c)

$\xi^{\sharp}H_{1}^{w}(\mathcal{F})\subseteq p_{K}^{\sharp}H_{1}^{w}(G)$.

###### Proof.

By virtue of (2.30) from Lemma 2.17, $\mathop{\rm
co}(\xi)^{\sharp}\pi_{1}(\Gamma\times
X)=\xi^{\sharp}\mathcal{F}_{z}^{\sharp}\pi_{1}(\Gamma)$. By applying also
Lemma 5.5 and Lemma 3.36, we obtain the equivalence of the following
conditions:

  * •

$\mathop{\rm co}(\xi)$ lifts across $p_{K}$ to an extension
$\mathcal{C}\in\mathop{\rm{\bf Z}_{\mathcal{F}}}(G)$,

  * •

$\mathop{\rm co}(\xi)$ lifts across $p_{K}$ to a continuous base point
preserving map $\Gamma\times X\to G$,

  * •

$\mathop{\rm co}(\xi)^{\sharp}\pi_{1}(\Gamma\times X)\subseteq
p_{K}^{\sharp}\pi_{1}(G)$,

  * •

$\xi^{\sharp}\mathcal{F}_{z}^{\sharp}\pi_{1}(\Gamma)\subseteq
p_{K}^{\sharp}\pi_{1}(G)$.

This verifies the equivalence of (a) and (b).

Before turning to the proof of the equivalence of (b) and (c) let us add some
comments on the notation. The symbols $p_{\Gamma}$ and $p_{G/K}$ stand for the
canonical quotient morphisms $\pi_{1}(\Gamma)\to H_{1}^{w}(\Gamma)$ and
$\pi_{1}(G/K)\to H_{1}^{w}(G/K)$, respectively, whereas the symbol $p_{K}$ is
used to denote the quotient morphism $G\to G/K$ according to the assumptions
of the corollary. Recall also that by definition of $H_{1}^{w}(\mathcal{F})$,
$H_{1}^{w}(\mathcal{F})=\mathcal{F}_{z}^{\sharp}H_{1}^{w}(\Gamma)=p_{X}\mathcal{F}_{z}^{\sharp}\pi_{1}(\Gamma)$,
where $p_{X}$ stands for the canonical quotient morphism $\pi_{1}(X)\to
H_{1}^{w}(X)$.

Now, since the group $\pi_{1}(G/K)$ is abelian and torsion-free, the morphism
$p_{G/K}$ is an isomorphism. Thus, in order to verify the equivalence of (b)
and (c), it suffices to check the following two identities:

  1. ($\alpha$)

$\xi^{\sharp}H_{1}^{w}(\mathcal{F})=p_{G/K}\xi^{\sharp}\mathcal{F}_{z}^{\sharp}\pi_{1}(\Gamma)$,

  2. ($\beta$)

$p_{K}^{\sharp}H_{1}^{w}(G)=p_{G/K}p_{K}^{\sharp}\pi_{1}(G)$.

First, we have

$\xi^{\sharp}H_{1}^{w}(\mathcal{F})=\xi^{\sharp}\mathcal{F}_{z}^{\sharp}H_{1}^{w}(\Gamma)=\xi^{\sharp}\mathcal{F}_{z}^{\sharp}p_{\Gamma}\pi_{1}(\Gamma)=p_{G/K}\xi^{\sharp}\mathcal{F}_{z}^{\sharp}\pi_{1}(\Gamma),$

which verifies ($\alpha$). A similar computation yields

$p_{K}^{\sharp}H_{1}^{w}(G)=p_{K}^{\sharp}p_{G}\pi_{1}(G)=p_{G/K}p_{K}^{\sharp}\pi_{1}(G),$

where $p_{G}$ denotes the quotient morphism $\pi_{1}(G)\to H_{1}^{w}(G)$. This
verifies ($\beta$). ∎

In order to formulate the next lemma we fix some notation. Let
$G\in\mathsf{CAGp}$ and $K\mathrel{\lhd}G$ be totally disconnected. Consider
the quotient morphism $p_{K}\colon G\to G/K$ and the induced morphism of
fundamental groups $p_{K}^{\sharp}\colon\pi_{1}(G)\to\pi_{1}(G/K)$, where, as
usual, the identities of the groups are taken as the base points for their
fundamental groups. Recall that the groups $\pi_{1}(G)$ and $\pi_{1}(G/K)$ are
abelian.

Given a subgroup $E$ of $K$, let $\mathcal{Q}_{K}(E)$ consist of those
elements of $\pi_{1}(G/K)$, which lift across $p_{K}$ to a path both starting
and ending in $E$ (or, equivalently, whose lift starting at $e$ ends at an
element of $E$). (Recall from Remark 5.6 that $\mathcal{Q}_{K}(E)$ is a well
defined object since we have a monodromy action of $\pi_{1}(G/K)$ on $K$ just
as in the case of a finite kernel $K$. Thus, the endpoint $\widetilde{f}(1)$
of the lift $\widetilde{f}$ of a loop $f$ in $G/K$ across $p_{K}$ with
$\widetilde{f}(0)=e$ depends indeed only on the path-homotopy class of $f$.)
Observe also that the identity $\mathcal{Q}_{K}(E)=\mathcal{Q}_{K}(E\cap
G_{a})$ holds by definition of $\mathcal{Q}_{K}$. Despite the latter identity
we shall not restrict ourselves to the subgroups of $K$ contained in $G_{a}$.
We have two reasons for doing so. First, $K\cap G_{a}$ need not be a closed
subgroup of $K$. Second, some of the subgroups $E$ of $K$, which we will have
to deal with later in our considerations, will not be contained in $G_{a}$.
(See Remarks 5.15 and 5.19 below.)

Conversely, given a subgroup $Q$ of $\pi_{1}(G/K)$, let $\mathcal{E}_{K}(Q)$
be the set of all $k\in K$, such that there exists a path in $G$ starting at
$e$, ending at $k$ and projecting by $p_{K}$ to an element of $Q$. Clearly,
$\mathcal{E}_{K}(Q)=\mathcal{E}_{K}(Q+p_{K}^{\sharp}\pi_{1}(G))$ and we shall
therefore often restrict ourselves to the subgroups $Q$ of $\pi_{1}(G/K)$
containing $p_{K}^{\sharp}\pi_{1}(G)$. Observe that $\mathcal{E}_{K}(Q)$ is
the orbit of the identity $e$ under the monodromy action of $Q$ on $K$. The
following lemma summarizes some of the properties of $\mathcal{Q}_{K}$ and
$\mathcal{E}_{K}$.

######  Lemma 5.8.

Let $G\in\mathsf{CAGp}$, $K\mathrel{\lhd}G$ be totally disconnected and
$p_{K}\colon G\to G/K$ be the quotient morphism. Fix a subgroup $E$ of $K$ and
a subgroup $Q$ of $\pi_{1}(G/K)$ containing $p_{K}^{\sharp}\pi_{1}(G)$. Then
the following statements hold:

  1. (i)

$\mathcal{Q}_{K}(E)$ is a subgroup of $\pi_{1}(G/K)$ containing
$p_{K}^{\sharp}\pi_{1}(G)$,

  2. (ii)

$\mathcal{E}_{K}(Q)$ is a subgroup of $K$ contained in $G_{a}$,

  3. (iii)

there are identities

$\mathcal{E}_{K}(\mathcal{Q}_{K}(E))=E\cap G_{a}\hskip
11.38109pt\text{and}\hskip 11.38109pt\mathcal{Q}_{K}(\mathcal{E}_{K}(Q))=Q,$

  4. (iv)

there is an isomorphism

$\mathcal{E}_{K}(Q)\cong Q/p_{K}^{\sharp}\pi_{1}(G).$

######  Remark 5.9.

We wish to mention the following facts.

  * •

Under the assumptions of the lemma there are identities
$\mathcal{E}_{K}(p_{K}^{\sharp}\pi_{1}(G))=e$ and
$\mathcal{E}_{K}(\pi_{1}(G/K))=K\cap G_{a}$. In particular, the latter
identity and statement (iv) together yield an isomorphism $K\cap
G_{a}\cong\pi_{1}(G/K)/p_{K}^{\sharp}\pi_{1}(G)$.

  * •

In Examples 5.10 and 5.12 below we describe the correspondence
$\mathcal{E}_{K}$ in the situations when $G=\mathbb{T}^{n}$ is a finite-
dimensional torus and when $G=S_{{\bf p}}$ is a solenoid.

###### Proof of Lemma 5.8.

We verify statement (i). Inclusion
$p_{K}^{\sharp}\pi_{1}(G)\subseteq\mathcal{Q}_{K}(E)$ is an immediate
consequence of the definition of $\mathcal{Q}_{K}$. To see that
$\mathcal{Q}_{K}(E)$ is a subgroup of $\pi_{1}(G/K)$, fix
$f_{1},f_{2}\in\mathcal{Q}_{K}(E)$ and let
$\widetilde{f}_{1},\widetilde{f}_{2}$ be the lifts of $f_{1},f_{2}$ across
$p_{K}$ starting at $e$. By definition of $\mathcal{Q}_{K}(E)$,
$k_{1}=\widetilde{f}_{1}(1)$ and $k_{2}=\widetilde{f}_{2}(1)$ are both
elements of $E$. Now, the path
$\widetilde{f}_{1}*\overline{k_{1}k_{2}^{-1}\widetilde{f}_{2}}$ is the lift of
$f_{1}*\overline{f_{2}}$ across $p_{K}$ starting at $e$ and its endpoint is
$(\widetilde{f}_{1}*\overline{k_{1}k_{2}^{-1}\widetilde{f}_{2}})(1)=k_{1}k_{2}^{-1}\widetilde{f}_{2}(0)=k_{1}k_{2}^{-1}\in
E$. Thus, $f_{1}*\overline{f_{2}}\in\mathcal{Q}_{K}(E)$, as was to be shown.

We verify statement (ii). Inclusion $\mathcal{E}_{K}(Q)\subseteq K\cap G_{a}$
follows directly from the definition of $\mathcal{E}_{K}$. To see that
$\mathcal{E}_{K}(Q)$ is a subgroup of $K$, fix
$k_{1},k_{2}\in\mathcal{E}_{K}(Q)$ and let
$\widetilde{f}_{1},\widetilde{f}_{2}$ be paths in $G$ from $e$ to
$k_{1},k_{2}$, projecting by $p_{K}$ to elements $f_{1},f_{2}$ of $Q$,
respectively. Since $Q$ is a subgroup of $\pi_{1}(G/K)$,
$f_{1}*\overline{f_{2}}$ is an element of $Q$ and its lift
$\widetilde{f}_{1}*\overline{k_{1}k_{2}^{-1}\widetilde{f}_{2}}$ across $p_{K}$
starting at $e$ thus ends at an element of $\mathcal{E}_{K}(Q)$. That is,
$k_{1}k_{2}^{-1}=(\widetilde{f}_{1}*\overline{k_{1}k_{2}^{-1}\widetilde{f}_{2}})(1)\in\mathcal{E}_{K}(Q)$,
as was to be shown.

The first identity from statement (iii) is an immediate consequence of the
definitions of $\mathcal{Q}_{K}$ and $\mathcal{E}_{K}$, and of the existence
and uniqueness of the lifts $\widetilde{f}$ across $p_{K}$ with
$\widetilde{f}(0)=e$ for the loops $f\in\pi_{1}(G/K)$, see Lemma 5.5. The
second identity from (iii) follows also from the definitions of
$\mathcal{Q}_{K}$ and $\mathcal{E}_{K}$, and from the assumption
$p_{K}^{\sharp}\pi_{1}(G)\subseteq Q$.

Consider the map $\varphi\colon Q\to\mathcal{E}_{K}(Q)$, associating with
every loop $f\in Q$ the endpoint of its lift $\widetilde{f}$ across $p_{K}$
with $\widetilde{f}(0)=e$. To verify statement (iv) it suffices to show that
$\varphi$ is an epimorphism of groups with kernel $p_{K}^{\sharp}\pi_{1}(G)$.
First, $\varphi$ is well defined due to the monodromy action of $\pi_{1}(G/K)$
on $K$, and it is surjective by definition of $\mathcal{E}_{K}(Q)$. To see
that $\varphi$ is a morphism of groups, fix $f_{1},f_{2}\in Q$ and consider
their lifts $\widetilde{f}_{1},\widetilde{f}_{2}$ across $p_{K}$ starting at
$e$. Then
$\varphi(f_{1}*f_{2})=(\widetilde{f}_{1}*\widetilde{f}_{1}(1)\widetilde{f}_{2})(1)=\widetilde{f}_{1}(1)\widetilde{f}_{2}(1)=\varphi(f_{1})\varphi(f_{2})$,
as was to be shown. Finally, the identity
$\ker(\varphi)=p_{K}^{\sharp}\pi_{1}(G)$ is clear and the proof is thus
finished. ∎

######  Example 5.10.

We describe the operation $\mathcal{E}_{K}$ in the case when the group $K$ is
finite and either $G$ or $G/K$ is a finite-dimensional torus. To justify the
setting used in this example recall the following well known facts.

  * •

Every quotient group of $\mathbb{T}^{n}$ modulo a finite group is
topologically isomorphic to $\mathbb{T}^{n}$ itself. Hence, every covering
morphism from $\mathbb{T}^{n}$ is equivalent to a surjective endomorphism $p$
of $\mathbb{T}^{n}$.

  * •

If $G\in\mathsf{CAGp}$ is connected and $\mathbb{T}^{n}$ is a quotient group
of $G$ modulo a finite subgroup of $G$ then $G$ is topologically isomorphic to
$\mathbb{T}^{n}$.

Now let $p$ be a surjective endomorphism of $\mathbb{T}^{n}$ induced by an
integer matrix $A=(a_{jl})$. That is, the matrix $A$ is regular and
$p(z_{m})_{m=1}^{n}=\left(z_{1}^{a_{m1}}\dots z_{n}^{a_{mn}}\right)_{m=1}^{n}$
for every $(z_{m})_{m=1}^{n}\in\mathbb{T}^{n}$. Recall the isomorphism
$\pi_{1}(\mathbb{T}^{n})\cong\mathbb{Z}^{n}$,

$\mathbb{Z}^{n}\ni(k_{1},\dots,k_{n})\mapsto
f_{k_{1},\dots,k_{n}}\colon[0,1]\ni t\mapsto\left(e^{i2\pi
k_{1}t},\dots,e^{i2\pi k_{n}t}\right)\in\mathbb{T}^{n},$

under which the induced morphism
$p^{\sharp}\colon\pi_{1}(\mathbb{T}^{n})\to\pi_{1}(\mathbb{T}^{n})$ takes the
form of the endomorphism of $\mathbb{Z}^{n}$ induced by the matrix $A$. That
is, $p^{\sharp}\colon\mathbb{Z}^{n}\to\mathbb{Z}^{n}$ acts by the rule
$p^{\sharp}(k_{m})_{m=1}^{n}=(a_{m1}k_{1}+\dots+a_{mn}k_{n})_{m=1}^{n}$. Let
$B=(b_{jl})$ be the inverse of $A$ in $GL_{n}(\mathbb{Q})$. Then for all
$k_{1},\dots,k_{n}\in\mathbb{Z}$, the loop $f_{k_{1},\dots,k_{n}}$ lifts
across $p$ to the path

$\widetilde{f}_{k_{1},\dots,k_{n}}\colon[0,1]\ni
t\mapsto\left(e^{i2\pi\left(\sum_{l=1}^{n}b_{1l}k_{l}\right)t},\dots,e^{i2\pi\left(\sum_{l=1}^{n}b_{nl}k_{l}\right)t}\right)\in\mathbb{T}^{n}.$

The path $\widetilde{f}_{k_{1},\dots,k_{n}}$ starts at the identity $1$ of
$\mathbb{T}^{n}$ and ends at

$\begin{split}\widetilde{f}_{k_{1},\dots,k_{n}}(1)&=\left(e^{i2\pi\sum_{l=1}^{n}b_{1l}k_{l}},\dots,e^{i2\pi\sum_{l=1}^{n}b_{nl}k_{l}}\right)=q\left(\sum_{l=1}^{n}b_{1l}k_{l},\dots,\sum_{l=1}^{n}b_{nl}k_{l}\right)\\\
&=q\left(\left(B(k_{1},\dots,k_{n})^{T}\right)^{T}\right),\end{split}$

where $(-)^{T}$ stands for the transposing operator and
$q\colon\mathbb{R}^{n}\to\mathbb{T}^{n}=\mathbb{R}^{n}/\mathbb{Z}^{n}$ for the
usual covering morphism. Consequently,
$p^{\sharp}\pi_{1}(\mathbb{T}^{n})=A\mathbb{Z}^{n}$ and the operation
$\mathcal{E}_{K}$ with $K=\ker(p)$ can be described as follows:

  * •

if $Q$ is a subgroup of $\mathbb{Z}^{n}$ containing $A\mathbb{Z}^{n}$ then
$\mathcal{E}_{K}(Q)=q(BQ)$.

With some additional information on $p$, the operation $\mathcal{E}_{K}$ can
be simplified as follows. Let $e_{1},\dots,e_{n}$ be the standard basis for
$\mathbb{Z}^{n}$. Then for every monomorphism
$h\colon\mathbb{Z}^{n}\to\mathbb{Z}^{n}$ there are automorphisms
$\varphi,\psi$ of $\mathbb{Z}^{n}$ and positive integers $d_{1},\dots,d_{n}$
with $d_{i}$ dividing $d_{i+1}$ for $i=1,\dots,n-1$, such that the composition
$k=\varphi h\psi$ acts by the rule $k(e_{i})=d_{i}e_{i}$ ($i=1,\dots,n$).
Consequently, every surjective endomorphism of $\mathbb{T}^{n}$ is equivalent
to a morphism of the form
$p(z_{1},\dots,z_{n})=(z_{1}^{d_{1}},\dots,z_{n}^{d_{n}})$ with
$d_{1},\dots,d_{n}$ as above. In such case the matrix $A$ of $p$ is the
diagonal matrix with $d_{1},\dots,d_{n}$ as its diagonal elements. Hence, the
inverse $B=A^{-1}$ of $A$ in $GL_{n}(\mathbb{Q})$ is also a diagonal matrix
with the diagonal elements $1/d_{1},\dots,1/d_{n}$. Having expressed $p$ in
such form we see that

$p^{\sharp}\pi_{1}(\mathbb{T}^{n})=A\mathbb{Z}^{n}=d_{1}\mathbb{Z}\oplus\dots\oplus
d_{n}\mathbb{Z},$

and for every subgroup $Q$ of $\mathbb{Z}^{n}$ containing
$d_{1}\mathbb{Z}\oplus\dots\oplus d_{n}\mathbb{Z}$,

$\mathcal{E}_{K}(Q)=q(BQ)=\left\\{\left(e^{i2\pi\frac{q_{1}}{d_{1}}},\dots,e^{i2\pi\frac{q_{n}}{d_{n}}}\right):(q_{1},\dots,q_{n})\in
Q\right\\}$

with $K=\ker(p)$.

######  Example 5.11.

Let $G\in\mathsf{CAGp}$, $K\mathrel{\lhd}G$ be totally disconnected and
$p_{K}\colon G\to G/K$ be the quotient morphism. We show that for every
positive integer $n$,

$\mathcal{E}_{K}\left(n\pi_{1}(G/K)\right)=n(K\cap G_{a}).$

Fix $k\in\mathcal{E}_{K}(n\pi_{1}(G/K))$. There is a loop $f\in
n\pi_{1}(G/K)$, whose lift $\widetilde{f}$ across $p_{K}$ starting at $e$ ends
at $k$. Choose $g\in\pi_{1}(G/K)$ with $f=ng$ and denote by $\widetilde{g}$
the lift of $g$ across $p_{K}$ starting at $e$. The endpoint $l$ of
$\widetilde{g}$ is clearly an element of $K\cap G_{a}$. Moreover, the
uniqueness of lifts yields
$\widetilde{f}=\widetilde{g}*l\widetilde{g}*\dots*l^{n-1}\widetilde{g}$ and
hence
$k=\widetilde{f}(1)=(\widetilde{g}*l\widetilde{g}*\dots*l^{n-1}\widetilde{g})(1)=l^{n}\in
n(K\cap G_{a})$. This verifies the inclusion
$\mathcal{E}_{K}(n\pi_{1}(G/K))\subseteq n(K\cap G_{a})$. To verify the
converse inclusion let $k\in n(K\cap G_{a})$ and write $k=l^{n}$ with $l\in
K\cap G_{a}$. Choose a path $\widetilde{g}$ in $G$ from $e$ to $l$ and set
$g=p_{K}\widetilde{g}\in\pi_{1}(G/K)$. Then
$\widetilde{g}*l\widetilde{g}*\dots*l^{n-1}\widetilde{g}$ is the lift of $ng$
across $p_{K}$ starting at $e$ and its endpoint is $l^{n}=k$. Thus,
$k\in\mathcal{E}_{K}(n\pi_{1}(G/K))$, as was to be shown.

######  Example 5.12.

We describe the operation $\mathcal{E}_{K}$ in the situation when $G$ is a
solenoid and $K$ is an arbitrary closed totally disconnected subgroup of $G$.
So let $S_{{\bf p}}$ be a solenoid and let $K\mathrel{\lhd}S_{{\bf p}}$ be
totally disconnected. The dual group $(S_{{\bf p}}/K)^{*}$ of $S_{{\bf p}}/K$
is isomorphic to $K^{\perp}\mathrel{\lhd}(S_{{\bf p}})^{*}\cong\mathbb{Z}/{\bf
p}$ and so it has rank $1$. Consequently, $S_{{\bf p}}/K$ is topologically
isomorphic either to a solenoid or to the circle group $\mathbb{T}^{1}$. In
the first case we have $\pi_{1}(S_{{\bf p}}/K)=0$ and hence
$\mathcal{E}_{K}=e$ is trivial. In the second case every subgroup $Q$ of
$\pi_{1}(S_{{\bf p}}/K)$ has the form $Q=n\pi_{1}(S_{{\bf p}}/K)$ for some
integer $n$, in which case
$\mathcal{E}_{K}(Q)=\mathcal{E}_{K}(n\pi_{1}(S_{{\bf p}}/K))=n(K\cap(S_{{\bf
p}})_{a})$ by Example 5.11.

Before formulating our main result of this section, namely Theorem 5.14, we
fix some more necessary notation. Let $G\in\mathsf{CAGp}$, $K\mathrel{\lhd}G$
be totally disconnected and $p_{K}\colon G\to G/K$ be the quotient morphism.
Given a subgroup $Q$ of $\pi_{1}(G/K)$, we let $\overline{\mathcal{E}_{K}}(Q)$
denote the closure in $K$ of the group $\mathcal{E}_{K}(Q)$. Since
$\mathcal{E}_{K}(Q)=\mathcal{E}_{K}(Q+p_{K}^{\sharp}\pi_{1}(G))$ for every
$Q$, the same identity holds also with $\mathcal{E}_{K}$ replaced by
$\overline{\mathcal{E}_{K}}$. Observe that $\overline{\mathcal{E}_{K}}(Q)$ is
the orbit closure of the identity $e$ under the monodromy action of $Q$ on
$K$. Moreover, by Lemma 5.8(iii),
$\overline{\mathcal{E}_{K}}(\mathcal{Q}_{K}(E))=\overline{E\cap G_{a}}$ for
every subgroup $E$ of $K$ and
$\mathcal{Q}_{K}(\overline{\mathcal{E}_{K}}(Q))\supseteq Q$ for every subgroup
$Q$ of $\pi_{1}(G/K)$ containing $p_{K}^{\sharp}\pi_{1}(G)$.

######  Remark 5.13.

Let $G\in\mathsf{CAGp}$ and $K\mathrel{\lhd}G$ be totally disconnected. Assume
that $K\subseteq G_{a}$. (This is true, for instance, if $G$ is a (finite- or
infinite-dimensional) torus, in which case $G_{a}=G$.) Then, by Lemma
5.8(iii), the following statements hold:

  1. (a)

$\mathcal{E}_{K}(\mathcal{Q}_{K}(E))=E$ for every subgroup $E$ of $K$,

  2. (b)

$\mathcal{Q}_{K}(\mathcal{E}_{K}(Q))=Q$ for every subgroup $Q$ of
$\pi_{1}(G/K)$ containing $p_{K}^{\sharp}\pi_{1}(G)$.

Consequently, $\mathcal{E}_{K}$ and $\mathcal{Q}_{K}$ consitute one-to-one
correspondences between the subgroups $E$ of $K$ on one side and the subgroups
$Q$ of $\pi_{1}(G/K)$ containing $p_{K}^{\sharp}\pi_{1}(G)$ on the other side.
As a consequence of these facts, the following observations emerge:

  * •

$\mathcal{E}_{K}(Q)$ need not be a closed subgroup of $K$; to see this, take
$Q=\mathcal{Q}_{K}(E)$ with $E$ a non-closed subgroup of $K$,

  * •

the equality $\mathcal{Q}_{K}(\overline{\mathcal{E}_{K}}(Q))=Q$ may fail; to
see this, take, as above, $Q=\mathcal{Q}_{K}(E)$ with $E$ a non-closed
subgroup of $K$, then

$\mathcal{Q}_{K}(\overline{\mathcal{E}_{K}}(Q))=\mathcal{Q}_{K}(\overline{\mathcal{E}_{K}}(\mathcal{Q}_{K}(E)))=\mathcal{Q}_{K}(\overline{E})\supsetneq\mathcal{Q}_{K}(E)=Q.$

######  Theorem 5.14.

Let $\mathop{\rm\mathcal{F}\colon\Gamma\curvearrowright X}$ be a minimal flow,
let $G\in\mathsf{CAGp}$ and $\mathcal{C}\in\mathop{\rm{\bf
Z}_{\mathcal{F}}}(G)$, and fix $z\in X$. Assume that $K\mathrel{\lhd}G$ is
totally disconnected with $p_{K}\mathcal{C}\in\mathop{\rm{\bf
B}_{\mathcal{F}}}(G/K)$, and denote by $\xi$ the transfer function of
$p_{K}\mathcal{C}$ with $\xi(z)=e$. If $X\in\mathsf{CLAC}$ and $\Gamma$ has no
non-trivial finite abelian quotient groups then

$F(\mathcal{C})=\overline{\mathcal{E}_{K}}\left(\xi^{\sharp}\pi_{1}(X)+p_{K}^{\sharp}\pi_{1}(G)\right)=\overline{\mathcal{E}_{K}}\left(\xi^{\sharp}\pi_{1}(X)\right).$
(5.1)

If, in addition, the group $K$ is finite then the following statements hold:

  1. (1)

$F(\mathcal{C})$ is given by the formula

$F(\mathcal{C})=\mathcal{E}_{K}\left(\xi^{\sharp}\pi_{1}(X)+p_{K}^{\sharp}\pi_{1}(G)\right)=\mathcal{E}_{K}\left(\xi^{\sharp}\pi_{1}(X)\right),$

  2. (2)

$F(\mathcal{C})$ satisfies

$\mathcal{Q}_{K}\left(F(\mathcal{C})\right)=\xi^{\sharp}\pi_{1}(X)+p_{K}^{\sharp}\pi_{1}(G),$

  3. (3)

$F(\mathcal{C})\subseteq G_{a}$ and

$F(\mathcal{C})\cong\left(\xi^{\sharp}\pi_{1}(X)+p_{K}^{\sharp}\pi_{1}(G)\right)\Big{/}p_{K}^{\sharp}\pi_{1}(G)\cong\pi_{1}(G/F(\mathcal{C}))\Big{/}p_{F(\mathcal{C})}^{\sharp}\pi_{1}(G).$

  4. (4)

the following statements are equivalent:

    1. (i)

$F(\mathcal{C})=K$,

    2. (ii)

$K\subseteq G_{a}$ and
$\xi^{\sharp}\pi_{1}(X)+p_{K}^{\sharp}\pi_{1}(G)=\pi_{1}(G/K)$.

######  Remark 5.15.

We wish to mention the following facts.

  * •

In the isomorphism (5.1) the symbol $\overline{\mathcal{E}_{K}}$ can not be
replaced by the symbol $\mathcal{E}_{K}$, for the group
$\xi^{\sharp}\pi_{1}(X)+p_{K}^{\sharp}\pi_{1}(G)$ need not correspond to a
closed subgroup of $K$ via the operation $\mathcal{E}_{K}$, see Remark 5.19
below.

  * •

If the group $K$ is not finite then $F(\mathcal{C})$ may fail to be a subgroup
of $G_{a}$ and so statement (3) is no longer true. Again, we leave an example
of such a situation to Remark 5.19.

  * •

An immediate consequence of (5.1) is that the section $F(p_{K}^{-1}\mathop{\rm
co}(\xi))$ of the lift $p_{K}^{-1}\mathop{\rm co}(\xi)$ of $\mathop{\rm
co}(\xi)$ across $p_{K}$ depends only on the homotopy class of $\xi\in
C_{z}(X,G/K)$. Consequently, if $C_{z}(X,G/K)_{a}$ is an open subgroup of
$C_{z}(X,G/K)$, which is the case if $X$ is a compact space and $G/K$ is a
finite-dimensional torus (see Subsection 2.4.8), then $F(p_{K}^{-1}\mathop{\rm
co}(\xi))$ is a locally constant (and hence continuous) function of the
variable $\xi\in C_{z}(X,G/K)$.

###### Proof of Theorem 5.14.

Let $E$ be a closed subgroup of $K$. Recall the notation $p_{K}$ and $p_{E}$
for the quotient morphisms $G\to G/K$ and $G\to G/E$, respectively, and
$q_{E}$ for the (unique quotient) morphism $G/E\to G/K$ with
$q_{E}p_{E}=p_{K}$. Since $E$ and $K$ are closed totally disconnected
subgroups of $G$, Lemma 5.5 applies to the quotient morphisms $p_{E}$ and
$p_{K}$. Moreover, it applies also to $q_{E}$, for $q_{E}$ is equivalent to
the quotient morphism $G/E\to(G/E)/(K/E)$, whose kernel $K/E\cong p_{E}(K)$ is
a closed totally disconnected subgroup of $G/E$.

We claim that $q_{E}^{\sharp}\pi_{1}(G/E)=\mathcal{Q}_{K}(G_{a}\cap E)$. To
see this let $f$ be a loop in $G/K$ based at $e$ and denote by $\widetilde{f}$
the lift of $f$ across $p_{K}$ starting at $e$. Then, clearly,
$p_{E}\widetilde{f}$ is the lift of $f$ across $q_{E}$ starting at $e$. Thus,
the equivalence of the following statements follows:

  * •

$f\in q_{E}^{\sharp}\pi_{1}(G/E)$,

  * •

$p_{E}\widetilde{f}$ is a loop,

  * •

$\widetilde{f}$ ends at an element of $E$,

  * •

$f\in\mathcal{Q}_{K}(E)$,

  * •

$f\in\mathcal{Q}_{K}(E\cap G_{a})$,

and that verifies the claim.

Further, we claim that the following statements are equivalent:

  1. (a)

$\xi$ lifts across $q_{E}$ to a continuous base point preserving map
$\eta\colon X\to G/E$,

  2. (b)

$\xi^{\sharp}\pi_{1}(X)\subseteq q_{E}^{\sharp}\pi_{1}(G/E)$,

  3. (c)

$\xi^{\sharp}\pi_{1}(X)\subseteq Q_{K}(G_{a}\cap E)$.

Indeed, (a) and (b) are equivalent by virtue of Lemma 5.5 and the equivalence
of (b) and (c) follows from the equality
$q_{E}^{\sharp}\pi_{1}(G/E)=\mathcal{Q}_{K}(G_{a}\cap E)$ verified above.

Now, from statement (i) in Proposition 5.3 and from the equivalence of (a) and
(c) it follows that $F(\mathcal{C})$ equals the smallest of all the closed
subgroups $E$ of $K$ with $\xi^{\sharp}\pi_{1}(X)\subseteq Q_{K}(G_{a}\cap
E)$. Since $\mathcal{Q}_{K}(G_{a}\cap E)=q_{E}^{\sharp}\pi_{1}(G/E)\supseteq
p_{K}^{\sharp}\pi_{1}(G)$, the inclusion $\xi^{\sharp}\pi_{1}(X)\subseteq
Q_{K}(G_{a}\cap E)$ is equivalent with
$\xi^{\sharp}\pi_{1}(X)+p_{K}^{\sharp}\pi_{1}(G)\subseteq\mathcal{Q}_{K}(G_{a}\cap
E)$, and the latter inclusion is equivalent with

$\mathcal{E}_{K}\left(\xi^{\sharp}\pi_{1}(X)+p_{K}^{\sharp}\pi_{1}(G)\right)\subseteq
G_{a}\cap E$ (5.2)

by virtue of Lemma 5.8(iii). Since the left-hand side of (5.2) is contained in
$G_{a}$ by Lemma 5.8(ii), the smallest closed subgroup $E$ of $K$ for which
(5.2) holds is the group

$F(\mathcal{C})=\overline{\mathcal{E}_{K}}\left(\xi^{\sharp}\pi_{1}(X)+p_{K}^{\sharp}\pi_{1}(G)\right)=\overline{\mathcal{E}_{K}}\left(\xi^{\sharp}\pi_{1}(X)\right).$

This verifies (5.1).

Assume, from now on, that the group $K$ is finite. Then every subgroup $E$ of
$K$ is closed and $\mathcal{E}_{K}$ therefore coincides with
$\overline{\mathcal{E}_{K}}$. Thus, statement (1) follows at once from (5.1).
Statement (2) follows from statement (1) by applying $\mathcal{Q}_{K}$ and by
using Lemma 5.8.

The inclusion $F(\mathcal{C})\subseteq G_{a}$ in statement (3) follows from
statement (1) and from Lemma 5.8(ii). The first isomorphism from (3) follows
by applying Lemma 5.8(iv) to
$Q=\xi^{\sharp}\pi_{1}(X)+p_{K}^{\sharp}\pi_{1}(G)$. The second isomorphism
from (3) is proved similarly, this time one applies Lemma 5.8(iv) to
$Q=\pi_{1}(G/F(\mathcal{C}))$, with $K$ replaced by $F(\mathcal{C})$, to
obtain

$F(\mathcal{C})=F(\mathcal{C})\cap
G_{a}=\mathcal{E}_{F(\mathcal{C})}(\pi_{1}(G/F(\mathcal{C})))\cong\pi_{1}(G/F(\mathcal{C}))/p_{F(\mathcal{C})}^{\sharp}\pi_{1}(G).$

We finish the proof of the theorem by verifying the equivalence of (i) and
(ii) from statement (4). Assume, first, that $F(\mathcal{C})=K$. Then
$K\subseteq G_{a}$ by virtue of (3). Moreover, from statement (1) and from
Lemma 5.8(iii) it follows that

$\pi_{1}(G/K)=\mathcal{Q}_{K}(K)=\mathcal{Q}_{K}\mathcal{E}_{K}(\xi^{\sharp}\pi_{1}(X)+p_{K}^{\sharp}\pi_{1}(G))=\xi^{\sharp}\pi_{1}(X)+p_{K}^{\sharp}\pi_{1}(G),$

which shows that (i) implies (ii). To verify that (ii) implies (i) we invoke
statement (1) once more to obtain

$F(\mathcal{C})=\mathcal{E}_{K}(\xi^{\sharp}\pi_{1}(X)+p_{K}^{\sharp}\pi_{1}(G))=\mathcal{E}_{K}(\pi_{1}(G/K))=K\cap
G_{a}=K.$

This verifies statement (4). ∎

Before formulating our next result let us recall some facts and notation.
Given $G\in\mathsf{CAGp}$ connected second countable, there is a topological
direct sum $G=\mathfrak{f}(G)\oplus\mathfrak{t}(G)$, where $\mathfrak{f}(G)$
is a torus and $\mathfrak{t}(G)$ is a torus-free group (that is, $\mathop{\rm
Hom}(\mathbb{T}^{1},\mathfrak{t}(G))=0$). The group $\mathfrak{f}(G)$ is
unique and is called the maximal torus of $G$. The group $\mathfrak{t}(G)$ is
unique up to a topological isomorphism. Given, in addition, a topological
space $X$ and a continuous map $\xi\colon X\to G$, we write
$\mathfrak{f}(\xi)$ and $\mathfrak{t}(\xi)$ for the projections of $\xi$ to
$\mathfrak{f}(G)$ and $\mathfrak{t}(G)$, respectively. The maps
$\mathfrak{f}(\xi)$, $\mathfrak{t}(\xi)$ are continuous and satisfy
$\xi=\mathfrak{f}(\xi)\oplus\mathfrak{t}(\xi)$. If $X$ is a pointed space and
the map $\xi$ is base point preserving, then so are the maps
$\mathfrak{f}(\xi)$ and $\mathfrak{t}(\xi)$.

######  Corollary 5.16.

Under the assumptions of Theorem 5.14, if $X$ is a compact connected manifold
and $G\in\mathsf{CAGp}$ is connected second countable then

$F(\mathcal{C})=\overline{\mathcal{E}_{K}}\left(\mathfrak{f}(\xi)^{\sharp}\pi_{1}(X)\right)\subseteq\left(p_{K}^{-1}\mathfrak{f}(G/K)\right)_{0}.$
(5.3)

If, in addition, the group $K$ is finite then

$F(\mathcal{C})\subseteq\mathfrak{f}(G)\cap\left(p_{K}^{-1}\mathfrak{f}(G/K)\right)_{a}.$
(5.4)

######  Remark 5.17.

It follows from the corollary that if $\mathfrak{f}(G/K)=e$ (that is, if $G/K$
contains no torus) then $F(\mathcal{C})=e$ and hence
$\mathcal{C}\in\mathop{\rm{\bf B}_{\mathcal{F}}}(G)$. Indeed, by virtue of
(5.3),

$F(\mathcal{C})\subseteq\left(p_{K}^{-1}\mathfrak{f}(G/K)\right)_{0}=\left(p_{K}^{-1}e\right)_{0}=K_{0}=e.$

We shall have an occasion to use this observation in the situation when $G$ is
a solenoid. In such a case every quotient group $G/K$ of $G$ modulo a totally
disconnected subgroup $K\mathrel{\lhd}G$ is either a solenoid or else is
isomorphic to $\mathbb{T}^{1}$. In the first case $\mathfrak{f}(G/K)=e$ and
hence $F(\mathcal{C})=e$. Thus, in order to obtain a non-trivial totally
disconnected section $F(\mathcal{C})$, it is necessary to have
$G/K\cong\mathbb{T}^{1}$ or, equivalently, $K^{\perp}\cong\mathbb{Z}$, where
$K^{\perp}$ stands for the annihilator of $K$ in $G^{*}$.

###### Proof of Corollary 5.16.

We begin the proof by showing that the maps $\xi,\mathfrak{f}(\xi)\colon X\to
G/K$ are homotopic. To this end, it is sufficient to verify that the map
$\mathfrak{t}(\xi)\colon X\to\mathfrak{t}(G/K)$ is null-homotopic. Since the
group
$\pi^{1}(\mathfrak{t}(G/K))\cong\mathfrak{t}(G/K)^{*}\cong\mathfrak{t}((G/K)^{*})$
is torsion-less (that is, it has no non-trivial free abelian quotient groups)
and the group $\pi^{1}(X)\cong H_{1}^{w}(X)$ is free abelian by our
assumptions on $X$ (see Lemma 5.20 for more details), it follows that the
induced morphism
$\mathfrak{t}(\xi)^{\flat}\colon\pi^{1}(\mathfrak{t}(G/K))\to\pi^{1}(X)$
vanishes. Consequently, the isomorphism (2.15) from Subsection 2.4.10 yields
that the map $\mathfrak{t}(\xi)$ is null-homotopic, as was to be shown. This
shows that $\xi$ and $\mathfrak{f}(\xi)$ are indeed homotopic and so the
induced morphisms
$\xi^{\sharp},\mathfrak{f}(\xi)^{\sharp}\colon\pi_{1}(X)\to\pi_{1}(G/K)$
coincide. Consequently, by virtue of (5.1) from Theorem 5.14,

$F(\mathcal{C})=\overline{\mathcal{E}_{K}}\left(\xi^{\sharp}\pi_{1}(X)\right)=\overline{\mathcal{E}_{K}}\left(\mathfrak{f}(\xi)^{\sharp}\pi_{1}(X)\right),$
(5.5)

which verifies the first part of (5.3). Further, since $\mathfrak{f}(\xi)$
takes its values in $\mathfrak{f}(G/K)$, the group
$\mathcal{E}_{K}(\mathfrak{f}(\xi)^{\sharp}\pi_{1}(X))$ is contained in the
identity arc-component $(p_{K}^{-1}\mathfrak{f}(G/K))_{a}$ of
$p_{K}^{-1}\mathfrak{f}(G/K)$ by definition of $\mathcal{E}_{K}$. Thus, by
virtue of (5.5),

$F(\mathcal{C})=\overline{\mathcal{E}_{K}\left(\mathfrak{f}(\xi)^{\sharp}\pi_{1}(X)\right)}\subseteq\overline{\left(p_{K}^{-1}\mathfrak{f}(G/K)\right)_{a}}=\left(p_{K}^{-1}\mathfrak{f}(G/K)\right)_{0},$

which verifies the second part of (5.3).

Now assume that the group $K$ is finite. Then, by Theorem 5.14(1),

$F(\mathcal{C})=\mathcal{E}_{K}\left(\xi^{\sharp}\pi_{1}(X)\right)=\mathcal{E}_{K}\left(\mathfrak{f}(\xi)^{\sharp}\pi_{1}(X)\right)\subseteq\left(p_{K}^{-1}\mathfrak{f}(G/K)\right)_{a}.$

Finally, to see that $F(\mathcal{C})\subseteq\mathfrak{f}(G)$, fix $g\in
F(\mathcal{C})$. Since $F(\mathcal{C})\subseteq G_{a}=\exp(\mathcal{L}(G))$ by
Theorem 5.14(3), there exist $h\in\mathop{\rm Hom}(\mathbb{R},G)$ and
$t\in\mathbb{R}\setminus 0$ with $g=h(t)$. Let $d$ be the largest torsion
coefficient of $K$. Then $e=g^{d}=h(dt)$ and hence
$dt\mathbb{Z}\subseteq\ker(h)$. Consequently, there is $r\in\mathop{\rm
Hom}(\mathbb{T}^{1},G)$ with $\mathop{\rm im}(r)=\mathop{\rm im}(h)\ni g$.
Since $\mathop{\rm im}(r)\subseteq\mathfrak{f}(G)$, it follows that
$g\in\mathfrak{f}(G)$. This verifies (5.4). ∎

As promised earlier, we conclude this section by justifying the first two
statements from Remark 5.15.

######  Proposition 5.18.

Let $G\in\mathsf{CAGp}$ and $K\mathrel{\lhd}G$ be a totally disconnected
subgroup of $G$. Given a (not necessarily closed) subgroup $E$ of $K$, the
following conditions are equivalent:

  1. 1.

there exist a minimal flow $\mathcal{F}\colon\mathbb{R}\curvearrowright X$ on
a compact (pointed) manifold $X$ and an extension
$\mathcal{C}\in\mathop{\rm{\bf Z}_{\mathcal{F}}}(G)$ with
$p_{K}\mathcal{C}\in\mathop{\rm{\bf B}_{\mathcal{F}}}(G/K)$ such that

$E=\mathcal{E}_{K}\left(\xi^{\sharp}\pi_{1}(X)+p_{K}^{\sharp}\pi_{1}(G)\right),$

where $\xi$ is the transfer function of $p_{K}\mathcal{C}$ with $\xi(z)=e$,

  2. 2.

$E$ is finitely generated in the algebraic sense and is contained in $G_{a}$.

######  Remark 5.19.

In Remark 5.15 we made two claims, which we now justify using Proposition
5.18. Our fist claim concerned the fact that in the situation of Theorem 5.14,
the group $\xi^{\sharp}\pi_{1}(X)+p_{K}^{\sharp}\pi_{1}(G)$ need not
correspond to a closed subgroup of $K$ under $\mathcal{E}_{K}$, and hence the
operations $\mathcal{E}_{K}$ and $\overline{\mathcal{E}_{K}}$ are distinct.
Our second claim concerned the fact that the inclusion
$F(\mathcal{C})\subseteq G_{a}$ from statement (3) of Theorem 5.14 may fail if
the group $K$ is not finite. We present an example justifying both these
statements. Let $G\mathrel{\lhd}\mathbb{T}^{\mathbb{N}}$ be the dyadic
solenoid and $K$ be the set of all $(z_{n})_{n\in\mathbb{N}}\in G$ with
$z_{1}=1$. Clearly, $K\mathrel{\lhd}G$ is totally disconnected. Recall that
the identity arc-component $G_{a}$ of $G$ consists of the sequences
$(z_{n})_{n\in\mathbb{N}}\in G$ satisfying $z_{n}\to 1$ in $\mathbb{T}^{1}$ as
$n\to\infty$. Consequently, the following statements hold:

  * •

$K$ is not contained in $G_{a}$,

  * •

$K\cap G_{a}$ contains a topological generator $k$ of $K$.

Let $E=\langle k\rangle$ be the subgroup of $K$ generated by $k$ in the
algebraic sense. Then $E$ satisfies both assumptions from statement (2) in
Proposition 5.18, and so there exist $\mathcal{F}$, $\mathcal{C}$ and $\xi$ as
in statement (1) of Proposition 5.18 such that
$E=\mathcal{E}_{K}(\xi^{\sharp}\pi_{1}(X)+p_{K}^{\sharp}\pi_{1}(G))$. Further,
by virtue of Theorem 5.14,
$F(\mathcal{C})=\overline{\mathcal{E}_{K}}(\xi^{\sharp}\pi_{1}(X)+p_{K}^{\sharp}\pi_{1}(G))=\overline{E}=K$.
We thus have the following conclusions:

  1. (i)

the group $\mathcal{E}_{K}(\xi^{\sharp}\pi_{1}(X)+p_{K}^{\sharp}\pi_{1}(G))$
is not closed in $K$; indeed, we have $E\subsetneq\overline{E}$, for $E$ is
contained in $G_{a}$, while $\overline{E}=K$ is not,

  2. (ii)

the group $F(\mathcal{C})=K$ is not contained in $G_{a}$.

###### Proof of Proposition 5.18.

Assume that condition (1) holds. Since $X$ is a compact connected manifold,
its fundamental group $\pi_{1}(X)$ is finitely generated and hence so is the
group $\xi^{\sharp}\pi_{1}(X)$. Let $f_{1},\dots,f_{m}$ be a family of
generators for $\xi^{\sharp}\pi_{1}(X)$. Further, for $1\leq i\leq m$, let
$k_{i}$ be the endpoint of the lift of $f_{i}$ across $p_{K}$ starting at $e$.
Then for every $f\in\xi^{\sharp}\pi_{1}(X)$, the endpoint of the lift of $f$
across $p_{K}$ starting at $e$ is expressible as a product of the integer
powers of the elements $k_{1},\dots,k_{m}$. This shows that $E\subseteq\langle
k_{1},\dots,k_{m}\rangle$ is finitely generated. Also, the inclusion
$E\subseteq G_{a}$ follows from Lemma 5.8(ii) and so (1) implies (2).

Now assume that condition (2) holds and let $k_{1},\dots,k_{m}$ be a family of
algebraic generators of $E$. Since $E\subseteq G_{a}$ by the assumptions, it
follows that for every $i=1,\dots,m$ there is $f_{i}\in\pi_{1}(G/K)$ such that
the lift of $f_{i}$ across $p_{K}$ starting at $e$ ends at $k_{i}$. By [38,
Theorem 8.57, p. 420] we may assume that $f_{i}$ is of the form
$f_{i}=h_{i}p$, where $p\colon[0,1]\to\mathbb{T}^{1}$ is the restriction of
the standard covering morphism $\mathbb{R}\to\mathbb{T}^{1}$ and
$h_{i}\in\mathop{\rm Hom}(\mathbb{T}^{1},G/K)$. Clearly,
$\mathcal{E}_{K}(\langle f_{1},\dots,f_{m}\rangle)=\langle
k_{1},\dots,k_{m}\rangle=E$.

Let $X=\mathbb{T}^{m}$ have as a base point its identity $e$ and consider the
base point preserving map $\xi\colon\mathbb{T}^{m}\to G/K$ be defined by
$\xi(z_{1},\dots,z_{m})=h_{1}(z_{1})\dots h_{m}(z_{m})$. We claim that
$\mathcal{E}_{K}(\xi^{\sharp}\pi_{1}(X)+p_{K}^{\sharp}\pi_{1}(G))=E$. Given
$i\in\\{1,\dots,m\\}$, let
$p_{i}=(1,\dots,1,p,1,\dots,1)\in\pi_{1}(\mathbb{T}^{m})$ be the loop having
$p$ as its $i^{\text{th}}$ coordinate. Then, clearly,
$\pi_{1}(\mathbb{T}^{m})=\langle p_{1},\dots,p_{m}\rangle$. Moreover,
$\xi^{\sharp}(p_{i})=\xi p_{i}=h_{i}p=f_{i}$ for every $i$ and hence
$\xi^{\sharp}\pi_{1}(\mathbb{T}^{m})=\langle f_{1},\dots,f_{m}\rangle$.
Consequently,

$\mathcal{E}_{K}\left(\xi^{\sharp}\pi_{1}(\mathbb{T}^{m})+p_{K}^{\sharp}\pi_{1}(G)\right)=\mathcal{E}_{K}\left(\xi^{\sharp}\pi_{1}(\mathbb{T}^{m})\right)=\mathcal{E}_{K}(\langle
f_{1},\dots,f_{m}\rangle)=E.$

To finish the proof of (1), let $\mathcal{F}$ be, say, an irrational
continuous flow on $\mathbb{T}^{m}$ and consider the coboundary $\mathop{\rm
co}(\xi)\in\mathop{\rm{\bf B}_{\mathcal{F}}}(G/K)$ over $\mathcal{F}$ with the
transfer function $\xi$. Since the acting group $\mathbb{R}$ of $\mathcal{F}$
is simply connected, we have
$\xi^{\sharp}\mathcal{F}_{z}^{\sharp}\pi_{1}(\mathbb{R})=0$. Thus, by
Corollary 5.7, $\mathop{\rm co}(\xi)$ lifts across $p_{K}$ to an extension
$\mathcal{C}\in\mathop{\rm{\bf Z}_{\mathcal{F}}}(G)$. Hence $\mathcal{F}$ and
$\mathcal{C}$ are as in (1), which shows that (2) implies (1). ∎

###  5.3 Existence of totally disconnected sections

Given a minimal flow $\mathop{\rm\mathcal{F}\colon\Gamma\curvearrowright X}$,
a group $G\in\mathsf{CAGp}$ and a closed subgroup $K$ of $G$, does there exist
an extension $\mathcal{C}\in\mathop{\rm{\bf Z}_{\mathcal{F}}}(G)$ with
$F(\mathcal{C})=K$? In this section we use our results from Section 5.2 to
give a necessary and sufficient condition for the existence of such an
extension $\mathcal{C}$ in the case when the group $K$ is totally disconnected
(see Thoerem 5.22) and, in particular, when $K$ is finite (see Corollary
5.24). The condition involves ranks of the groups $H_{1}^{w}(\mathcal{F})$ and
$H_{1}^{w}(X)$, the elementary divisors of $H_{1}^{w}(\mathcal{F})$ in
$H_{1}^{w}(X)$, and the number and orders of topological generators of $K$
lying on the arc-component $G_{a}$ of $G$.

######  Lemma 5.20.

Let $X$ be a pointed compact connected manifold with the base point $z$.
Consider the map $\lambda\colon\pi_{1}(X)\ni l\mapsto\lambda(l)\in\mathop{\rm
Hom}(\pi^{1}(X),\pi_{1}(\mathbb{T}^{1}))$ with $\lambda(l)\colon\pi^{1}(X)\ni
f\mapsto f^{\sharp}(l)=fl\in\pi_{1}(\mathbb{T}^{1})$. Then $\lambda$ is an
epimorphism of groups with the kernel $\ker(\lambda)=\ker(p_{X})$, where
$p_{X}\colon\pi_{1}(X)\to H_{1}^{w}(X)$ is the canonical quotient morphism.
Thus, $\lambda$ induces an isomorphism $\nu\colon H_{1}^{w}(X)\to\mathop{\rm
Hom}(\pi^{1}(X),\pi_{1}(\mathbb{T}^{1}))$ with $\nu p_{X}=\lambda$.

###### Proof.

Before turning to the proof let us mention that we shall use the standard
identifications $\pi_{1}(\mathbb{T}^{1})\cong
H_{1}^{w}(\mathbb{T}^{1})\cong\mathbb{Z}$ and that the group operation of
$\pi^{1}(X)$ will be written additively. Moreover, given $f\in
C_{z}(X,\mathbb{T}^{1})$, we shall use the same symbol $f^{\sharp}$ for the
induced morphisms $\pi_{1}(X)\to\mathbb{Z}$ and $H_{1}^{w}(X)\to\mathbb{Z}$.
Finally, a map $f\in C_{z}(X,\mathbb{T}^{1})$ will be sometimes identified
with its homotopy class in $\pi^{1}(X)$. We divide the proof of the lemma into
three steps.

_1st step._ We show that the mapping

$\sigma\colon\pi^{1}(X)\ni f\mapsto f^{\sharp}\in\mathop{\rm
Hom}(\pi_{1}(X),\mathbb{Z})$ (5.6)

is an isomorphism of groups.

Recall from Subsection 2.4.9 that there is a morphism of groups

$\varrho\colon C_{z}(X,\mathbb{T}^{1})\ni f\mapsto f^{\sharp}\in\mathop{\rm
Hom}(\pi_{1}(X),\mathbb{Z}).$

Since the induced morphism $f^{\sharp}\colon\pi_{1}(X)\to\mathbb{Z}$ depends
only on the homotopy class of $f\in C_{z}(X,\mathbb{T}^{1})$, $\varrho$ gives
rise to a morphism $\sigma$ as in (5.6). By our assumptions on $X$, $\varrho$
is an epimorphism and hence so is $\sigma$. To see that $\sigma$ is in fact an
isomorphism observe that if $f\in C_{z}(X,\mathbb{T}^{1})$ satisfies
$\varrho(f)=f^{\sharp}=0$ then $f$ lifts across the usual covering morphism
$p\colon\mathbb{R}\to\mathbb{T}^{1}$ to an element of $C_{z}(X,\mathbb{R})$
and hence $f=0$ as an element of $\pi^{1}(X)$.

_2nd step._ By our assumptions on $X$, $\pi_{1}(X)$ is a finitely generated
group and $H_{1}^{w}(X)$ is thus a free abelian group with a finite rank $n$.
Assume that $n\neq 0$ and choose $l_{1},\dots,l_{n}\in\pi_{1}(X)$ in such a
way that $p_{X}(l_{1}),\dots,p_{X}(l_{n})$ form a basis for $H_{1}^{w}(X)$.
Use the first step of the proof to find $f_{1},\dots,f_{n}\in\pi^{1}(X)$ with
$f_{i}^{\sharp}(l_{j})=\delta_{ij}$ for $i,j=1,\dots,n$. We show that
$f_{1},\dots,f_{n}$ form a basis for $\pi^{1}(X)$. (Observe that, by the first
step of the proof,

$\pi^{1}(X)\cong\mathop{\rm Hom}(\pi_{1}(X),\mathbb{Z})\cong\mathop{\rm
Hom}(H_{1}^{w}(X),\mathbb{Z})\cong\mathop{\rm
Hom}(\mathbb{Z}^{n},\mathbb{Z})\cong\mathbb{Z}^{n}$

is a free abelian group with rank $n$.)

To see that $f_{1},\dots,f_{n}$ are independent in $\pi^{1}(X)$, let
$k_{1},\dots,k_{n}$ be integers with $\sum_{i=1}^{n}k_{i}f_{i}=0$. By applying
the isomorphism $\sigma$ we obtain $\sum_{i=1}^{n}k_{i}f_{i}^{\sharp}=0$. By
our choice of the maps $f_{i}$ it follows at once that $k_{1}=\dots=k_{n}=0$.

To see that $f_{1},\dots,f_{n}$ generate $\pi^{1}(X)$, fix $f\in\pi^{1}(X)$.
For $i=1,\dots,n$ write $k_{i}=f^{\sharp}(l_{i})$ and set
$g=\sum_{i=1}^{n}k_{i}f_{i}\in\pi^{1}(X)$. Then
$g^{\sharp}(l_{j})=k_{j}=f^{\sharp}(l_{j})$ for every $j=1,\dots,n$ and hence,
by our choice of the loops $l_{1},\dots,l_{n}$,
$\sigma(g)=g^{\sharp}=f^{\sharp}=\sigma(f)$. Since $\sigma$ is an isomorphism
by the first step of the proof, it follows that
$f=g=\sum_{i=1}^{n}k_{i}f_{i}$.

_3rd step._ We finish the proof of the lemma by showing that $\lambda$ is an
epimorphism with the kernel $\ker(\lambda)=\ker(p_{X})$.

First, it is clear that $\lambda(l)\in\mathop{\rm Hom}(\pi^{1}(X),\mathbb{Z})$
for every $l\in\pi_{1}(X)$ and so $\lambda$ is well defined. An elementary
argument also shows that $\lambda$ is a morphism of groups. To see that
$\lambda$ is an epimorphism assume, first, that $H_{1}^{w}(X)=0$. Then
$f^{\sharp}=0$ for every $f\in C_{z}(X,\mathbb{T}^{1})$ and hence
$\pi^{1}(X)=0$. Thus, in this case, $\lambda=0$ is indeed an epimorphism. So
assume that $\mathop{\rm rank}(H_{1}^{w}(X))=n\neq 0$ and fix $h\in\mathop{\rm
Hom}(\pi^{1}(X),\mathbb{Z})$. We invoke the notation introduced in the second
step of the proof. Write $m_{i}=h(f_{i})$ for $i=1,\dots,n$ and set
$l=l_{1}^{m_{1}}*\dots*l_{n}^{m_{n}}\in\pi_{1}(X)$. Then

$\lambda(l)(f_{i})=f_{i}^{\sharp}(l)=f_{i}^{\sharp}(l_{1}^{m_{1}}*\dots*l_{n}^{m_{n}})=\sum_{j=1}^{n}m_{j}f_{i}^{\sharp}(l_{j})=m_{i}=h(f_{i})$

for every $i=1,\dots,n$ and, since $f_{1},\dots,f_{n}$ generate $\pi^{1}(X)$
by the second step of the proof, it follows that $h=\lambda(l)$. Thus,
$\lambda$ is indeed an epimorphism.

We show that $\ker(\lambda)=\ker(p_{X})$. To this end, fix $l\in\pi_{1}(X)$.
By our choice of $l_{1},\dots,l_{n}$, there are integers $k_{1},\dots,k_{n}$
with $p_{X}(l)=\sum_{i=1}^{n}k_{i}p_{X}(l_{i})$ in $H_{1}^{w}(X)$. Then the
following conditions are equivalent by definition of $f_{1},\dots,f_{n}$ and
$l_{1},\dots,l_{n}$:

  * •

$l\in\ker(\lambda)$,

  * •

$f_{i}^{\sharp}(l)=0$ for every $i$,

  * •

$f_{i}^{\sharp}(p_{X}(l))=0$ for every $i$,

  * •

$k_{i}=0$ for every $i$,

  * •

$p_{X}(l)=0$,

  * •

$l\in\ker(p_{X})$.

Thus, $\ker(\lambda)=\ker(p_{X})$, as was to be shown. ∎

######  Lemma 5.21.

Let $X$ be a pointed compact connected manifold with the base point $z$ and
let $G\in\mathsf{CAGp}$ be connected. Given $h\in\mathop{\rm
Hom}(\pi_{1}(X),\pi_{1}(G))$, there exists $\xi\in C_{z}(X,G)$ with
$h=\xi^{\sharp}$. Such a map $\xi$ is unique up to a homotopy.

###### Proof.

In the proof of the lemma we shall use the usual identification
$\pi_{1}(\mathbb{T}^{1})\cong\mathbb{Z}$. Further, recall from Subsection
2.4.10 that by connectedness of $G$, every element of $\pi^{1}(G)$ (considered
as a homotopy class) contains one and only one character of $G$. Hence
$G^{*}\cong\pi^{1}(G)$. Finally, we invoke the isomorphism
$\mu\colon\pi_{1}(G)\to\mathop{\rm Hom}(G^{*},\mathbb{Z})$, acting by the rule
$\mu(l)\colon\chi\mapsto\chi^{\sharp}(l)=\chi l$ for $l\in\pi_{1}(G)$ and
$\chi\in G^{*}$.

Now we turn to the proof of the lemma, proceeding in four steps. Since the
group $\pi_{1}(G)\cong\mathop{\rm Hom}(G^{*},\mathbb{Z})$ is abelian and
torsion-free, there is an isomorphism of groups $\mathop{\rm
Hom}(\pi_{1}(X),\pi_{1}(G))\cong\mathop{\rm Hom}(H_{1}^{w}(X),H_{1}^{w}(G))$,
and the statement of the lemma is thus true if $H_{1}^{w}(X)=0$. We shall
therefore assume throughout the whole proof that

$\mathop{\rm rank}(H_{1}^{w}(X))=n\neq 0.$

_1st step._ By Lemma 5.20, the map $\lambda$ defined therein is an epimorphism
of groups with the kernel $\ker(\lambda)=\ker(p_{X})$. Since $\mathop{\rm
Hom}(G^{*},\mathbb{Z})$ is an abelian torsion-free group, we have $\ker(\mu
h)\supseteq\ker(p_{X})=\ker(\lambda)$ and so there is a morphism
$\kappa\colon\mathop{\rm Hom}(\pi^{1}(X),\mathbb{Z})\to\mathop{\rm
Hom}(G^{*},\mathbb{Z})$ with $\kappa\lambda=\mu h$. We show that
$\kappa=\mathop{\rm Hom}(\varrho,\mathbb{Z})$ for an appropriate
$\varrho\in\mathop{\rm Hom}(G^{*},\pi^{1}(X))$.

It follows from our proof of Lemma 5.20 that $\pi^{1}(X)\cong H_{1}^{w}(X)$ is
a free abelian group with rank $n\neq 0$. Let $f_{1},\dots,f_{n}$ be a basis
for $\pi^{1}(X)$ and $\varphi_{1},\dots,\varphi_{n}$ be the corresponding dual
basis for $\mathop{\rm Hom}(\pi^{1}(X),\mathbb{Z})$. (That is,
$\varphi_{i}(f_{j})=\delta_{ij}$ for $i,j=1,\dots,n$.) Write
$h_{i}=\kappa(\varphi_{i})\in\mathop{\rm Hom}(G^{*},\mathbb{Z})$ for
$i=1,\dots,n$ and set $\varrho\colon
G^{*}\ni\chi\mapsto\sum_{i=1}^{n}h_{i}(\chi)f_{i}\in\pi^{1}(X)$. Then
$\varrho$ is clearly a morphism of groups and we claim that
$\kappa=\mathop{\rm Hom}(\varrho,\mathbb{Z})$. Given $j\in\\{1,\dots,n\\}$ and
$\chi\in G^{*}$, we have

$\mathop{\rm
Hom}(\varrho,\mathbb{Z})(\varphi_{j})\colon\chi\mapsto\varphi_{j}(\varrho(\chi))=\varphi_{j}\left(\sum_{i=1}^{n}h_{i}(\chi)f_{i}\right)=h_{j}(\chi)=\kappa(\varphi_{j})(\chi).$

Hence $\mathop{\rm Hom}(\varrho,\mathbb{Z})(\varphi_{j})=\kappa(\varphi_{j})$
for every $j=1,\dots,n$ and this shows that $\mathop{\rm
Hom}(\varrho,\mathbb{Z})=\kappa$.

_2nd step._ We show that there is $\xi\in C_{z}(X,G)$ with
$\xi^{\flat}=\varrho$, where $\pi^{1}(G)$ is identified with $G^{*}$ in the
usual way.

Recall from Subsection 2.4.7 that there is an isomorphism of groups

$C_{z}(X,G)\ni\xi\mapsto\xi^{*}\in\mathop{\rm
Hom}(G^{*},C_{z}(X,\mathbb{T}^{1})),$ (5.7)

where $\xi^{*}$ acts by the rule $\xi^{*}(\chi)=\chi\xi\in
C_{z}(X,\mathbb{T}^{1})$ for $\chi\in G^{*}$. Since $\pi^{1}(X)$ can be viewed
(by one of its equivalent definitions) as a subgroup of
$C_{z}(X,\mathbb{T}^{1})$, we have $\varrho=\xi^{*}$ for an appropriate
$\xi\in C_{z}(X,G)$. Then $\varrho=\xi^{*}=\xi^{\flat}$, as was to be shown.

_3rd step._ We show that the map $\xi$ from the second step of the proof
satisfies the conclusion of the lemma.

By our choice of $\xi$ and by definition of $\kappa=\mathop{\rm
Hom}(\varrho,\mathbb{Z})$, we have $\mathop{\rm
Hom}(\xi^{\flat},\mathbb{Z})\lambda=\mu h$. Further, for every
$l\in\pi_{1}(X)$,

$\mathop{\rm Hom}(\xi^{\flat},\mathbb{Z})\lambda\colon
l\mapsto\lambda(l)\xi^{\flat}\colon
G^{*}\ni\chi\mapsto\chi^{\sharp}\xi^{\sharp}(l)\in\mathbb{Z}\cong\pi_{1}(\mathbb{T}^{1})$

and

$\mu h\colon l\mapsto\mu(h(l))\colon
G^{*}\ni\chi\mapsto\chi^{\sharp}(h(l))\in\mathbb{Z}\cong\pi_{1}(\mathbb{T}^{1}).$

To summarize, if $l\in\pi_{1}(X)$ then
$\chi^{\sharp}\xi^{\sharp}(l)=\chi^{\sharp}h(l)$ holds for every $\chi\in
G^{*}$, and hence, since the morphisms $\chi^{\sharp}$ ($\chi\in G^{*}$)
separate the points of $\pi_{1}(G)$ (see Subsection 2.4.10), it follows that
$\xi^{\sharp}(l)=h(l)$. Thus, $\xi^{\sharp}=h$, as was to be shown.

_4th step._ We finish the proof of the lemma by verifying the uniqueness
statement.

First recall the following facts:

  1. (i)

there is an epimorphism $C_{z}(X,\mathbb{T}^{1})\ni f\mapsto
f^{\sharp}\in\mathop{\rm Hom}(\pi_{1}(X),\pi_{1}(\mathbb{T}^{1}))$, whose
kernel consists of the group $C_{z}(X,\mathbb{T}^{1})_{0}$ of the null-
homotopic maps; consequently, if two maps $f_{1},f_{2}\in
C_{z}(X,\mathbb{T}^{1})$ satisfy $f_{1}^{\sharp}=f_{2}^{\sharp}$ then they are
homotopic,

  2. (ii)

there is an isomorphism $[X,G]\ni[\xi]\mapsto\xi^{\flat}\in\mathop{\rm
Hom}(G^{*},\pi^{1}(X))$; consequently, if $\xi_{1},\xi_{2}\in C_{z}(X,G)$ are
such that $\chi\xi_{1},\chi\xi_{2}$ are homotopic for every $\chi\in G^{*}$,
then $\xi_{1},\xi_{2}$ are homotopic.

Now let $\xi_{1},\xi_{2}\in C_{z}(X,G)$ satisfy
$\xi_{1}^{\sharp}=\xi_{2}^{\sharp}$. Then, for every $\chi\in G^{*}$,
$(\chi\xi_{1})^{\sharp}=\chi^{\sharp}\xi_{1}^{\sharp}=\chi^{\sharp}\xi_{2}^{\sharp}=(\chi\xi_{2})^{\sharp}$
and hence $\chi\xi_{1}$, $\chi\xi_{2}$ are homotopic by virtue of (i). Thus,
by virtue of (ii), the maps $\xi_{1},\xi_{2}$ are homotopic. This verifies the
uniqueness statement of the lemma. ∎

######  Theorem 5.22.

Let $\mathop{\rm\mathcal{F}\colon\Gamma\curvearrowright X}$ be a minimal flow
with $\Gamma\in\mathsf{LieGp}$ connected and with $X$ a compact connected
manifold. Write $n=\mathop{\rm rank}(H_{1}^{w}(\mathcal{F}))$,
$n+m=\mathop{\rm rank}(H_{1}^{w}(X))$ and let $d_{1},\dots,d_{n}$ be the
elementary divisors of $H_{1}^{w}(\mathcal{F})$ in $H_{1}^{w}(X)$. Assume that
$G\in\mathsf{CAGp}$ is connected and $K\mathrel{\lhd}G$ is totally
disconnected. Then the following conditions are equivalent:

  1. (1)

there is $\mathcal{C}\in\mathop{\rm{\bf Z}_{\mathcal{F}}}(G)$ with
$F(\mathcal{C})=K$,

  2. (2)

there exist $k_{1},\dots,k_{n+m}\in K\cap G_{a}$ with $K=\overline{\langle
k_{1},\dots,k_{n+m}\rangle}$ and $k_{i}^{d_{i}}=e$ for $i=1,\dots,n$.

######  Remark 5.23.

We wish to add the following observations.

  * •

The points $k_{1},\dots,k_{n+m}$ from condition (2) are not required to be
independent in any sense (in particular, some of them may equal the identity
$e$ of $K$). The condition requires that the group $K$ possess a subset $S$ of
$G_{a}$ of cardinality at most $n+m$, which generates $K$ topologically. There
are no restrictions on $m$ of these generators, but the remaining ones (if
any) must be torsion elements of $K$ with orders dividing the elementary
divisors of $H_{1}^{w}(\mathcal{F})$ in $H_{1}^{w}(X)$.

  * •

It follows from the theorem that if the flow $\mathcal{F}$ does not possess a
free cycle (that is, if $m=0$) then every totally disconnected section
$F(\mathcal{C})$ of an extension $\mathcal{C}\in\mathop{\rm{\bf
Z}_{\mathcal{F}}}(G)$ is necessarily a finite subgroup of $G_{a}$. Moreover,
in such a situation the group $F(\mathcal{C})$ is a quotient group of
$\mathbb{Z}_{d_{1}}\oplus\dots\oplus\mathbb{Z}_{d_{n}}$, see Corollary 5.24.

  * •

If, on the other hand, we have $n=0$ (that is, $H_{1}^{w}(\mathcal{F})=0$),
then $K=F(\mathcal{C})$ for some $\mathcal{C}\in\mathop{\rm{\bf
Z}_{\mathcal{F}}}(G)$ if and only if $K\cap G_{a}$ has a subset with
cardinality at most $m$, which generates $K$ topologically.

  * •

Assume that $n+m=0$, that is, $H_{1}^{w}(X)=0$. Given $\xi\in C_{z}(X,G/K)$,
we have $\xi^{\sharp}\pi_{1}(X)=0$ and hence, by Lemma 5.5, $\xi$ lifts across
$p_{K}$ to an element of $C_{z}(X,G)$. Consequently, if
$\mathcal{C}\in\mathop{\rm{\bf Z}_{\mathcal{F}}}(G)$ satisfies
$F(\mathcal{C})\subseteq K$ then $\mathcal{C}\in\mathop{\rm{\bf
B}_{\mathcal{F}}}(G)$ by virtue of Lemma 5.1. Thus, in this case, condition
(2) from Theorem 5.22 takes the form of the identity $K=e$. In the proof of
the theorem we shall restrict to the (non-trivial) case of $H_{1}^{w}(X)\neq
0$.

###### Proof of Theorem 5.22.

We divide the proof into three steps. To simplify notation we shall
occasionally identify every loop $f\in\pi_{1}(X)$ with its underlying cycle
$p_{X}(f)\in H_{1}^{w}(X)$. Also, by virtue of part four of Remark 5.23, we
shall assume that $H_{1}^{w}(X)\neq 0$, that is, $n+m\neq 0$.

_1st step._ Let $p_{K}\colon G\to G/K$ denote the canonical quotient morphism.
Fix loops $f_{1},\dots,f_{n+m}\in\pi_{1}(X)$ in such a way that
$p_{X}(f_{1}),\dots,p_{X}(f_{n+m})$ form a basis for $H_{1}^{w}(X)$ and
$d_{1}p_{X}(f_{1}),\dots,d_{n}p_{X}(f_{n})$ form a basis for
$H_{1}^{w}(\mathcal{F})$. Given a continuous base point preserving map
$\xi\colon X\to G/K$, let $k_{i}$ ($i=1,\dots,n+m$) be the endpoint of the
lift of $\xi f_{i}$ across $p_{K}$ starting at $e$. (Recall from Lemma 5.5 and
Remark 5.6 that the paths in $G/K$ do lift across the morphism $p_{K}$.) We
claim that the following conditions are equivalent:

  1. ($\alpha$)

$\mathop{\rm co}(\xi)$ lifts across $p_{K}$ to an extension
$\mathcal{C}\in\mathop{\rm{\bf Z}_{\mathcal{F}}}(G)$,

  2. ($\beta$)

$k_{i}^{d_{i}}=e$ for $i=1,\dots,n$.

Given $i\in\\{1,\dots,n\\}$, let $\widetilde{f}_{i}$ be the lift of $\xi
f_{i}$ across $p_{K}$ starting at $e$. Then
$\widetilde{g}_{i}:=\widetilde{f}_{i}*k_{i}\widetilde{f}_{i}*\dots*k_{i}^{d_{i}-1}\widetilde{f}_{i}$
is the lift of $d_{i}(\xi f_{i})$ across $p_{K}$ starting at $e$ and its
endpoint is $k_{i}^{d_{i}}$. Thus, by virtue of Corollary 5.7, Lemma 5.5 and
Lemma 2.17, the following conditions are equivalent:

  * •

$\mathop{\rm co}(\xi)$ lifts across $p_{K}$ to an element of $\mathop{\rm{\bf
Z}_{\mathcal{F}}}(G)$,

  * •

$\xi^{\sharp}H_{1}^{w}(\mathcal{F})\subseteq p_{K}^{\sharp}H_{1}^{w}(G)$,

  * •

$\xi^{\sharp}(d_{i}p_{X}(f_{i}))\in p_{K}^{\sharp}H_{1}^{w}(G)$ for
$i=1,\dots,n$,

  * •

$d_{i}\xi^{\sharp}(f_{i})=\xi^{\sharp}(d_{i}f_{i})\in
p_{K}^{\sharp}\pi_{1}(G)$ for $i=1,\dots,n$,

  * •

$d_{i}(\xi f_{i})$ lifts across $p_{K}$ to a loop based at $e$ for
$i=1,\dots,n$,

  * •

$\widetilde{g}_{i}(1)=e$ for $i=1,\dots,n$,

  * •

$k_{i}^{d_{i}}=e$ for $i=1,\dots,n$.

This verifies the equivalence of ($\alpha$) and ($\beta$).

_2nd step._ We show that (2) follows from (1).

Let $\mathcal{C}\in\mathop{\rm{\bf Z}_{\mathcal{F}}}(G)$ satisfy
$F(\mathcal{C})=K$ and denote by $\xi$ the continuous base point preserving
map $X\to G/K$ with $p_{K}\mathcal{C}=\mathop{\rm co}(\xi)$. For
$i=1,\dots,n+m$ let $k_{i}$ be the endpoint of the lift of $\xi f_{i}$ across
$p_{K}$ starting at $e$. Then $k_{1},\dots,k_{n+m}\in K\cap G_{a}$ and, by the
equivalence of ($\alpha$) and ($\beta$) from the first step of the proof,
$k_{i}^{d_{i}}=e$ for $i=1,\dots,n$. Finally, since the group $\pi_{1}(G/K)$
is abelian torsion-free, we have
$\xi^{\sharp}\pi_{1}(X)=\langle\xi^{\sharp}(f_{1}),\dots,\xi^{\sharp}(f_{n+m})\rangle$
and hence, by virtue of Theorem 5.14,

$K=F(\mathcal{C})=\overline{\mathcal{E}_{K}}(\xi^{\sharp}\pi_{1}(X))=\overline{\mathcal{E}_{K}}(\langle\xi^{\sharp}(f_{1}),\dots,\xi^{\sharp}(f_{n+m})\rangle)=\overline{\langle
k_{1},\dots,k_{n+m}\rangle},$

which verifies condition (2).

_3rd step._ We show that (1) follows from (2).

Fix elements $k_{1},\dots,k_{n+m}$ of $K$ as in (2). For $i=1,\dots,n+m$ let
$\widetilde{f}_{i}$ be a path in $G$ from $e$ to $k_{i}$. Since $H_{1}^{w}(X)$
is a free abelian group with basis $f_{1},\dots,f_{n+m}$ and the group
$\pi_{1}(G/K)\cong H_{1}^{w}(G/K)$ is abelian, there is a morphism
$h\colon\pi_{1}(X)\to\pi_{1}(G/K)$ with $h(f_{i})=p_{K}\widetilde{f}_{i}$ for
$i=1,\dots,n+m$. By Lemma 5.21, $h=\xi^{\sharp}$ for an appropriate continuous
base point preserving map $\xi\colon X\to G/K$. Clearly, $k_{i}$
($i=1,\dots,n+m$) is the endpoint of the lift of $\xi f_{i}$ across $p_{K}$
starting at $e$. Moreover, given $i\in\\{1,\dots,n\\}$, we have
$k_{i}^{d_{i}}=e$ by the assumptions. Consequently, it follows from the
equivalence of ($\alpha$) and ($\beta$) from the first step of the proof that
$\mathop{\rm co}(\xi)$ lifts across $p_{K}$ to an extension
$\mathcal{C}\in\mathop{\rm{\bf Z}_{\mathcal{F}}}(G)$. By virtue of Theorem
5.14,

$F(\mathcal{C})=\overline{\mathcal{E}_{K}}\left(\xi^{\sharp}\pi_{1}(X)\right)=\overline{\mathcal{E}_{K}}(\langle
p_{K}\widetilde{f}_{1},\dots,p_{K}\widetilde{f}_{n+m}\rangle)=\overline{\langle
k_{1},\dots,k_{n+m}\rangle}=K.$

This verifies condition (1). ∎

######  Corollary 5.24.

Let $\mathop{\rm\mathcal{F}\colon\Gamma\curvearrowright X}$ be a minimal flow
with $\Gamma\in\mathsf{LieGp}$ connected and with $X$ a compact connected
manifold. Write $n=\mathop{\rm rank}(H_{1}^{w}(\mathcal{F}))$,
$n+m=\mathop{\rm rank}(H_{1}^{w}(X))$ and let $d_{1},\dots,d_{n}$ be the
elementary divisors of $H_{1}^{w}(\mathcal{F})$ in $H_{1}^{w}(X)$. Assume that
$G\in\mathsf{CAGp}$ is connected and $K\mathrel{\lhd}G$ is finite. Then the
following conditions are equivalent:

  1. (i)

there is $\mathcal{C}\in\mathop{\rm{\bf Z}_{\mathcal{F}}}(G)$ with
$F(\mathcal{C})=K$,

  2. (ii)

$K\subseteq G_{a}$ is a (finite) quotient group of
$\mathbb{Z}_{d_{1}}\oplus\dots\oplus\mathbb{Z}_{d_{n}}\oplus\mathbb{Z}^{m}$.

###### Proof.

Similarly to the proof of Theorem 5.22, we may restrict to the case of
$H_{1}^{w}(X)\neq 0$. Assume that $\mathcal{C}\in\mathop{\rm{\bf
Z}_{\mathcal{F}}}(G)$ satisfies $F(\mathcal{C})=K$. Then, by Theorem 5.14(3),
$K\subseteq G_{a}$. Moreover, by Theorem 5.22, $K$ possesses a family of
(algebraic) generators $k_{1},\dots,k_{n+m}$ with $k_{i}^{d_{i}}=e$ for
$i=1,\dots,n$. This means that $K$ is a quotient group of
$\mathbb{Z}_{d_{1}}\oplus\dots\oplus\mathbb{Z}_{d_{n}}\oplus\mathbb{Z}^{m}$.
Thus, (ii) follows from (i).

Now assume that $K\subseteq G_{a}$ is a quotient group of
$\mathbb{Z}_{d_{1}}\oplus\dots\oplus\mathbb{Z}_{d_{n}}\oplus\mathbb{Z}^{m}$.
Then there is a family of generators $k_{1},\dots,k_{n+m}\in G_{a}$ for $K$
with $k_{i}^{d_{i}}=e$ for $i=1,\dots,n$. By applying Theorem 5.22 once more
we see that $F(\mathcal{C})=K$ holds for an appropriate extension
$\mathcal{C}\in\mathop{\rm{\bf Z}_{\mathcal{F}}}(G)$. Thus, (i) follows from
(ii). ∎

###  5.4 Existence of prescribed sections, part 1

In this section we continue to pursue the problem formulated at the beginning
of Section 5.3. First, we relate the existence of finite (or arbitrary)
sections for extensions with values in a torus $G=\mathbb{T}^{k}$ to the
difference of ranks of the weak homology groups $\mathop{\rm
rank}(H_{1}^{w}(X))-\mathop{\rm rank}(H_{1}^{w}(\mathcal{F}))$; this is done
in Theorem 5.25. Then, in Theorem 5.30, we concentrate on the situation when
$G=S_{{\bf p}}$ is the ${\bf p}$-adic solenoid with an arbitrary sequence
${\bf p}=(p_{n})_{n\in\mathbb{N}}$. First we show that the flow $\mathcal{F}$
must possess a free cycle in order to possess an extension
$\mathcal{C}\in\mathop{\rm{\bf Z}_{\mathcal{F}}}(S_{\bf p})$ with a non-
trivial totally disconnected section $F(\mathcal{C})$. Then, under the
assumption that $\mathcal{F}$ does possess a free cycle, we show that
$\mathop{\rm{\bf Z}_{\mathcal{F}}}(S_{\bf p})$ contains an extension with a
prescribed non-trivial totally disconnected section $K\mathrel{\lhd}S_{\bf p}$
if and only if $K^{\perp}\cong\mathbb{Z}$. Finally, we show that for a flow
$\mathcal{F}$ with a free cycle, the group $\mathop{\rm{\bf
Z}_{\mathcal{F}}}(S_{\bf p})$ contains also minimal extensions.

######  Theorem 5.25.

Let $\mathop{\rm\mathcal{F}\colon\Gamma\curvearrowright X}$ be a minimal flow
with $\Gamma\in\mathsf{LieGp}$ connected and with $X$ a compact connected
manifold, and let $k\in\mathbb{N}$. Then the following conditions are
equivalent:

  1. (1)

for every $K\mathrel{\lhd}\mathbb{T}^{k}$ there is
$\mathcal{C}\in\mathop{\rm{\bf Z}_{\mathcal{F}}}(\mathbb{T}^{k})$ with
$F(\mathcal{C})=K$,

  2. (2)

for every $K\mathrel{\lhd}\mathbb{T}^{k}$ finite there is
$\mathcal{C}\in\mathop{\rm{\bf Z}_{\mathcal{F}}}(\mathbb{T}^{k})$ with
$F(\mathcal{C})=K$,

  3. (3)

$\mathop{\rm rank}(H_{1}^{w}(X))-\mathop{\rm rank}(H_{1}^{w}(\mathcal{F}))\geq
k$.

###### Proof.

Write $n=\mathop{\rm rank}(H_{1}^{w}(\mathcal{F}))$, $n+m=\mathop{\rm
rank}(H_{1}^{w}(X))$ and let $d_{1},\dots,d_{n}=d$ be the elementary divisors
of $H_{1}^{w}(\mathcal{F})$ in $H_{1}^{w}(X)$ written, as usual, in the
increasing order. The implication (1)$\Rightarrow$(2) is clear. To verify that
(2) implies (3), fix an integer $l\geq 2$ co-prime with $d$ and set
$K=(\mathbb{Z}_{l})^{k}\mathrel{\lhd}\mathbb{T}^{k}$. By virtue of (2), there
is $\mathcal{C}\in\mathop{\rm{\bf Z}_{\mathcal{F}}}(\mathbb{T}^{k})$ with
$F(\mathcal{C})=K$. Set
$A=\mathbb{Z}_{d_{1}}\oplus\dots\oplus\mathbb{Z}_{d_{n}}\oplus\mathbb{Z}^{m}$.
By Corollary 5.24, $K$ is a quotient group of $A$. Denote the underlying
quotient morphism $A\to K$ by $q$. Then, by our choice of $l$,
$K=dK=dq(A)=q(dA)=q(d\mathbb{Z}^{m})$. It follows that $m\geq k$, which
verifies (3).

We show that (3) implies (2). So assume that $m\geq k$ and fix a finite
subgroup $K$ of $\mathbb{T}^{k}$. The Pontryagin dual to the inclusion
morphism $K\to\mathbb{T}^{k}$ is (equivalent to) a quotient morphism
$\mathbb{Z}^{k}\to K$. Since $m\geq k$, it follows that $K$ is a quotient
group of $\mathbb{Z}^{m}$ and hence also of
$\mathbb{Z}_{d_{1}}\oplus\dots\oplus\mathbb{Z}_{d_{n}}\oplus\mathbb{Z}^{m}$.
Thus, by Corollary 5.24, there is an extension $\mathcal{C}\in\mathop{\rm{\bf
Z}_{\mathcal{F}}}(\mathbb{T}^{k})$ with $F(\mathcal{C})=K$. This verifies (2).

We finish the proof by showing that (3) implies (1). So assume that (3) holds
and fix a closed subgroup $K$ of $\mathbb{T}^{k}$. We distinguish two cases.

_Case 1._ The group $K\mathrel{\lhd}\mathbb{T}^{k}$ is connected.

In this case $K$ is a torus with dimension at most $k$ and it is therefore a
quotient group of $\mathbb{T}^{k}$. Let $r\colon\mathbb{T}^{k}\to K$ be a
quotient morphism and assume that $\mathcal{C}\in\mathop{\rm{\bf
Z}_{\mathcal{F}}}(\mathbb{T}^{k})$ is minimal (that is, it satisfies
$F(\mathcal{C})=\mathbb{T}^{k}$). Then $r\mathcal{C}\in\mathop{\rm{\bf
Z}_{\mathcal{F}}}(K)\subseteq\mathop{\rm{\bf
Z}_{\mathcal{F}}}(\mathbb{T}^{k})$ and
$F(r\mathcal{C})=rF(\mathcal{C})=r(\mathbb{T}^{k})=K$. Thus, in order to
verify condition (1) in Case 1, it suffices to construct a minimal extension
$\mathcal{C}\in\mathop{\rm{\bf Z}_{\mathcal{F}}}(\mathbb{T}^{k})$.

Condition (3) yields an epimorphism $h\colon H_{1}^{w}(X)\to
H_{1}^{w}(\mathbb{T}^{k})$ with $h(H_{1}^{w}(\mathcal{F}))=0$. By virtue of
Lemma 5.21, $h=\xi^{\sharp}$ for an appropriate continuous base point
preserving map $\xi\colon X\to\mathbb{T}^{k}$. Let
$p\colon\mathbb{R}^{k}\to\mathbb{T}^{k}$ be the usual covering morphism with
kernel $\mathbb{Z}^{k}$. Since
$\xi^{\sharp}\mathcal{F}_{z}^{\sharp}\pi_{1}(\Gamma)=0$ by our choice of
$\xi$, statement (2.30) from Lemma 2.17 yields $\mathop{\rm
co}(\xi)^{\sharp}\pi_{1}(\Gamma\times X)=0$. Consequently, $\mathop{\rm
co}(\xi)$ lifts across $p$ to an extension $\mathcal{D}\in\mathop{\rm{\bf
Z}_{\mathcal{F}}}(\mathbb{R}^{k})$. For every $l\in\mathbb{N}$ set
$\mathcal{C}_{l}=p((1/l)\mathcal{D})\in\mathop{\rm{\bf
Z}_{\mathcal{F}}}(\mathbb{T}^{k})$. Then
$\kappa_{l}\mathcal{C}_{l}=\kappa_{l}p((1/l)\mathcal{D})=p\mathcal{D}=\mathcal{C}$,
where $\kappa_{l}$ stands for the $l$-endomorphism of $\mathbb{T}^{k}$.

We claim that the following statements hold:

  1. (a)

$F(\mathcal{C}_{l})=(\mathbb{Z}_{l})^{k}$ for every $l\in\mathbb{N}$; indeed,
by virtue of Theorem 5.14(1),

$F(\mathcal{C}_{l})=\mathcal{E}_{\ker(\kappa_{l})}(\xi^{\sharp}\pi_{1}(X))=\mathcal{E}_{\ker(\kappa_{l})}(\pi_{1}(\mathbb{T}^{k}))=\ker(\kappa_{l})=(\mathbb{Z}_{l})^{k},$

  2. (b)

$\mathcal{C}_{l}\stackrel{{\scriptstyle ucs}}{{\longrightarrow}}1$ as
$l\to\infty$; this follows from our definition of the extensions
$\mathcal{C}_{l}$ ($l\in\mathbb{N}$) and from an obvious convergence
$(1/l)\mathcal{D}\stackrel{{\scriptstyle ucs}}{{\longrightarrow}}0$ as
$l\to\infty$.

Further, since the groups $F(\mathcal{C}_{l})=(\mathbb{Z}_{l})^{k}$
($l\in\mathbb{N}$) become mutually group-disjoint if we restrict to the prime
numbers $l$, we may apply Theorem 3.23 to obtain an increasing sequence of
prime numbers $(p_{l})_{l\in\mathbb{N}}$ and an extension
$\mathcal{C}\in\mathop{\rm{\bf Z}_{\mathcal{F}}}(\mathbb{T}^{k})$ with

  * •

$\mathcal{C}=\sum_{l=1}^{\infty}\mathcal{C}_{p_{l}}$ u.c.s. in
$\mathop{\rm{\bf Z}_{\mathcal{F}}}(\mathbb{T}^{k})$, and

  * •

$F(\mathcal{C})\supseteq\overline{\sum_{l\in\mathbb{N}}F(\mathcal{C}_{p_{l}})}$.

Since $\mathbb{T}^{k}\supseteq
F(\mathcal{C})\supseteq\overline{\sum_{l\in\mathbb{N}}F(\mathcal{C}_{p_{l}})}=\overline{\sum_{l\in\mathbb{N}}(\mathbb{Z}_{p_{l}})^{k}}=\mathbb{T}^{k}$,
it follows that $F(\mathcal{C})=\mathbb{T}^{k}$ and so the extension
$\mathcal{C}\in\mathop{\rm{\bf Z}_{\mathcal{F}}}(\mathbb{T}^{k})$ is minimal.
This verifies condition (1) in Case 1.

_Case 2._ The group $K\mathrel{\lhd}\mathbb{T}^{k}$ is arbitrary.

The group $K$ can be expressed as a topological direct sum $K=K_{1}\oplus
K_{2}$, where $K_{1}\mathrel{\lhd}\mathbb{T}^{k}$ is connected and
$K_{2}\mathrel{\lhd}\mathbb{T}^{k}$ is finite. From Case 1 we know that
$K_{1}=F(\mathcal{C}_{1})$ for an appropriate
$\mathcal{C}_{1}\in\mathop{\rm{\bf Z}_{\mathcal{F}}}(\mathbb{T}^{k})$.
Moreover, since the equivalence of (2) and (3) has already been established in
an earlier part of the proof, we may apply (2) to obtain
$\mathcal{C}_{2}\in\mathop{\rm{\bf Z}_{\mathcal{F}}}(\mathbb{T}^{k})$ with
$F(\mathcal{C}_{2})=K_{2}$. Now, by virtue of Corollary 3.20,
$F(\mathcal{C}_{1}+\mathcal{C}_{2})=F(\mathcal{C}_{1})+F(\mathcal{C}_{2})=K_{1}\oplus
K_{2}=K$. This verifies statement (1) also in Case 2. ∎

By applying Theorem 5.25 to minimal equicontinuous flows $\mathcal{F}$, we
will be able to enlarge ranges of dense Lie group morphisms in Corollary 5.28.
Before doing so, we prove an auxiliary lemma.

######  Lemma 5.26.

Let $G$ be a compact connected Lie group. Then the map

$\varphi\colon\mathop{\rm
Hom}(G,\mathbb{T}^{1})\ni\xi\mapsto\xi^{\sharp}\in\mathop{\rm
Hom}(\pi_{1}(G),\pi_{1}(\mathbb{T}^{1}))$ (5.8)

is an isomorphism of groups. In particular, given $\varrho\in\mathop{\rm
Hom}(\pi_{1}(G),\pi_{1}(\mathbb{T}^{1}))$, there is $\xi\in\mathop{\rm
Hom}(G,\mathbb{T}^{1})$ with $\varrho=\xi^{\sharp}$.

######  Remark 5.27.

We wish to add the following remarks.

  * •

Recall from Subsection 2.4.9 that there is an epimorphism

$C_{e}(G,\mathbb{T}^{1})\ni f\mapsto f^{\sharp}\in\mathop{\rm
Hom}(\pi_{1}(G),\pi_{1}(\mathbb{T}^{1})),$

which induces an isomorphism between the group $\pi^{1}(G)$ of the homotopy
classes of continuous base point preserving maps $f\colon G\to\mathbb{T}^{1}$
and the group of morphisms
$\varrho\colon\pi_{1}(G)\to\pi_{1}(\mathbb{T}^{1})$. By combining this fact
with the statement of Lemma 5.26, we infer that every continuous base point
preserving map $f\colon G\to\mathbb{T}^{1}$ is homotopic to a unique
topological morphism $\xi\colon G\to\mathbb{T}^{1}$.

  * •

We were not able to find the statement of this lemma explicitly stated in the
literature. Therefore, we present its detailed proof based on results from the
structure theory of compact (Lie) groups.

###### Proof of Lemma 5.26.

A rough idea of the proof is to reduce the problem to the case when $G$ is a
torus, in which case the statement of the lemma clearly holds (see, for
instance, our discussion in Subsections 2.4.9 and 2.4.10). We divide the proof
into three steps.

_1st step._ We start by collecting some preliminary observations.

Let $G^{\prime}$ be the commutator subgroup of $G$. By virtue of [38, Theorem
6.18(i), p. 206], $G^{\prime}$ is a compact connected semi-simple Lie group
and its fundamental group $\pi_{1}(G^{\prime})$ is thus finite by the Weyl’s
theorem (see [5, Corollary 4, p. 285]). Further, by the second structure
theorem for connected compact Lie groups (see [38, Theorem 6.41(i), p. 221]),
there is a (finite-dimensional) torus $T$ such that $G$ is the topological
semi-direct product $T\ltimes G^{\prime}$ of $T$ and $G^{\prime}$; we denote
by $p\colon G\to T$ the associated projection morphism. Under the usual
identification $\pi_{1}(G)=\pi_{1}(T)\oplus\pi_{1}(G^{\prime})$, the induced
morphism $p^{\sharp}$ takes the form of the projection of $\pi_{1}(G)$ onto
$\pi_{1}(T)$ and so it is an epimorphism with the kernel
$\ker(p^{\sharp})=\pi_{1}(G^{\prime})$. Finally, given $\xi\in\mathop{\rm
Hom}(G,\mathbb{T}^{1})$, we have $G^{\prime}\subseteq\ker(\xi)$ by
commutativity of $\mathbb{T}^{1}$ and so $\xi=\xi^{\prime}p$ for a unique
$\xi^{\prime}\in\mathop{\rm Hom}(T,\mathbb{T}^{1})$.

_2nd step._ We show that $\varphi$ is a monomorphism.

Fix $\xi\in\mathop{\rm Hom}(G,\mathbb{T}^{1})$ and assume that
$\varphi(\xi)=\xi^{\sharp}=0$. Then
$0=\xi^{\sharp}=(\xi^{\prime}p)^{\sharp}=(\xi^{\prime})^{\sharp}p^{\sharp}$
and hence $(\xi^{\prime})^{\sharp}=0$ by surjectivity of $p^{\sharp}$. Since
the domain of $\xi^{\prime}$ is a torus, it follows that $\xi^{\prime}=1$ and
so $\xi=\xi^{\prime}p=1$. Thus, $\varphi$ is indeed a monomorphism.

_3rd step._ We finish the proof by showing that $\varphi$ is an epimorphism.

So let $\varrho\in\mathop{\rm Hom}(\pi_{1}(G),\pi_{1}(\mathbb{T}^{1}))$; we
shall find $\xi\in\mathop{\rm Hom}(G,\mathbb{T}^{1})$ with
$\varrho=\xi^{\sharp}$. Since the group $\pi_{1}(G^{\prime})$ is finite and
the group $\pi_{1}(\mathbb{T}^{1})$ is torsion-free,
$\ker(\varrho)\supseteq\pi_{1}(G^{\prime})=\ker(p^{\sharp})$. Consequently,
there is $\vartheta\in\mathop{\rm Hom}(\pi_{1}(T),\pi_{1}(\mathbb{T}^{1}))$
with $\varrho=\vartheta p^{\sharp}$. Let $\zeta\in\mathop{\rm
Hom}(T,\mathbb{T}^{1})$ be such that $\zeta^{\sharp}=\vartheta$ and set
$\xi=\zeta p$. Then $\xi\in\mathop{\rm Hom}(G,\mathbb{T}^{1})$ and
$\varrho=\vartheta p^{\sharp}=\zeta^{\sharp}p^{\sharp}=(\zeta
p)^{\sharp}=\xi^{\sharp}$. Thus, $\varrho\in\mathop{\rm im}(\varphi)$, as was
to be shown. ∎

######  Corollary 5.28.

Let $\Gamma,G$ be connected Lie groups, $\Gamma$ non-compact, $G$ compact, and
let $h\in\mathop{\rm Hom}(\Gamma,G)$ be a monomorphism with a dense image.
Then for every $k\in\mathbb{N}$ there is $r\in\mathop{\rm
Hom}(\Gamma,\mathbb{T}^{k})$ such that the morphism $h\oplus r\colon\Gamma\to
G\oplus\mathbb{T}^{k}$ has a dense image, that is, $\overline{\mathop{\rm
im}(h\oplus r)}=G\oplus\mathbb{T}^{k}$.

######  Remark 5.29.

We wish to add the following observations.

  * •

In connection with Remark 2.11 notice the following immediate consequence of
the corollary: if for a given connected group $\Gamma\in\mathsf{LieGp}$ there
exist a compact group $G\in\mathsf{LieGp}$ and a monomorphism
$h\colon\Gamma\to G$ with a dense image (in other words, if $\Gamma$ possesses
a faithful finite-dimensional unitary representation) then $\Gamma$ possesses
also a non-trivial one-dimensional unitary representation, that is, a non-
trivial character $\Gamma\to\mathbb{T}^{1}$.

  * •

The statement of the corollary can be reformulated by saying that the minimal
equicontinuous flow $\mathcal{F}\colon\Gamma\curvearrowright G$ induced by $h$
possesses a minimal pure quasi-coboundary $\mathcal{Q}_{r}\in\mathop{\rm{\bf
Z}_{\mathcal{F}}}(\mathbb{T}^{k})$ for every $k\in\mathbb{N}$.

###### Proof of Corollary 5.28.

It is clearly sufficient to verify the statement of the corollary in case when
$k=1$; the general case then follows by a repeated application of this
particular one. Our proof will follow the line of that of Case 1 in the proof
of Theorem 5.25, but we shall present it with all the details.

Let $\mathcal{F}\colon\Gamma\curvearrowright G$ be the minimal equicontinuous
flow induced by $h$ (see (2.22)). We choose the identity $z=e$ as the base
point of $G$; this leads to $\mathcal{F}_{z}=h$. Since $h\colon\Gamma\to G$ is
not an isomorphism, the flow $\mathcal{F}$ possesses a free cycle by virtue of
Proposition 2.12; that is, $\mathop{\rm
rank}(H_{1}^{w}(\mathcal{F}))<\mathop{\rm rank}(H_{1}^{w}(G))$. Consequently,
there is an epimorphism $\varrho\colon H_{1}^{w}(G)\to
H_{1}^{w}(\mathbb{T}^{1})$ with
$H_{1}^{w}(\mathcal{F})\subseteq\ker(\varrho)$. By virtue of Lemma 5.26,
$\varrho=\xi^{\sharp}$ for some $\xi\in\mathop{\rm Hom}(G,\mathbb{T}^{1})$.

Now, under the usual identification
$\pi_{1}(\mathbb{T}^{1})=H_{1}^{w}(\mathbb{T}^{1})$, Lemma 2.17 yields

$\mathop{\rm co}(\xi)^{\sharp}\pi_{1}(\Gamma\times
G)=\xi^{\sharp}\mathcal{F}_{z}^{\sharp}\pi_{1}(\Gamma)=\xi^{\sharp}\mathcal{F}_{z}^{\sharp}H_{1}^{w}(\Gamma)=\xi^{\sharp}H_{1}^{w}(\mathcal{F})=\varrho
H_{1}^{w}(\mathcal{F})=0.$

Thus, $\mathop{\rm co}(\xi)$ lifts across the usual covering morphism
$p\colon\mathbb{R}\to\mathbb{T}^{1}$ to an extension
$\mathcal{D}\in\mathop{\rm{\bf Z}_{\mathcal{F}}}(\mathbb{R})$. For
$l\in\mathbb{N}$ set $\mathcal{C}_{l}=p((1/l)\mathcal{D})$; clearly,
$\mathcal{C}_{l}\in\mathop{\rm{\bf Z}_{\mathcal{F}}}$ is the lift of
$\mathop{\rm co}(\xi)$ across the $l$-endomorphism $\kappa_{l}$ of
$\mathbb{T}^{1}$. Also, since $(1/l)\mathcal{D}\stackrel{{\scriptstyle
ucs}}{{\longrightarrow}}0$ as $l\to\infty$, it follows that
$\mathcal{C}_{l}\stackrel{{\scriptstyle ucs}}{{\longrightarrow}}1$ as
$l\to\infty$.

Further, by our choice of $\xi$ and by Theorem 5.14(1),

$F(\mathcal{C}_{l})=\mathcal{E}_{\ker(\kappa_{l})}\left(\xi^{\sharp}\pi_{1}(G)\right)=\mathcal{E}_{\ker(\kappa_{l})}(\pi_{1}(\mathbb{T}^{1}))=\ker(\kappa_{l})=\mathbb{Z}_{l}$

for every $l\in\mathbb{N}$. Consequently, Theorem 3.23 yields an increasing
sequence of prime numbers $(p_{l})_{l\in\mathbb{N}}$ and an extension
$\mathcal{C}\in\mathop{\rm{\bf Z}_{\mathcal{F}}}$, such that

  * •

$\mathcal{C}=\sum_{l=1}^{\infty}\mathcal{C}_{p_{l}}$ and

  * •

$F(\mathcal{C})\supseteq\overline{\sum_{l\in\mathbb{N}}F(\mathcal{C}_{p_{l}})}$.

Moreover, since
$\overline{\sum_{l\in\mathbb{N}}F(\mathcal{C}_{p_{l}})}=\overline{\sum_{l\in\mathbb{N}}\mathbb{Z}_{p_{l}}}=\mathbb{T}^{1}$,
it follows that the extension $\mathcal{C}$ of $\mathcal{F}$ is minimal.

We claim that $\mathcal{C}$ is a pure quasi-coboundary over $\mathcal{F}$.
First, given $\gamma\in\Gamma$ and $g\in G$, we have

$p\mathcal{D}(\gamma,g)=\mathop{\rm
co}(\xi)(\gamma,g)=\xi(h(\gamma)g)\xi(g)^{-1}=\xi h(\gamma),$

since $\xi\colon G\to\mathbb{T}^{1}$ is a morphism of groups. Consequently,
since $G$ is connected and $\ker(p)=\mathbb{Z}$ is totally disconnected, it
follows that $\mathcal{D}(\gamma,-)$ is constant on $G$ for every
$\gamma\in\Gamma$. This means that $\mathcal{D}$ is a pure quasi-coboundary
over $\mathcal{F}$ and hence so is $\mathcal{C}_{l}$ for every
$l\in\mathbb{N}$. Since the pure quasi-coboundaries form a closed subgroup of
$\mathop{\rm{\bf Z}_{\mathcal{F}}}$, the extension
$\mathcal{C}=\sum_{l=1}^{\infty}\mathcal{C}_{p_{l}}$ is also a pure quasi-
coboundary over $\mathcal{F}$, which verifies the claim.

Let $r\in\mathop{\rm Hom}(\Gamma,\mathbb{T}^{1})$ be the morphism with
$\mathcal{C}=\mathcal{Q}_{r}$; we show that $r$ satisfies the conclusion of
the corollary. Indeed, by minimality of $\mathcal{C}$,

$G\oplus\mathbb{T}^{1}=\overline{\mathcal{O}}_{\mathcal{F}_{\mathcal{C}}}(e,1)=\overline{\\{(h(\gamma),r(\gamma)):\gamma\in\Gamma\\}}=\overline{\mathop{\rm
im}(h\oplus r)},$

as was to be shown. ∎

######  Theorem 5.30.

Let $\mathop{\rm\mathcal{F}\colon\Gamma\curvearrowright X}$ be a minimal flow
with $\Gamma\in\mathsf{LieGp}$ connected and with $X$ a compact connected
manifold, and let $d_{1},\dots,d_{n}$ be the elementary divisors of
$H_{1}^{w}(\mathcal{F})$ in $H_{1}^{w}(X)$. Fix a solenoid $S_{\bf{p}}$ and a
closed non-trivial proper subgroup $K$ of $S_{\bf{p}}$. Then the following
statements hold:

  1. (A)

If the flow $\mathcal{F}$ does not possess a free cycle then there is no
extension $\mathcal{C}\in\mathop{\rm{\bf Z}_{\mathcal{F}}}(S_{{\bf p}})$ with
$F(\mathcal{C})=K$.

  2. (B)

If the flow $\mathcal{F}$ does possess a free cycle then the following
conditions are equivalent:

    1. (i)

$F(\mathcal{C})=K$ for an appropriate extension $\mathcal{C}\in\mathop{\rm{\bf
Z}_{\mathcal{F}}}(S_{\bf{p}})$,

    2. (ii)

$K^{\perp}$ is an infinite cyclic group, that is, $K^{\perp}\cong\mathbb{Z}$,

    3. (iii)

$K=\mathop{\rm pr}_{m}^{-1}(\mathbb{Z}_{k})$ for an appropriate pair of
integers $m\geq 1$ and $k\geq 2$.

  3. (C)

If the flow $\mathcal{F}$ possesses a free cycle then there is
$\mathcal{C}\in\mathop{\rm{\bf Z}_{\mathcal{F}}}(S_{{\bf p}})$ with
$F(\mathcal{C})=S_{{\bf p}}$, that is, $\mathop{\rm{\bf
Z}_{\mathcal{F}}}(S_{{\bf p}})$ contains a minimal extension.

######  Remark 5.31.

We wish to add the following remarks.

  * •

In connection with statements (A) and (C) observe that if $\mathcal{F}$ does
not possess a free cycle then there may or may not exist
$\mathcal{C}\in\mathop{\rm{\bf Z}_{\mathcal{F}}}(S_{{\bf p}})$ with
$F(\mathcal{C})=S_{{\bf p}}$. Firstly, let
$\mathcal{F}\colon\mathbb{R}\curvearrowright X$ be a minimal continuous flow
on a simply connected manifold $X$, say, on
$X=\mathbb{S}^{3}\times\mathbb{S}^{3}$. (The existence of a minimal continuous
flow on $\mathbb{S}^{3}\times\mathbb{S}^{3}$ follows from [18].) Then
$H_{1}^{w}(X)=0$ (hence $\mathcal{F}$ does not possess a free cycle) and the
group $\mathop{\rm{\bf Z}_{\mathcal{F}}}(G)$ contains a minimal extension for
every connected group $G\in\mathsf{CAGp}$ with $\mathop{\rm
card}(G^{*})\leq\mathfrak{c}$. (This follows from our Theorem 6.3 in Section
6.1.) Secondly, let $\mathcal{F}\colon\mathbb{T}^{1}\curvearrowright X$ be the
natural action of $\mathbb{T}^{1}$ on $X=\mathbb{T}^{1}$ by rotations. Then
$H_{1}^{w}(\mathcal{F})=H_{1}^{w}(X)\cong\mathbb{Z}$ (hence $\mathcal{F}$ does
not possess a free cycle) and for every non-trivial group $G\in\mathsf{CAGp}$,
the group $\mathop{\rm{\bf Z}_{\mathcal{F}}}(G)$ contains no minimal
extensions whatsoever. (Indeed, recall that, up to a homeomorphism, there is a
unique non-degenerate space - namely $\mathbb{T}^{1}$, on which the group
$\mathbb{T}^{1}$ acts in a minimal way.)

  * •

Condition (ii) from statement (B) is clearly equivalent to the isomorphism
$S_{{\bf p}}/K\cong\mathbb{T}^{1}$. This occurs if and only if $p_{K}$ is
equivalent to a non-trivial character of $S_{{\bf p}}$.

  * •

From statement (A) and from condition (ii) in statement (B) it follows that a
closed non-trivial proper subgroup $K$ of $S_{{\bf p}}$ must be infinite in
order to be the section $F(\mathcal{C})$ of an extension $\mathcal{C}$ from
$\mathop{\rm{\bf Z}_{\mathcal{F}}}(S_{{\bf p}})$. Consequently, finite non-
trivial sections for extensions in $\mathop{\rm{\bf Z}_{\mathcal{F}}}(S_{{\bf
p}})$ are not allowed. This observation follows also from Theorem 5.14(3), for
every finite subgroup of a solenoid $S_{{\bf p}}$ intersects the identity arc-
component $(S_{{\bf p}})_{a}$ of $S_{\bf p}$ only at the identity.
Alternatively, one could also use Corollary 5.16 and Remark 5.17.

###### Proof of Theorem 5.30.

We divide the proof into four steps.

_1st step._ We show that the following conditions are equivalent for a non-
trivial proper closed subgroup $K$ of $S_{{\bf p}}$:

  1. (a)

$K^{\perp}$ is an infinite cyclic group, that is, $K^{\perp}\cong\mathbb{Z}$,

  2. (b)

$K=\mathop{\rm pr}_{m}^{-1}(\mathbb{Z}_{k})$ for an appropriate pair of
integers $m,k\geq 1$,

  3. (c)

$K=\mathop{\rm pr}_{m}^{-1}(\mathbb{Z}_{k})$ for an appropriate pair of
integers $m\geq 1$ and $k\geq 2$.

Given integers $m,k\geq 1$, the following statements are equivalent:

  * •

$K=\mathop{\rm pr}_{m}^{-1}(\mathbb{Z}_{k})$,

  * •

$K=\ker(\chi_{k,m})$,

  * •

$K^{\perp}=\langle\chi_{k,m}\rangle$,

  * •

$\lambda^{-1}(K^{\perp})=k\mathbb{Z}/p_{0}\dots p_{m-1}$,

where $\lambda$ is the isomorphism $\mathbb{Z}/{\bf p}\to(S_{\bf p})^{*}$
defined by (2.19). Since the infinite cyclic subgroups of $\mathbb{Z}/{\bf p}$
are precisely the groups of the form $k\mathbb{Z}/p_{0}\dots p_{m-1}$ with
$m,k\in\mathbb{N}$, the equivalence of (a) and (b) follows. Implication
(c)$\Rightarrow$(b) is clear and the converse follows from the identity
$\mathop{\rm pr}_{m}^{-1}(\mathbb{Z}_{k})=\mathop{\rm
pr}_{m+1}^{-1}(\mathbb{Z}_{kp_{m}})$, which holds for all $m,k\in\mathbb{N}$.

_2nd step._ We verify statement (A).

Assume, contrary to (A), that $K=F(\mathcal{C})$ for some extension
$\mathcal{C}\in\mathop{\rm{\bf Z}_{\mathcal{F}}}(S_{\bf p})$. By Theorem 5.22,
there exist $k_{1},\dots,k_{n}\in K\cap(S_{\bf p})_{a}$ with
$K=\overline{\langle k_{1},\dots,k_{n}\rangle}$ and such that
$k_{i}^{d_{i}}=e$ for $i=1,\dots,n$. It follows that the group $K$ is finite
and $K\subseteq(S_{\bf p})_{a}$. Consequently, $K\subseteq(S_{\bf
p})_{a}\cap\mathop{\rm tor}(S_{{\bf p}})=e$, which contradicts the assumptions
of the theorem.

_3rd step._ We verify statement (B).

First, the equivalence of (ii) and (iii) has been established in the first
part of the proof. Further, implication (i)$\Rightarrow$(ii) follows
immediately from Remark 5.17. Finally, to show that (i) follows from (iii),
let $K$ have the form $K=\mathop{\rm pr}_{m}^{-1}(\mathbb{Z}_{k})$ with $m\geq
1$ and $k\geq 2$. Consider the morphism $h\in\mathcal{L}(S_{{\bf p}})$ defined
by $h(t)=(\exp(i2\pi t/p_{0}\dots p_{n-1}))_{n\in\mathbb{N}}$. Then
$h(p_{0}\dots p_{m-1}/k)$ is a topological generator of $K$, which lies on the
identity arc-component $(S_{{\bf p}})_{a}$ of $S_{{\bf p}}$. Since the flow
$\mathcal{F}$ possesses a free cycle by the assumptions, we may apply Theorem
5.22 to obtain $\mathcal{C}\in\mathop{\rm{\bf Z}_{\mathcal{F}}}(S_{{\bf p}})$
with $F(\mathcal{C})=K$. This verifies statement (i) and finishes the proof of
(B).

_4th step._ We verify statement (C). The argument will be similar to the one
used to verify Case 1 in the proof of Theorem 5.25.

Since the flow $\mathcal{F}$ possesses a free cycle by the assumptions, Lemma
5.21 yields a continuous base point preserving map $\xi\colon
X\to\mathbb{T}^{1}$ with
$\xi^{\sharp}\mathcal{F}_{z}^{\sharp}\pi_{1}(\Gamma)=0$ and
$\xi^{\sharp}\pi_{1}(X)=\pi_{1}(\mathbb{T}^{1})$. Let
$p\colon\mathbb{R}\to\mathbb{T}^{1}$ be the standard covering morphism with
the kernel $\mathbb{Z}$. By virtue of (2.30) from Lemma 2.17, $\mathop{\rm
co}(\xi)$ lifts across $p$ to an extension $\mathcal{D}\in\mathop{\rm{\bf
Z}_{\mathcal{F}}}(\mathbb{R})$. Consider the morphism $h\colon\mathbb{R}\to
S_{{\bf p}}$ from the third step of the proof and set
$\mathcal{C}_{l}=h((1/l)\mathcal{D})\in\mathop{\rm{\bf
Z}_{\mathcal{F}}}(S_{{\bf p}})$ for $l\in\mathbb{N}$. Since
$(1/l)\mathcal{D}\stackrel{{\scriptstyle ucs}}{{\longrightarrow}}0$ as
$l\to\infty$, it follows that $\mathcal{C}_{l}\stackrel{{\scriptstyle
ucs}}{{\longrightarrow}}e$ as $l\to\infty$. Consequently, there exists an
increasing sequence of prime numbers $(r_{l})_{l\in\mathbb{N}}$ such that the
series $\sum_{l=1}^{\infty}\mathcal{C}_{q_{l}}$ converges u.c.s. in
$\mathop{\rm{\bf Z}_{\mathcal{F}}}(S_{{\bf p}})$ for every subsequence
$(q_{l})_{l\in\mathbb{N}}$ of $(r_{l})_{l\in\mathbb{N}}$.

Given $l\in\mathbb{N}$, we show that $F(\mathcal{C}_{l})=\mathop{\rm
pr}_{1}^{-1}(\mathbb{Z}_{l})$. First, from the identity $\mathop{\rm
pr}_{1}h=p$ it follows that

$(\kappa_{l}\text{pr}_{1})\mathcal{C}_{l}=\kappa_{l}\text{pr}_{1}h((1/l)\mathcal{D})=\kappa_{l}p((1/l)\mathcal{D})=p\mathcal{D}=\mathop{\rm
co}(\xi).$

Moreover, since $K_{l}:=\ker(\kappa_{l}\mathop{\rm pr}_{1})=\mathop{\rm
pr}_{1}^{-1}(\mathbb{Z}_{l})$ is a (closed) totally disconnected subgroup of
$S_{{\bf p}}$, we may apply (5.1) from Theorem 5.14 to obtain

$K_{l}\supseteq
F(\mathcal{C}_{l})=\overline{\mathcal{E}_{K_{l}}}(\xi^{\sharp}\pi_{1}(X))=\overline{\mathcal{E}_{K_{l}}}(\pi_{1}(\mathbb{T}^{1}))\supseteq\overline{\langle
h(1/l)\rangle}=K_{l};$

where we have used the following facts:

  * •

$\mathcal{E}_{K_{l}}(\pi_{1}(\mathbb{T}^{1}))\ni h(1/l)$; this follows from
the fact that $h(t/l)$ ($0\leq t\leq 1$) is a path in $S_{\bf p}$ from $e$ to
$h(1/l)$, which projects to the element $p(t)$ ($0\leq t\leq 1$) of
$\pi_{1}(\mathbb{T}^{1})$ via $\kappa_{l}\mathop{\rm pr}_{1}$,

  * •

$h(1/l)$ is a topological generator of $K_{l}$; this is clear (see also the
third step of the proof).

Thus it follows that $F(\mathcal{C}_{l})=K_{l}=\mathop{\rm
pr}_{1}^{-1}(\mathbb{Z}_{l})$, as was to be shown. As an immediate corollary
of this fact we get $F(\mathop{\rm pr}_{1}\mathcal{C}_{l})=\mathop{\rm
pr}_{1}F(\mathcal{C}_{l})=\mathbb{Z}_{l}$.

From our discussion so far the following facts emerge:

  * •

$\mathop{\rm pr}_{1}\mathcal{C}_{r_{l}}\stackrel{{\scriptstyle
ucs}}{{\longrightarrow}}1$ as $l\to\infty$,

  * •

$F(\mathop{\rm pr}_{1}\mathcal{C}_{r_{l}})=\mathbb{Z}_{r_{l}}$
($l\in\mathbb{N}$) are mutually group-disjoint.

Consequently, by Theorem 3.23, there is a subsequence
$(q_{l})_{l\in\mathbb{N}}$ of $(r_{l})_{l\in\mathbb{N}}$ such that the series
$\sum_{l=1}^{\infty}\mathop{\rm pr}_{1}\mathcal{C}_{q_{l}}$ converges u.c.s.
in $\mathop{\rm{\bf Z}_{\mathcal{F}}}$ and

$\mathbb{T}^{1}\supseteq
F\left(\sum_{l=1}^{\infty}\text{pr}_{1}\mathcal{C}_{q_{l}}\right)\supseteq\overline{\sum_{l=1}^{\infty}F(\text{pr}_{1}\mathcal{C}_{q_{l}})}=\overline{\sum_{l=1}^{\infty}\mathbb{Z}_{q_{l}}}=\mathbb{T}^{1}.$
(5.9)

Now set $\mathcal{C}=\sum_{l=1}^{\infty}\mathcal{C}_{q_{l}}\in\mathop{\rm{\bf
Z}_{\mathcal{F}}}(S_{{\bf p}})$. (Recall that the series converges u.c.s. in
$\mathop{\rm{\bf Z}_{\mathcal{F}}}(S_{\bf p})$ by our choice of the sequence
$(r_{l})_{l\in\mathbb{N}}$ and so the extension $\mathcal{C}$ is well
defined.) Moreover, by virtue of (5.9),

$\text{pr}_{1}F(\mathcal{C})=F(\text{pr}_{1}\mathcal{C})=F\left(\sum_{l=1}^{\infty}\text{pr}_{1}\mathcal{C}_{q_{l}}\right)=\mathbb{T}^{1},$

which shows that $F(\mathcal{C})=S_{{\bf p}}$. This verifies statement (C). ∎

###  5.5 Relation to the first cohomotopy group

Let $\mathop{\rm\mathcal{F}\colon\Gamma\curvearrowright X}$ be a minimal flow.
In this section we continue our study of the problem of determining the
section $F(\mathcal{C})$ of an extension
$\mathcal{C}\in\mathop{\rm\mathsf{CAGpZ}_{\mathcal{F}}}$ in terms of morphisms
induced by transfer functions of certain quotients of $\mathcal{C}$ on one-
dimensional algebraic-topological invariants. Now we concentrate on the
situation when $X$ is an arbitrary compact connected space. When dealing with
general, not necessarily locally connected continua, the first cohomotopy
group $\pi^{1}$ is a more appropriate tool than the first homology group
$H_{1}$. In Theorems 5.32 and 5.40 we explain how the functors $F$ and
$\pi^{1}$ are related, and show that $\pi^{1}$ can be used to obtain the
values of $F$ in some important situations.

Before formulating and proving our results, let us mention that in the whole
present section we shall use notation introduced at the beginning of Section
5.1.

######  Theorem 5.32.

Let $\mathop{\rm\mathcal{F}\colon\Gamma\curvearrowright X}$ be a minimal flow,
$G\in\mathsf{CAGp}$ be connected and $\mathcal{C}\in\mathop{\rm{\bf
Z}_{\mathcal{F}}}(G)$. Fix $z\in X$. Assume that $K\mathrel{\lhd}G$ is finite
with $p_{K}\mathcal{C}\in\mathop{\rm{\bf B}_{\mathcal{F}}}(G/K)$ (that is,
with $F(\mathcal{C})\subseteq K$), and denote by $\xi$ the transfer function
of $p_{K}\mathcal{C}$ with $\xi(z)=e$. If the space $X$ is compact connected
and the group $\Gamma$ has no non-trivial finite abelian quotient groups then
the following statements hold under the usual identifications
$\pi^{1}(G/K)\cong(G/K)^{*}\cong K^{\perp}$:

  1. (1)

the group $F(\mathcal{C})$ is the smallest of all the subgroups $E$ of $K$,
such that the morphism $\xi^{\flat}\colon(G/K)^{*}\to\pi^{1}(X)$ extends
through $(q_{E})^{*}\colon(G/K)^{*}\to(G/E)^{*}$ to a morphism
$\varrho\colon(G/E)^{*}\to\pi^{1}(X)$,

  2. (2)

if $d$ stands for the largest torsion coefficient of $K$ and $\eta=|K|$
denotes the cardinality of $K$ then

$F(\mathcal{C})^{\perp}=\frac{1}{d}\left(\xi^{\flat}\right)^{-1}\left(d\pi^{1}(X)\right)=\frac{1}{\eta}\left(\xi^{\flat}\right)^{-1}\left(\eta\pi^{1}(X)\right),$
(5.10)

see also _( 5.11)_ below,

  3. (3)

the following statements are equivalent:

    1. ($\iota$)

$F(\mathcal{C})=K$,

    2. ($\iota\iota$)

$dG^{*}\cap\left(\xi^{\flat}\right)^{-1}(d\pi^{1}(X))=dK^{\perp}$,

    3. ($\iota\iota\iota$)

$\xi^{\flat}\left(dG^{*}\setminus dK^{\perp}\right)\cap
d\pi^{1}(X)=\emptyset$.

######  Remark 5.33.

We wish to mention the following facts.

  * •

The connectedness assumption of $G$ is not at all restrictive in the situation
of the theorem, for under our assumptions on $\Gamma$ and $X$, every extension
$\mathcal{C}\in\mathop{\rm{\bf Z}_{\mathcal{F}}}(G)$ with $G\in\mathsf{CAGp}$
takes its values in the identity component $G_{0}$ of $G$. Indeed, since the
group $G/G_{0}$ is totally disconnected, its Pontryagin dual $(G/G_{0})^{*}$
is a torsion group. Further, by Theorem 4.10, $\mathop{\rm{\bf
Z}_{\mathcal{F}}}$ is a torsion-free group. Hence, by Theorem 3.1,

$\mathop{\rm{\bf Z}_{\mathcal{F}}}(G/G_{0})\cong\mathop{\rm
Hom}((G/G_{0})^{*},\mathop{\rm{\bf Z}_{\mathcal{F}}})=0.$

Thus, if $p_{0}\colon G\to G/G_{0}$ denotes the canonical quotient morphism
then $p_{0}\mathcal{C}\in\mathop{\rm{\bf Z}_{\mathcal{F}}}(G/G_{0})$ is
trivial and hence $\mathcal{C}$ takes its values in $G_{0}$.

  * •

We would like to clarify formula (5.10) from statement (2) of the theorem. By
definition, $\xi^{\flat}$ is a morphism $\pi^{1}(G/K)\to\pi^{1}(X)$. Since the
group $G/K\in\mathsf{CAGp}$ is connected, there are isomorphisms
$\pi^{1}(G/K)\cong(G/K)^{*}\cong K^{\perp}$, see Subsection 2.4.10. Thus,
$(\xi^{\flat})^{-1}(d\pi^{1}(X))$ can be interpreted as a subgroup of
$K^{\perp}\subseteq G^{*}$ and the expression
$(1/d)(\xi^{\flat})^{-1}(d\pi^{1}(X))$ then stands for the group of all
$\chi\in G^{*}$ with $\chi^{d}\in(\xi^{\flat})^{-1}(d\pi^{1}(X))$. If one
prefers not to use the identification $(G/K)^{*}\cong K^{\perp}$ and to regard
$\xi^{\flat}$ as a morphism $(G/K)^{*}\to\pi^{1}(X)$ then the formula (5.10)
takes the form

$F(\mathcal{C})^{\perp}=\frac{1}{d}\,p_{K}^{*}\left(\xi^{\flat}\right)^{-1}\left(d\pi^{1}(X)\right)=\frac{1}{\eta}\,p_{K}^{*}\left(\xi^{\flat}\right)^{-1}\left(\eta\pi^{1}(X)\right).$
(5.11)

More generally, if the morphism $p_{K}$ is replaced by a general quotient
morphism $q\colon G\to H$ with the kernel $K$ (and the map $\xi$ thus takes
its values in $H$) then

$F(\mathcal{C})^{\perp}=\frac{1}{d}\,q^{*}\left(\xi^{\flat}\right)^{-1}\left(d\pi^{1}(X)\right)=\frac{1}{\eta}\,q^{*}\left(\xi^{\flat}\right)^{-1}\left(\eta\pi^{1}(X)\right),$
(5.12)

where $\xi^{\flat}$ is interpreted as a morphism $H^{*}\to\pi^{1}(X)$. This
follows from (5.11) and from the equivalence of $q$ and $p_{K}$.

  * •

If the group $\pi^{1}(X)$ is $d$-divisible (that is, if
$d\pi^{1}(X)=\pi^{1}(X)$) then formula (5.10) yields
$F(\mathcal{C})^{\perp}=(1/d)K^{\perp}=G^{*}$, where the last equality holds
by definition of $d$. Thus, in this situation, $F(\mathcal{C})=e$ and hence
$\mathcal{C}\in\mathop{\rm{\bf B}_{\mathcal{F}}}(G)$.

  * •

Recall that under the assumptions of Theorem 5.14, $F(\mathcal{C})\subseteq
G_{a}$ as soon as $F(\mathcal{C})$ is finite. Under the assumptions of Theorem
5.32 this is no longer the case, not even when the acting group $\Gamma$ of
$\mathcal{F}$ is a (simply) connected Lie group. For an example of such a
situation we refer to Section 5.6, where finite subgroups of solenoids $S_{\bf
p}$ appear as sections $F(\mathcal{C})$ of appropriate extensions
$\mathcal{C}\in\mathop{\rm{\bf Z}_{\mathcal{F}}}(S_{\bf p})$. (Recall that a
finite subgroup of a solenoid $S_{\bf p}$ intersects the identity arc-
component $(S_{\bf p})_{a}$ of $S_{\bf p}$ only at the identity $e$.)

###### Proof of Theorem 5.32.

We begin the proof by verifying statement (1). By virtue of statement (ii)
from Proposition 5.3, it suffices to show that for every subgroup $E$ of $K$,
the following conditions are equivalent:

  1. ($\alpha$)

there is $\sigma\in\mathop{\rm Hom}((G/E)^{*},C_{z}(X,\mathbb{T}^{1}))$ with
$\sigma(q_{E})^{*}=\xi^{*}$,

  2. ($\beta$)

there is $\varrho\in\mathop{\rm Hom}((G/E)^{*},\pi^{1}(X))$ with
$\varrho(q_{E})^{*}=\xi^{\flat}$.

First recall that there is an isomorphism $\varphi\colon
C_{z}(X,\mathbb{T}^{1})\to C_{z}(X,\mathbb{R})\oplus\pi^{1}(X)$ with
$\mathop{\rm pr}_{2}\varphi=\pi$, where $\pi$ stands for the canonical
quotient morphism $C_{z}(X,\mathbb{T}^{1})\to\pi^{1}(X)$. Set $\nu=\mathop{\rm
pr}_{1}\varphi\xi^{*}$. Then $\nu\in\mathop{\rm
Hom}((G/K)^{*},C_{z}(X,\mathbb{R}))$ and, since $\pi\xi^{*}=\xi^{\flat}$,
$\varphi\xi^{*}=\nu\oplus\xi^{\flat}$. Consequently, we may use
identifications $C_{z}(X,\mathbb{T}^{1})\cong
C_{z}(X,\mathbb{R})\oplus\pi^{1}(X)$ and $\xi^{*}\cong\nu\oplus\xi^{\flat}$.

Now let $E$ be a subgroup of $K$. We show that ($\alpha$) and ($\beta$) are
equivalent. First, ($\beta$) follows from ($\alpha$) at once, for if
$\sigma(q_{E})^{*}=\xi^{*}$ then
$(\pi\sigma)(q_{E})^{*}=\pi(\sigma(q_{E})^{*})=\pi\xi^{*}=\xi^{\flat}$ and so
we may take $\varrho=\pi\sigma$ to see that ($\beta$) holds. Conversely,
assume that a morphism $\varrho\colon(G/E)^{*}\to\pi^{1}(X)$ satisfies
$\varrho(q_{E})^{*}=\xi^{\flat}$. Since the group $C_{z}(X,\mathbb{R})$ is
divisible, $\nu$ extends through $(q_{E})^{*}$ to a morphism
$\mu\colon(G/E)^{*}\to C_{z}(X,\mathbb{R})$. That is, $\mu(q_{E})^{*}=\nu$.
Set $\sigma=\mu\oplus\varrho\colon(G/E)^{*}\to C_{z}(X,\mathbb{T}^{1})$. Then
$\sigma(q_{E})^{*}=\mu(q_{E})^{*}\oplus\varrho(q_{E})^{*}=\nu\oplus\xi^{\flat}=\xi^{*}$
and this verifies statement ($\alpha$).

We verify formula (5.10) from statement (2). In order to simplify notation,
write $D=(1/d)\left(\xi^{\flat}\right)^{-1}(d\pi^{1}(X))$. Given a subgroup
$E$ of $K$, we shall use the standard identification of the group $(G/E)^{*}$
with the annihilator $E^{\perp}$ of $E$ in $G^{*}$, the isomorphism between
them being the Pontryagin dual $(p_{E})^{*}\colon(G/E)^{*}\to
E^{\perp}\subseteq G^{*}$ to the quotient morphism $p_{E}\colon G\to G/E$.
Under this identification the morphism $(q_{E})^{*}$ takes the form of the
inclusion morphism $K^{\perp}\to E^{\perp}$. Further, via the annihilator
mechanism in $G^{*}$, the subgroups $E$ of $K$ are in a one-to-one
correspondence with the subgroups $A$ of $G^{*}$ containing $K^{\perp}$. Thus,
by virtue of statement (1) of the theorem, $F(\mathcal{C})^{\perp}$ is the
largest (with respect to inclusion) of all the subgroups $A$ of $G^{*}$
containing $K^{\perp}$, such that $\xi^{\flat}$ extends to a morphism
$\varrho\colon A\to\pi^{1}(X)$. Consequently, in order to prove the first
equality in (5.10), we need to show that the following statements hold:

  1. (i)

$K^{\perp}\subseteq D$,

  2. (ii)

for every $\chi\in G^{*}$, $\chi$ is an element of $D$ if and only if
$\xi^{\flat}$ extends to a morphism
$K^{\perp}+\langle\chi\rangle\to\pi^{1}(X)$.

Before proving the two statements let us describe when a given character
$\chi\in G^{*}$ belongs to $D$. Given $\chi\in G^{*}$, we have $\chi^{d}\in
K^{\perp}$ by definition of $d$ and so $\chi^{d}=\Upsilon p_{K}$ for a unique
$\Upsilon\in(G/K)^{*}$. Thus, under our identifications
$\pi^{1}(G/K)\cong(G/K)^{*}\cong K^{\perp}$, $\chi$ belongs to $D$ if and only
if $\xi^{\flat}(\Upsilon)\in d\pi^{1}(X)$.

Now, given $\chi\in K^{\perp}$, fix $\Upsilon\in(G/K)^{*}$ with $\chi=\Upsilon
p_{K}$. Then $\chi^{d}=\Upsilon^{d}p_{K}$ and, since
$\xi^{\flat}(\Upsilon^{d})=d\xi^{\flat}(\Upsilon)\in d\pi^{1}(X)$, statement
(i) follows. To verify statement (ii), fix $\chi\in G^{*}$ and let
$\Upsilon\in(G/K)^{*}$ be such that $\chi^{d}=\Upsilon p_{K}$. We need to show
that the following conditions are equivalent:

  1. (a)

$\xi^{\flat}$ extends to a morphism $\varrho\colon
K^{\perp}+\langle\chi\rangle\to\pi^{1}(X)$,

  2. (b)

there is $f\in\pi^{1}(X)$ with $\xi^{\flat}(\Upsilon)=df$.

Assume that condition (a) is satisfied and set $f=\varrho(\chi)$. Then
$f\in\pi^{1}(X)$ and
$df=d\varrho(\chi)=\varrho(\chi^{d})=\xi^{\flat}(\chi^{d})=\xi^{\flat}(\Upsilon)$,
which verifies (b). Conversely, let $f\in\pi^{1}(X)$ be as in (b) and denote
by $p$ the smallest positive integer with $\chi^{p}\in K^{\perp}$. Since
$\chi^{d}\in K^{\perp}$, it follows that $p$ divides $d$ and so $d=pq$ for an
appropriate positive integer $q$. We claim that $\xi^{\flat}(\chi^{p})=pf$.
Indeed, by our choice of $f$,
$q\xi^{\flat}(\chi^{p})=\xi^{\flat}(\chi^{pq})=\xi^{\flat}(\Upsilon)=df=q(pf)$
and hence, since $\pi^{1}(X)$ is a torsion-free group,
$\xi^{\flat}(\chi^{p})=pf$. Using now the additive notation in $G^{*}$, define
a map $\varrho\colon K^{\perp}+\langle\chi\rangle\to\pi^{1}(X)$ by the rule
$\varrho(\chi^{\prime}+k\chi)=\xi^{\flat}(\chi^{\prime})+kf$, where
$\chi^{\prime}\in K^{\perp}$ and $k\in\mathbb{Z}$. We show that $\varrho$ is
well defined and it will immediately follow that $\varrho$ is in fact a
morphism extending $\xi^{\flat}$. So let
$\chi_{1}^{\prime},\chi_{2}^{\prime}\in K^{\perp}$ and $k,l\in\mathbb{Z}$, and
assume that $\chi_{1}^{\prime}+k\chi=\chi_{2}^{\prime}+l\chi$. Then
$(l-k)\chi=\chi_{1}^{\prime}-\chi_{2}^{\prime}\in K^{\perp}$ and hence, by our
definition of $p$, $l-k=pr$ for an appropriate integer $r$. Thus,

$\begin{split}\xi^{\flat}(\chi_{1}^{\prime})+kf&=\xi^{\flat}(\chi_{2}^{\prime}+r(p\chi))+kf=\xi^{\flat}(\chi_{2}^{\prime})+r\xi^{\flat}(p\chi)+kf\\\
&=\xi^{\flat}(\chi_{2}^{\prime})+rpf+kf=\xi^{\flat}(\chi_{2}^{\prime})+lf,\end{split}$

as was to be shown. This finishes the proof of the first equality from (5.10).

We verify the second equality from (5.10). Write $\eta=ld$ with
$l\in\mathbb{N}$. Since the group $\pi^{1}(X)$ is torsion-free, the
equivalence of the following conditions follows immediately for every $\chi\in
G^{*}$:

  * •

$\chi\in(1/\eta)\left(\xi^{\flat}\right)^{-1}(\eta\pi^{1}(X))$,

  * •

$\xi^{\flat}(\chi^{\eta})\in\eta\pi^{1}(X)$,

  * •

$l\xi^{\flat}(\chi^{d})\in l(d\pi^{1}(X))$,

  * •

$\xi^{\flat}(\chi^{d})\in d\pi^{1}(X)$,

  * •

$\chi\in(1/d)\left(\xi^{\flat}\right)^{-1}(d\pi^{1}(X))$.

This verifies the second equality from (5.10).

We finish the proof of the theorem be verifying statement (3). By virtue of
statement (2), condition ($\iota$) is equivalent with
$K^{\perp}=(1/d)\left(\xi^{\flat}\right)^{-1}(d\pi^{1}(X))$. Moreover, the
inclusion $K^{\perp}\subseteq(1/d)\left(\xi^{\flat}\right)^{-1}(d\pi^{1}(X))$
has been verified above (see condition (i)) and the inclusion
$dG^{*}\cap\left(\xi^{\flat}\right)^{-1}(d\pi^{1}(X))\supseteq dK^{\perp}$
from ($\iota\iota$) is clear. Consequently, it remains to verify the
equivalence of the following statements:

  * •

$(1/d)\left(\xi^{\flat}\right)^{-1}(d\pi^{1}(X))\subseteq K^{\perp}$,

  * •

$dG^{*}\cap\left(\xi^{\flat}\right)^{-1}(d\pi^{1}(X))\subseteq dK^{\perp}$,

  * •

$\xi^{\flat}(dG^{*}\setminus dK^{\perp})\cap d\pi^{1}(X)=\emptyset$.

The equivalence of the above statements follows by elementary arguments, with
the help of the fact that the group $G^{*}$ is torsion-free by the assumptions
of the theorem. ∎

######  Corollary 5.34.

Let $\mathop{\rm\mathcal{F}\colon\Gamma\curvearrowright X}$ be a minimal flow
on a compact connected space $X$. Consider the following statements:

  1. (i)

for every $\mathcal{C}\in\mathop{\rm\mathsf{CAGpZ}_{\mathcal{F}}}$, the group
$F(\mathcal{C})$ is connected,

  2. (ii)

the group $\pi^{1}(X)$ is divisible.

If $\Gamma$ has no non-trivial finite abelian quotient groups then _(i)_
follows from _(ii)_. If $\Gamma\in\mathsf{CLAC}$ is simply connected then
_(ii)_ follows from _(i)_.

######  Remark 5.35.

Recall from Remark 4.40 that the following conditions are equivalent for every
non-trivial connected group $G\in\mathsf{CAGp}$:

  * •

$G$ has no proper non-trivial closed connected subgroup,

  * •

$\mathop{\rm rank}(G^{*})=1$.

These conditions are satisfied by the circle group $G=\mathbb{T}^{1}$ and by
the solenoids $G=S_{{\bf p}}$. Consequently, if the acting group $\Gamma$ of
$\mathcal{F}$ has no non-trivial finite abelian quotient groups, the space $X$
has a divisible first cohomotopy group $\pi^{1}(X)$ and $G\in\mathsf{CAGp}$ is
a connected group with $\mathop{\rm rank}(G^{*})=1$, then every extension
$\mathcal{C}\in\mathop{\rm{\bf Z}_{\mathcal{F}}}(G)$ is either minimal or a
coboundary.

###### Proof of Corollary 5.34.

We show that (i) follows from (ii). To this end, assume that $\Gamma$ has no
non-trivial finite abelian quotient groups and that the group $\pi^{1}(X)$ is
divisible. Fix $G\in\mathsf{CAGp}$ and $\mathcal{C}\in\mathop{\rm{\bf
Z}_{\mathcal{F}}}(G)$. We need to show that $F(\mathcal{C})$ is connected or,
equivalently, that $F(\mathcal{C})^{*}$ is torsion-free. So let $\chi$ be a
torsion element of $F(\mathcal{C})^{*}$. Then $\chi
F(\mathcal{C})\subseteq\mathbb{Z}_{d}$ for some positive integer $d$. The
morphism $\chi\colon F(\mathcal{C})\to\mathbb{T}^{1}$ extends to a morphism of
topological groups $G\to\mathbb{T}^{1}$, denote the latter one by the same
symbol $\chi$. Since $F(\chi\mathcal{C})=\chi
F(\mathcal{C})\subseteq\mathbb{Z}_{d}$, it follows that
$F(\kappa_{d}(\chi\mathcal{C}))=\kappa_{d}F(\chi\mathcal{C})=1$ and hence
$\kappa_{d}(\chi\mathcal{C})\in\mathop{\rm{\bf B}_{\mathcal{F}}}$, where
$\kappa_{d}$ is the $d$-endomorphism of $\mathbb{T}^{1}$. Let $\xi\in
C_{z}(X,\mathbb{T}^{1})$ be the transfer function for
$\kappa_{d}(\chi\mathcal{C})$. Then by virtue of statement (2) from Theorem
5.32 and the second part of Remark 5.33, we obtain

$\begin{split}F(\chi\mathcal{C})^{\perp}&=\frac{1}{d}(\kappa_{d})^{*}\left(\xi^{\flat}\right)^{-1}(d\pi^{1}(X))=\frac{1}{d}(\kappa_{d})^{*}\left(\xi^{\flat}\right)^{-1}(\pi^{1}(X))\\\
&=\frac{1}{d}(\kappa_{d})^{*}(\mathbb{Z})=\frac{1}{d}(d\mathbb{Z})=\mathbb{Z}\end{split}$

with the usual identifications
$\pi^{1}(\mathbb{T}^{1})\cong(\mathbb{T}^{1})^{*}\cong\mathbb{Z}$. This shows
that $\chi F(\mathcal{C})=F(\chi\mathcal{C})=1$ and so $\chi=1$ as an element
of $F(\mathcal{C})^{*}$. Thus, the group $F(\mathcal{C})^{*}$ is torsion-free,
as was to be shown.

We show that (ii) follows from (i). So assume that $\Gamma\in\mathsf{CLAC}$ is
simply connected. Since $C_{z}(X,\mathbb{T}^{1})\cong
C_{z}(X,\mathbb{R})\oplus\pi^{1}(X)$ and the group $C_{z}(X,\mathbb{R})$ is
divisible, the divisibility of $\pi^{1}(X)$ is equivalent to the divisibility
of $C_{z}(X,\mathbb{T}^{1})$. Fix $f\in C_{z}(X,\mathbb{T}^{1})$ and
$d\in\mathbb{N}$. By Theorem 3.38, $\mathop{\rm co}(f)\in\mathop{\rm{\bf
B}_{\mathcal{F}}}$ lifts across
$\kappa_{d}\colon\mathbb{T}^{1}\to\mathbb{T}^{1}$ to an extension
$\mathcal{C}\in\mathop{\rm{\bf Z}_{\mathcal{F}}}$ and
$F(\mathcal{C})\subseteq\ker(\kappa_{d})=\mathbb{Z}_{d}$. Since the group
$F(\mathcal{C})$ is connected by the assumptions, $F(\mathcal{C})=1$ and hence
$\mathcal{C}\in\mathop{\rm{\bf B}_{\mathcal{F}}}$. Let $g\in
C_{z}(X,\mathbb{T}^{1})$ be the transfer function for $\mathcal{C}$. Then $f$
lifts across $\kappa_{d}$ to $g$, which means that $f=g^{d}$ holds in
$C_{z}(X,\mathbb{T}^{1})$. Since $f\in C_{z}(X,\mathbb{T}^{1})$ and
$d\in\mathbb{N}$ were arbitrary, this shows that the group
$C_{z}(X,\mathbb{T}^{1})$ is divisible and hence so is the group $\pi^{1}(X)$.
∎

######  Corollary 5.36.

Let $\mathop{\rm\mathcal{F}\colon\Gamma\curvearrowright X}$ be a minimal flow
with $\Gamma\in\mathsf{CLAC}$ simply connected and with $X$ compact. Given
$G\in\mathsf{CAGp}$ connected and $K\mathrel{\lhd}G$ finite with the largest
torsion coefficient $d$, the following conditions are equivalent:

  1. (a)

there is $\mathcal{C}\in\mathop{\rm{\bf Z}_{\mathcal{F}}}(G)$ with
$F(\mathcal{C})=K$,

  2. (b)

there is $\varrho\in\mathop{\rm Hom}(K^{\perp},\pi^{1}(X))$ with
$dG^{*}\cap\varrho^{-1}(d\pi^{1}(X))=dK^{\perp}$.

###### Proof.

To see that (b) follows from (a), let $\xi$ be the base point preserving
transfer function for $p_{K}\mathcal{C}$. Then by Theorem 5.32(3), condition
(b) holds with $\varrho=\xi^{\flat}$. To see that (a) follows from (b), fix
$\varrho$ as in (b). By the isomorphism (2.15) from Subsection 2.4.10,
$\varrho=\xi^{\flat}$ for an appropriate continuous base point preserving map
$\xi\colon X\to G/K$. By the second part of Remark 3.39, Theorem 4.10 and
Theorem 4.14(f), $\mathop{\rm co}(\xi)$ lifts across $p_{K}$ to an extension
$\mathcal{C}\in\mathop{\rm{\bf Z}_{\mathcal{F}}}(G)$. Using Theorem 5.32(3)
once more now yields $F(\mathcal{C})=K$, which verifies condition (a). ∎

######  Example 5.37.

Suppose that the assumptions of Theorem 5.14 and Theorem 5.32 are satisfied.
Then one can use both (5.1) and (5.10) to determine the section
$F(\mathcal{C})$ of a given extension $\mathcal{C}$. As a matter of fact, the
two formulas follow from each other. To show this, we verify the following
statement which does not involve any dynamics on $X$ at all:

  1. ($*$)

Let $X$ be a compact connected manifold, $G\in\mathsf{CAGp}$ be connected and
$K\mathrel{\lhd}G$ be finite with the largest torsion coefficient $d$. Given a
continuous base point preserving map $\xi\colon X\to G/K$, the following
identity holds:

$\mathcal{E}_{K}\left(\xi^{\sharp}\pi_{1}(X)\right)^{\perp}=\frac{1}{d}\left(\xi^{\flat}\right)^{-1}(d\pi^{1}(X)),$
(5.13)

where the annihilator on the left is taken in $G^{*}$.

###### Proof of statement ($*$).

We shall prove ($*$) by verifying the equivalence of several pairs of
auxiliary statements. This might not be the fastest way to arrive at (5.13)
but we believe it is (one of) the most transparent.

Given $f\in\pi_{1}(G/K)$, let $\widetilde{f}$ be the lift of $f$ across
$p_{K}$ starting at $e$. Fix $\chi\in G^{*}$. It follows immediately from the
definition of $\mathcal{E}_{K}$ that the following statements are equivalent:

  1. ($\alpha$)

$\chi\in\mathcal{E}_{K}(\xi^{\sharp}\pi_{1}(X))^{\perp}$,

  2. ($\beta$)

$\chi\widetilde{f}$ is a loop for every $f\in\xi^{\sharp}\pi_{1}(X)$.

Further, let $\kappa_{d}$ be the $d$-endomorphism of $\mathbb{T}^{1}$. By
definition of $d$, there is a unique character $\Upsilon\in(G/K)^{*}$ of $G/K$
with $\kappa_{d}\chi=\Upsilon p_{K}$. We show that ($\beta$) is equivalent
with

  1. ($\gamma$)

$\Upsilon^{\sharp}\xi^{\sharp}\pi_{1}(X)\subseteq\kappa_{d}^{\sharp}\pi_{1}(\mathbb{T}^{1})$.

Assume, first, that ($\beta$) holds and fix $f\in\xi^{\sharp}\pi_{1}(X)$. Then
$\Upsilon^{\sharp}(f)=\Upsilon f=\Upsilon
p_{K}\widetilde{f}=\kappa_{d}\chi\widetilde{f}\in\kappa_{d}^{\sharp}\pi_{1}(\mathbb{T}^{1})$,
since $\chi\widetilde{f}$ is a loop. Conversely, let condition ($\gamma$) be
fulfilled and fix $f\in\xi^{\sharp}\pi_{1}(X)$. By virtue of ($\gamma$),
$\Upsilon f$ lifts across $\kappa_{d}$ to a loop $l$ in $\mathbb{T}^{1}$ based
at $1$. That is, $\kappa_{d}l=\Upsilon f=\Upsilon
p_{K}\widetilde{f}=\kappa_{d}\chi\widetilde{f}$ and hence
$l=\chi\widetilde{f}$. This shows that $\chi\widetilde{f}$ is a loop and so
statement ($\beta$) holds.

We continue the proof by showing that ($\gamma$) is equivalent with

  1. ($\delta$)

$\Upsilon\xi\in dC_{z}(X,\mathbb{T}^{1})$.

Assume, first, that ($\delta$) holds and let $\eta\in C_{z}(X,\mathbb{T}^{1})$
satisfy $\Upsilon\xi=\eta^{d}$. Then
$(\Upsilon\xi)^{\sharp}=(\kappa_{d}\eta)^{\sharp}=\kappa_{d}^{\sharp}\eta^{\sharp}$
and so
$\Upsilon^{\sharp}\xi^{\sharp}\pi_{1}(X)=(\Upsilon\xi)^{\sharp}\pi_{1}(X)=\kappa_{d}^{\sharp}\eta^{\sharp}\pi_{1}(X)\subseteq\kappa_{d}^{\sharp}\pi_{1}(\mathbb{T}^{1})$,
which verifies ($\gamma$). Conversely, let condition ($\gamma$) be fulfilled
and define a morphism $\varrho\colon\pi_{1}(X)\to\pi_{1}(\mathbb{T}^{1})$ by
the formula $\varrho=(1/d)(\Upsilon\xi)^{\sharp}$. By our assumptions on $X$,
$\varrho=\eta^{\sharp}$ for an appropriate $\eta\in C_{z}(X,\mathbb{T}^{1})$.
Then $((\Upsilon\xi).(\eta^{d})^{-1})^{\sharp}=0$ and so there is $\zeta\in
C_{z}(X,\mathbb{T}^{1})$ with $(\Upsilon\xi).(\eta^{d})^{-1}=\zeta^{d}$.
Consequently, $\Upsilon\xi=(\eta.\zeta)^{d}\in dC_{z}(X,\mathbb{T}^{1})$ and
statement ($\delta$) follows.

We finish the proof by showing that ($\delta$) is equivalent with

  1. ($\epsilon$)

$\chi\in(1/d)(\xi^{\flat})^{-1}(d\pi^{1}(X))$;

the equivalence of ($\alpha$) and ($\epsilon$) will then yield (5.13). Let
$\pi$ be the canonical projection $C_{z}(X,\mathbb{T}^{1})\to\pi^{1}(X)$.
Since the complementary summand of $\pi^{1}(X)$ in $C_{z}(X,\mathbb{T}^{1})$
is divisible, the following statements are equivalent:

  * •

$\Upsilon\xi\in dC_{z}(X,\mathbb{T}^{1})$,

  * •

$\xi^{\flat}(\Upsilon)=\pi(\Upsilon\xi)\in d\pi^{1}(X)$,

  * •

$\chi\in(1/d)(\xi^{\flat})^{-1}(d\pi^{1}(X))$.

This verifies the equivalence of ($\delta$) and ($\epsilon$). ∎

In order to formulate our next result we fix some notation. Let
$K\in\mathsf{CAGp}$ be a totally disconnected group and let $(H_{j})_{j\in J}$
be a family of open subgroups of $K$ indexed by a directed set $J$, such that
$H_{j}\subseteq H_{i}$ for $i\leq j$ and $\bigcap_{j\in J}H_{j}=e$. For $j\in
J$ set $K_{j}=K/H_{j}$ and let $q_{j}\colon K\to K_{j}$ be the quotient
morphism. Clearly, $K_{j}$ is a finite abelian group; denote by $d_{j}$ its
largest torsion coefficient. Given $i\leq j$, there is a unique (quotient)
morphism $q_{ij}\colon K_{j}\to K_{i}$ with $q_{ij}q_{j}=q_{i}$ and it follows
that $d_{i}$ divides $d_{j}$. We call $(d_{j})_{j\in J}$ a _generating net of
torsions for $K$_. If there exist $d\in\mathbb{N}$ and $i\in J$ such that
$d_{j}=d$ for all $j\geq i$ then we say that $(d_{j})_{j\in J}$ _terminates at
$d$_. Finally, since $\bigcap_{j\in J}H_{j}=e$, the inverse limit of the
inverse system $(K_{i}\stackrel{{\scriptstyle
q_{ij}}}{{\longleftarrow}}K_{j})_{i\leq j}$ in $\mathsf{CAGp}$ is the group
$K$ along with the morphisms $q_{j}$ as the limit projections.

######  Remark 5.38.

Notice the following facts.

  * •

It is possible for two non-isomorphic groups to have a common generating net
of torsions. For example, if $d\geq 2$ is an integer then both
$\mathbb{Z}_{d}$ and $\mathbb{Z}_{d}\oplus\mathbb{Z}_{d}$ have as their
generating net of torsions the singleton consisting of $d$ alone.

  * •

A given totally disconnected group $K\in\mathsf{CAGp}$ has in general many
distinct generating nets of torsions. Nevertheless, if $(d_{j})_{j\in J}$ and
$(d_{l}^{\prime})_{l\in L}$ are two such nets then they are related by the
following (symmetric) rules:

    * $\circ$

for every $j\in J$ there is $l\in L$ such that $d_{j}$ divides
$d_{l}^{\prime}$,

    * $\circ$

for every $l\in L$ there is $j\in J$ such that $d_{l}^{\prime}$ divides
$d_{j}$.

In particular, if $K$ possesses a generating net of torsions terminating at
$d$ then each of its generating nets of torsions terminates at $d$.

  * •

Assume that $(d_{j})_{j\in J}$ is a net in $\mathbb{N}$ with the property that
$d_{i}$ divides $d_{j}$ for $i\leq j$. Then there exists a totally
disconnected group $K\in\mathsf{CAGp}$, of which $(d_{j})_{j\in J}$ is a
generating net of torsions. Indeed, for $i\leq j$ let $q_{ij}$ be the natural
quotient morphism $\mathbb{Z}_{d_{j}}\to\mathbb{Z}_{d_{i}}$ (that is, under
the usual identifications
$\mathbb{Z}_{d_{i}}\subseteq\mathbb{Z}_{d_{j}}\subseteq\mathbb{T}^{1}$,
$q_{ij}$ acts by the rule $q_{ij}(z)=z^{d_{j}/d_{i}}$). Then the inverse limit
$K=\lim_{\leftarrow}(\mathbb{Z}_{d_{i}}\stackrel{{\scriptstyle
q_{ij}}}{{\longleftarrow}}\mathbb{Z}_{d_{j}})$ in $\mathsf{CAGp}$ is totally
disconnected and it has $(d_{j})_{j\in J}$ as its generating net of torsions.

We would like to make now one more remark in order to make some of our later
arguments more transparent.

######  Remark 5.39.

Let $G\in\mathsf{CAGp}$ and $K\mathrel{\lhd}G$ be totally disconnected. Fix a
generating net of torsions $(d_{j})_{j\in J}$ for $K$. Then

  * ($*$)

for every $\chi\in G^{*}$ there is $j\in J$ with $\chi^{d_{j}}\in K^{\perp}$.

To verify statement ($*$), fix $\chi\in G^{*}$. Recall that $V=\\{\exp(i2\pi
t):-1/3<t<1/3\\}$ is a neighbourhood of $1$ in $\mathbb{T}^{1}$ which contains
no non-trivial subgroup of $\mathbb{T}^{1}$. Fix a neighbourhood $U$ of $e$ in
$G$ with $\chi(U)\subseteq V$. By definition of $(d_{j})_{j\in J}$, there
exist $j\in J$ and an open subgroup $H_{j}$ of $K$ such that $H_{j}\subseteq
U$ and $K/H_{j}$ has $d_{j}$ as its largest torsion coefficient. Since
$\chi(H_{j})\subseteq\chi(U)\subseteq V$ is a subgroup of $\mathbb{T}^{1}$, it
follows that $\chi\in H_{j}^{\perp}$. Consequently, the restriction
$\chi|_{K}$ of $\chi$ to $K$ equals $\chi|_{K}=\chi_{j}p_{j}$ for some
$\chi_{j}\in(K/H_{j})^{*}$, where $p_{j}$ denotes the quotient morphism $K\to
K/H_{j}$. Then $\chi_{j}^{d_{j}}=1$ and hence $\chi^{d_{j}}\in K^{\perp}$, as
was to be shown.

######  Theorem 5.40.

Let $\mathop{\rm\mathcal{F}\colon\Gamma\curvearrowright X}$ be a minimal flow
and $G\in\mathsf{CAGp}$ be connected. Let $\mathcal{C}\in\mathop{\rm{\bf
Z}_{\mathcal{F}}}(G)$ and fix $z\in X$. Assume that $K\mathrel{\lhd}G$ is
totally disconnected with $p_{K}\mathcal{C}\in\mathop{\rm{\bf
B}_{\mathcal{F}}}(G/K)$ and denote by $\xi$ the transfer function of
$p_{K}\mathcal{C}$ with $\xi(z)=e$. If $X$ is compact connected, $\Gamma$ has
no non-trivial finite abelian quotient groups and $(d_{j})_{j\in J}$ is a
generating net of torsions for $K$ then

$\begin{split}F(\mathcal{C})^{\perp}&=\bigcup_{j\in
J}\frac{1}{d_{j}}\left(\xi^{\flat}\right)^{-1}\left(d_{j}\pi^{1}(X)\right)=\sum_{j\in
J}\frac{1}{d_{j}}\left(\xi^{\flat}\right)^{-1}\left(d_{j}\pi^{1}(X)\right)\\\
&=\lim_{\longrightarrow}\frac{1}{d_{j}}\left(\xi^{\flat}\right)^{-1}\left(d_{j}\pi^{1}(X)\right),\end{split}$
(5.14)

where $\xi^{\flat}$ stands for the morphism $G^{*}\supseteq
K^{\perp}\to\pi^{1}(X)$ under the usual identifications
$\pi^{1}(G/K)\cong(G/K)^{*}\cong K^{\perp}$.

######  Remark 5.41.

We wish to add the following remarks.

  * •

In the third step of the proof of Theorem 5.40 we shall observe that the
inclusion
$(1/d_{i})(\xi^{\flat})^{-1}(d_{i}\pi^{1}(X))\subseteq(1/d_{j})(\xi^{\flat})^{-1}(d_{j}\pi^{1}(X))$
holds for all $i,j\in J$ with $i\leq j$. Consequently, if the group $K$
possesses a generating net of torsions terminating at $d\in\mathbb{N}$ then
the formula (5.14) takes the form

$F(\mathcal{C})^{\perp}=\frac{1}{d}\left(\xi^{\flat}\right)^{-1}(d\pi^{1}(X)),$

just as in Theorem 5.32(2), where the group $K$ was assumed to be finite with
the largest torsion coefficient $d$.

  * •

It follows from the identity (5.14) that the inclusion

$K^{\perp}\subseteq\bigcup_{j\in
J}\frac{1}{d_{j}}\left(\xi^{\flat}\right)^{-1}(d_{j}\pi^{1}(X))$

holds under the assumptions of Theorem 5.40. Consequently, the equality
$F(\mathcal{C})=K$ holds if and only if
$(1/d_{j})(\xi^{\flat})^{-1}(d_{j}\pi^{1}(X))\subseteq K^{\perp}$ for every
$j\in J$.

###### Proof of Theorem 5.40.

We divide the proof of the theorem into three steps. A rough idea of the proof
is to use Theorem 5.32 together with the continuity of the functor
$F\colon\mathop{\rm\mathsf{CAGpZ}_{\mathcal{F}}}\to\mathsf{CAGp}$ from Theorem
3.7.

_1st step._ We begin by fixing some notation and by collecting some useful
observations.

Let $H_{j},K_{j},q_{j}$ ($j\in J$) and $q_{ij}$ ($i\leq j$) be as in the
paragraph preceding Remark 5.38. Further, for $j\in J$ let $p_{j}\colon G\to
G/H_{j}$ be the quotient morphism and for $i\leq j$ let $p_{ij}$ be the uniqe