Z}_{\mathcal{F}}}$ is a Polish group with the topology of u.c.s. convergence
and hence its cardinality is at most $\mathfrak{c}$. Consequently,

$\mathfrak{k}=\mathop{\rm rank}(\mathop{\rm{\bf
Z}_{\mathcal{F}}})\leq\mathop{\rm card}(\mathop{\rm{\bf
Z}_{\mathcal{F}}})\leq\mathfrak{c}.$ (4.2)

Further, by our assumptions on $X$, $C_{z}(X,\mathbb{R})\neq 0$. Therefore,
$\mathop{\rm dim_{\mathbb{R}LS}}(C_{z}(X,\mathbb{R}))\geq 1$ and hence
$\mathop{\rm dim_{\mathbb{Q}LS}}(C_{z}(X,\mathbb{R}))\geq\mathfrak{c}$. Since
the space $X$ is connected by virtue of Theorem 4.10,
$C_{z}(X,\mathbb{T}^{1})\cong\mathop{\rm{\bf
B}_{\mathcal{F}}}\subseteq\mathop{\rm{\bf Z}_{\mathcal{F}}}$ contains an
isomorphic copy of $C_{z}(X,\mathbb{R})$. Consequently,

$\mathfrak{k}=\mathop{\rm rank}(\mathop{\rm{\bf
Z}_{\mathcal{F}}})\geq\mathop{\rm rank}(C_{z}(X,\mathbb{R}))=\mathop{\rm
dim_{\mathbb{Q}LS}}(C_{z}(X,\mathbb{R}))\geq\mathfrak{c}.$ (4.3)

From (4.2) and (4.3) we infer that $\mathfrak{k}=\mathfrak{c}$. This yields
isomorphisms of groups $\mathop{\rm{\bf
Z}_{\mathcal{F}}}\cong\mathbb{Q}^{(\mathfrak{k})}=\mathbb{Q}^{(\mathfrak{c})}\cong\mathbb{R}$
and topological isomorphisms
$\mathop{\rm\gimel_{\mathcal{F}}}\cong(\mathop{\rm{\bf
Z}_{\mathcal{F}}})_{d}^{*}\cong(\mathbb{R}_{d})^{*}\cong b\mathbb{R}$. ∎

###  4.4 The non-existence of free minimal extensions

Let $\mathop{\rm\mathcal{F}\colon\Gamma\curvearrowright X}$ be a minimal flow.
In Section 4.2 we have seen that the category
$\mathop{\rm\mathsf{CAGpZ}_{\mathcal{F}}}$ always possesses a free object. Our
aim in this section is to show that this is not the case for the subcategory
of $\mathop{\rm\mathsf{CAGpZ}_{\mathcal{F}}}$ formed by the minimal extensions
of $\mathcal{F}$. The main results of this section are Theorems 4.23 and 4.27.
The first of them states that if both $\mathop{\rm{\bf MinZ}_{\mathcal{F}}}$
and $\mathop{\rm{\bf B}_{\mathcal{F}}}$ are non-trivial then a free object for
the category of the minimal extensions from
$\mathop{\rm\mathsf{CAGpZ}_{\mathcal{F}}}$ does not exist. (This applies, for
instance, to flows satisfying one of the three assumptions listed at the
beginning of Section 6.1 in Chapter 6.) The second theorem states that every
minimal extension $\mathcal{C}$ from
$\mathop{\rm\mathsf{CAGpZ}_{\mathcal{F}}}$ is an epimorphic image of a minimal
extension $\beth$, which is maximal in the sense that it can not be expressed
as a non-isomorphic epimorphic image of another minimal extension
$\beth^{\prime}$.

Let $\mathcal{F}$ be a minimal flow. Throughout this whole section the group
$\mathop{\rm{\bf Z}_{\mathcal{F}}}$ is assumed to carry the discrete topology.
We use notation
$(\mathop{\rm\gimel_{\mathcal{F}}},\mathop{\rm\daleth_{\mathcal{F}}})$ for the
model of the free extension of $\mathcal{F}$ constructed in Section 4.2. Given
$G\in\mathsf{CAGp}$ and $\mathcal{C}\in\mathop{\rm{\bf Z}_{\mathcal{F}}}(G)$,
we let $h_{\mathcal{C}}$ stand for the unique morphism
$\mathop{\rm\gimel_{\mathcal{F}}}\to G$ with
$h_{\mathcal{C}}\mathop{\rm\daleth_{\mathcal{F}}}=\mathcal{C}$. We also write
$H_{\mathcal{C}}=\ker(h_{\mathcal{C}})$ and $G_{\mathcal{C}}=\mathop{\rm
im}(h_{\mathcal{C}})$. Recall from Proposition 4.9($\iota$) that
$G_{\mathcal{C}}$ is the closed subgroup of $G$ generated by the values of
$\mathcal{C}$. Consequently, if
$\mathcal{C},\mathcal{D}\in\mathop{\rm\mathsf{CAGpZ}_{\mathcal{F}}}$ and $h$
is a morphism $\mathcal{C}\to\mathcal{D}$ then
$G_{\mathcal{D}}=h(G_{\mathcal{C}})$. Finally, given a subgroup $\Lambda$ of
$\mathop{\rm{\bf Z}_{\mathcal{F}}}$, we use the symbol $p_{\Lambda}$ to denote
the quotient morphism
$\mathop{\rm\gimel_{\mathcal{F}}}\to\mathop{\rm\gimel_{\mathcal{F}}}/\Lambda^{\perp}$.

######  Lemma 4.17.

Let $\mathop{\rm\mathcal{F}\colon\Gamma\curvearrowright X}$ be a minimal flow
and $(\mathop{\rm\gimel_{\mathcal{F}}},\mathop{\rm\daleth_{\mathcal{F}}})$ be
its free extension. Then the following statements hold.

  1. (1)

For every subgroup $\Lambda$ of $\mathop{\rm{\bf Z}_{\mathcal{F}}}$, the
values of $p_{\Lambda}\mathop{\rm\daleth_{\mathcal{F}}}$ generate a dense
subgroup of $\mathop{\rm\gimel_{\mathcal{F}}}/\Lambda^{\perp}$; that is,
$G_{p_{\Lambda\mathop{\rm\daleth_{\mathcal{F}}}}}=\mathop{\rm\gimel_{\mathcal{F}}}/\Lambda^{\perp}$.
In particular,
$G_{\mathop{\rm\daleth_{\mathcal{F}}}}=\mathop{\rm\gimel_{\mathcal{F}}}$.

  2. (2)

Given a group $G\in\mathsf{CAGp}$ and an extension
$\mathcal{C}\in\mathop{\rm{\bf Z}_{\mathcal{F}}}(G)$ with $G_{\mathcal{C}}=G$,
there is a unique subgroup $\Lambda$ of $\mathop{\rm{\bf Z}_{\mathcal{F}}}$
such that $\mathcal{C}$ is isomorphic to
$p_{\Lambda}\mathop{\rm\daleth_{\mathcal{F}}}$ in
$\mathop{\rm\mathsf{CAGpZ}_{\mathcal{F}}}$.

###### Proof.

We begin by verifying the identity
$G_{\mathop{\rm\daleth_{\mathcal{F}}}}=\mathop{\rm\gimel_{\mathcal{F}}}$. To
this end, we need to show that for every
$\Upsilon\in(\mathop{\rm\gimel_{\mathcal{F}}})^{*}$,
$\Upsilon\mathop{\rm\daleth_{\mathcal{F}}}=1$ implies $\Upsilon=1$. First,
denote by $\omega$ the Pontryagin isomorphism $\mathop{\rm{\bf
Z}_{\mathcal{F}}}\to(\mathop{\rm{\bf
Z}_{\mathcal{F}}})^{**}=(\mathop{\rm\gimel_{\mathcal{F}}})^{*}$ corresponding
to $\mathop{\rm{\bf Z}_{\mathcal{F}}}$. Now, fix
$\Upsilon\in(\mathop{\rm\gimel_{\mathcal{F}}})^{*}$ with
$\Upsilon\mathop{\rm\daleth_{\mathcal{F}}}=1$ and write
$\mathcal{C}=\omega^{-1}(\Upsilon)\in\mathop{\rm{\bf Z}_{\mathcal{F}}}$. Then
for all $\gamma\in\Gamma$ and $x\in X$,

$\mathcal{C}(\gamma,x)=\mathop{\rm\daleth_{\mathcal{F}}}(\gamma,x)(\mathcal{C})=\Upsilon(\mathop{\rm\daleth_{\mathcal{F}}}(\gamma,x))=\Upsilon\mathop{\rm\daleth_{\mathcal{F}}}(\gamma,x)=1.$

Thus, $\mathcal{C}=1$ and hence $\Upsilon=1$. Now we turn to the proof of the
main statement from (1). So fix a subgroup $\Lambda$ of $\mathop{\rm{\bf
Z}_{\mathcal{F}}}$. Then

$G_{p_{\Lambda}\mathop{\rm\daleth_{\mathcal{F}}}}=p_{\Lambda}(G_{\mathop{\rm\daleth_{\mathcal{F}}}})=p_{\Lambda}(\mathop{\rm\gimel_{\mathcal{F}}})=\mathop{\rm\gimel_{\mathcal{F}}}/\Lambda^{\perp},$

as was to be shown.

Now fix $G$ and $\mathcal{C}$ as in (2). First we check the existence part of
statement (2). Let $h_{\mathcal{C}}$ be the morphism
$\mathop{\rm\gimel_{\mathcal{F}}}\to G$ with
$h_{\mathcal{C}}\mathop{\rm\daleth_{\mathcal{F}}}=\mathcal{C}$. We claim that
$h_{\mathcal{C}}$ is an epimorphism; indeed, by virtue of (1),

$G=G_{\mathcal{C}}=G_{h_{\mathcal{C}}\mathop{\rm\daleth_{\mathcal{F}}}}=h_{\mathcal{C}}(G_{\mathop{\rm\daleth_{\mathcal{F}}}})=h_{\mathcal{C}}(\mathop{\rm\gimel_{\mathcal{F}}}).$

Consequently, there exist a subgroup $\Lambda$ of the group $\mathop{\rm{\bf
Z}_{\mathcal{F}}}$ and a topological isomorphism
$\varphi\colon\mathop{\rm\gimel_{\mathcal{F}}}/\Lambda^{\perp}\to G$ such that
$\varphi p_{\Lambda}=h_{\mathcal{C}}$. Then
$\varphi(p_{\Lambda}\mathop{\rm\daleth_{\mathcal{F}}})=h_{\mathcal{C}}\mathop{\rm\daleth_{\mathcal{F}}}=\mathcal{C}$,
whence it follows that $\varphi$ is an isomorphism
$p_{\Lambda}\mathop{\rm\daleth_{\mathcal{F}}}\to\mathcal{C}$.

Now we check the uniqueness part of statement (2). So let $\Lambda$ satisfy
the conclusion of (2) and fix an underlying isomorphism $\varphi\colon
p_{\Lambda}\mathop{\rm\daleth_{\mathcal{F}}}\to\mathcal{C}$. Then $(\varphi
p_{\Lambda})\mathop{\rm\daleth_{\mathcal{F}}}=\mathcal{C}=h_{\mathcal{C}}\mathop{\rm\daleth_{\mathcal{F}}}$,
and so $\varphi p_{\Lambda}$ and $h_{\mathcal{C}}$ agree on the set of all
values of $\mathop{\rm\daleth_{\mathcal{F}}}$. By virtue of (1), this means
that $\varphi p_{\Lambda}=h_{\mathcal{C}}$ on the whole
$\mathop{\rm\gimel_{\mathcal{F}}}$. Since $\varphi$ is an isomorphism by the
assumptions, we obtain
$\Lambda^{\perp}=\ker(p_{\Lambda})=\ker(h_{\mathcal{C}})$. Thus,
$\Lambda=\omega^{-1}(\Lambda^{\perp\perp})=\omega^{-1}(\ker(h_{\mathcal{C}})^{\perp})$.
This verifies the uniqueness part of (2). ∎

######  Lemma 4.18.

Let $\mathop{\rm\mathcal{F}\colon\Gamma\curvearrowright X}$ be a minimal flow.
Then the following statements hold.

  1. (i)

Given groups $H,K\in\mathsf{CAGp}$ and extensions
$\mathcal{C}\in\mathop{\rm{\bf Z}_{\mathcal{F}}}(H)$,
$\mathcal{D}\in\mathop{\rm{\bf Z}_{\mathcal{F}}}(K)$ with $G_{\mathcal{C}}=H$
and $G_{\mathcal{D}}=K$, the set $\mathop{\rm Hom}(\mathcal{C},\mathcal{D})$
is either a singleton or empty. In the first case the corresponding morphism
from $\mathop{\rm Hom}(\mathcal{C},\mathcal{D})$ is an epimorphism.

  2. (ii)

Given subgroups $\Lambda$ and $\Xi$ of $\mathop{\rm{\bf Z}_{\mathcal{F}}}$,
the set $\mathop{\rm
Hom}(p_{\Lambda}\mathop{\rm\daleth_{\mathcal{F}}},p_{\Xi}\mathop{\rm\daleth_{\mathcal{F}}})$
is non-empty if and only if $\Xi\subseteq\Lambda$.

######  Remark 4.19.

Recall that if $G\in\mathsf{CAGp}$ and $\mathcal{C}\in\mathop{\rm{\bf
Z}_{\mathcal{F}}}(G)$ is minimal then $G_{\mathcal{C}}=G$. Consequently,
statement (i) from Lemma 4.18, as well as statement (2) from Lemma 4.17,
applies to all minimal extensions from
$\mathop{\rm\mathsf{CAGpZ}_{\mathcal{F}}}$.

###### Proof of Lemma 4.18.

Fix $H$, $K$, $\mathcal{C}$ and $\mathcal{D}$ as in (i) and assume that
$\varphi$ is a morphism $\mathcal{C}\to\mathcal{D}$. Consider the morphisms
$h_{\mathcal{C}}\colon\mathop{\rm\gimel_{\mathcal{F}}}\to H$ and
$h_{\mathcal{D}}\colon\mathop{\rm\gimel_{\mathcal{F}}}\to K$ with
$h_{\mathcal{C}}\mathop{\rm\daleth_{\mathcal{F}}}=\mathcal{C}$ and
$h_{\mathcal{D}}\mathop{\rm\daleth_{\mathcal{F}}}=\mathcal{D}$, respectively.
Then $(\varphi
h_{\mathcal{C}})\mathop{\rm\daleth_{\mathcal{F}}}=\varphi\mathcal{C}=\mathcal{D}=h_{\mathcal{D}}\mathop{\rm\daleth_{\mathcal{F}}}$
and hence, by Lemma 4.17(1), $\varphi h_{\mathcal{C}}=h_{\mathcal{D}}$. Since
both $h_{\mathcal{C}}$ and $h_{\mathcal{D}}$ are epimorphisms by our
assumptions on $\mathcal{C}$ and $\mathcal{D}$, it follows that such a
morphism $\varphi$ is at most one and, if it exists, it is in fact an
epimorphism. This verifies statement (i).

Now fix subgroups $\Lambda$ and $\Xi$ of $\mathop{\rm{\bf Z}_{\mathcal{F}}}$.
First we verify the “if” part of statement (ii). So assume that
$\Xi\subseteq\Lambda$. Then $\Lambda^{\perp}\subseteq\Xi^{\perp}$ and so there
is a morphism
$\varphi\colon\mathop{\rm\gimel_{\mathcal{F}}}/\Lambda^{\perp}\to\mathop{\rm\gimel_{\mathcal{F}}}/\Xi^{\perp}$
with $\varphi p_{\Lambda}=p_{\Xi}$. Consequently,
$\varphi(p_{\Lambda}\mathop{\rm\daleth_{\mathcal{F}}})=p_{\Xi}\mathop{\rm\daleth_{\mathcal{F}}}$
and hence $\varphi$ is a desired morphism
$p_{\Lambda}\mathop{\rm\daleth_{\mathcal{F}}}\to
p_{\Xi}\mathop{\rm\daleth_{\mathcal{F}}}$. Now we turn to proving the “only
if” part of statement (ii). So let $\varphi$ be a morphism of extensions
$p_{\Lambda}\mathop{\rm\daleth_{\mathcal{F}}}\to
p_{\Xi}\mathop{\rm\daleth_{\mathcal{F}}}$. Then $\varphi$ is a morphism of
topological groups
$\mathop{\rm\gimel_{\mathcal{F}}}/\Lambda^{\perp}\to\mathop{\rm\gimel_{\mathcal{F}}}/\Xi^{\perp}$
with $(\varphi
p_{\Lambda})\mathop{\rm\daleth_{\mathcal{F}}}=p_{\Xi}\mathop{\rm\daleth_{\mathcal{F}}}$.
Thus, by Lemma 4.17(1), $\varphi p_{\Lambda}=p_{\Xi}$. This shows that
$\Lambda^{\perp}\subseteq\Xi^{\perp}$ and hence $\Xi\subseteq\Lambda$. ∎

######  Lemma 4.20.

Let $\mathop{\rm\mathcal{F}\colon\Gamma\curvearrowright X}$ be a minimal flow,
$(\mathop{\rm\gimel_{\mathcal{F}}},\mathop{\rm\daleth_{\mathcal{F}}})$ be the
free extension of $\mathcal{F}$ and $\Lambda$ be a subgroup of
$\mathop{\rm{\bf Z}_{\mathcal{F}}}$. Then
$p_{\Lambda}\mathop{\rm\daleth_{\mathcal{F}}}\in\mathop{\rm{\bf
Z}_{\mathcal{F}}}(\mathop{\rm\gimel_{\mathcal{F}}}/\Lambda^{\perp})$ is
minimal if and only if $\Lambda\cap\mathop{\rm{\bf B}_{\mathcal{F}}}=1$.

###### Proof.

By Lemma 4.17(1),
$G_{p_{\Lambda}\mathop{\rm\daleth_{\mathcal{F}}}}=\mathop{\rm\gimel_{\mathcal{F}}}/\Lambda^{\perp}$.
Moreover,
$H_{p_{\Lambda}\mathop{\rm\daleth_{\mathcal{F}}}}=\ker(p_{\Lambda})=\Lambda^{\perp}$
and, by the first part of Remark 4.8,
$F(\mathop{\rm\daleth_{\mathcal{F}}})=(\mathop{\rm{\bf
B}_{\mathcal{F}}})^{\perp}$. Consequently, it follows from Proposition
4.9($\iota\iota\iota$) that the following conditions are equivalent:

  * •

$p_{\Lambda}\mathop{\rm\daleth_{\mathcal{F}}}$ is minimal,

  * •

$(H_{p_{\Lambda}\mathop{\rm\daleth_{\mathcal{F}}}})^{\perp}\cap
F(\mathop{\rm\daleth_{\mathcal{F}}})^{\perp}=1$,

  * •

$\Lambda^{\perp\perp}\cap(\mathop{\rm{\bf B}_{\mathcal{F}}})^{\perp\perp}=1$

  * •

$\Lambda\cap\mathop{\rm{\bf B}_{\mathcal{F}}}=1$.

∎

Let $\mathcal{F}$ be a minimal flow,
$\mathop{\rm\gimel_{\mathcal{F}}^{m}}\in\mathsf{CAGp}$ and
$\mathop{\rm\daleth_{\mathcal{F}}^{m}}\in\mathop{\rm{\bf
Z}_{\mathcal{F}}}(\mathop{\rm\gimel_{\mathcal{F}}^{m}})$. We say that
$(\mathop{\rm\gimel_{\mathcal{F}}^{m}},\mathop{\rm\daleth_{\mathcal{F}}^{m}})$
is a _free minimal extension of $\mathcal{F}$_ if the following condition
holds:

  1. ($m$)

for every $G\in\mathsf{CAGp}$ and every minimal extension
$\mathcal{C}\in\mathop{\rm{\bf Z}_{\mathcal{F}}}(G)$ there exists
$q\in\mathop{\rm Hom}(\mathop{\rm\gimel_{\mathcal{F}}^{m}},G)$ with
$q\mathop{\rm\daleth_{\mathcal{F}}^{m}}=\mathcal{C}$.

Observe that the morphism $q$ from ($m$) is unique by Lemma 4.18(i).
Consequently, if a free minimal extension of $\mathcal{F}$ does exist then it
is unique up to an isomorphism. We shall therefore speak of _the_ free minimal
extension of $\mathcal{F}$.

######  Lemma 4.21.

Let $\mathop{\rm\mathcal{F}\colon\Gamma\curvearrowright X}$ be a minimal flow.
Then the following conditions are equivalent:

  1. (a)

the free minimal extension of $\mathcal{F}$ exists,

  2. (b)

there is a largest subgroup $\Omega$ of $\mathop{\rm{\bf Z}_{\mathcal{F}}}$
with $\Omega\cap\mathop{\rm{\bf B}_{\mathcal{F}}}=1$.

######  Remark 4.22.

Assume that the group $\mathop{\rm{\bf Z}_{\mathcal{F}}}$ is torsion-free;
that is, $X$ is connected and $\Gamma$ has no finite abelian quotient groups.
Then a subgroup $\Omega$ of $\mathop{\rm{\bf Z}_{\mathcal{F}}}$ satisfies
$\Omega\cap\mathop{\rm{\bf B}_{\mathcal{F}}}=1$ if and only if it is a
subgroup of the groupoid $\mathop{\rm{\bf MinZ}_{\mathcal{F}}}$.

###### Proof of Lemma 4.21.

We show that (b) follows from (a). So assume that
$(\mathop{\rm\gimel_{\mathcal{F}}^{m}},\mathop{\rm\daleth_{\mathcal{F}}^{m}})$
is the free minimal extension of $\mathcal{F}$ and denote by $h$ the unique
morphism
$\mathop{\rm\gimel_{\mathcal{F}}}\to\mathop{\rm\gimel_{\mathcal{F}}^{m}}$ with
$h\mathop{\rm\daleth_{\mathcal{F}}}=\mathop{\rm\daleth_{\mathcal{F}}^{m}}$. By
Lemma 4.17(2), we may assume, without loss of generality, that
$\mathop{\rm\gimel_{\mathcal{F}}^{m}}=\mathop{\rm\gimel_{\mathcal{F}}}/\Omega^{\perp}$
and
$\mathop{\rm\daleth_{\mathcal{F}}^{m}}=p_{\Omega}\mathop{\rm\daleth_{\mathcal{F}}}$
for an appropriate subgroup $\Omega$ of $\mathop{\rm{\bf Z}_{\mathcal{F}}}$.
We show that $\Omega$ fulfills condition (b). First, by virtue of Lemma 4.20,
the minimality of
$\mathop{\rm\daleth_{\mathcal{F}}^{m}}=p_{\Omega}\mathop{\rm\daleth_{\mathcal{F}}}$
yields $\Omega\cap\mathop{\rm{\bf B}_{\mathcal{F}}}=1$. Further, fix
$\Lambda\mathrel{\lhd}\mathop{\rm{\bf Z}_{\mathcal{F}}}$ with
$\Lambda\cap\mathop{\rm{\bf B}_{\mathcal{F}}}=1$; we show that
$\Lambda\subseteq\Omega$. By virtue of Lemma 4.20,
$p_{\Lambda}\mathop{\rm\daleth_{\mathcal{F}}}\in\mathop{\rm{\bf
Z}_{\mathcal{F}}}(\mathop{\rm\gimel_{\mathcal{F}}}/\Lambda^{\perp})$ is a
minimal extension. Since $p_{\Omega}\mathop{\rm\daleth_{\mathcal{F}}}$ is the
free minimal extension of $\mathcal{F}$, we have $\mathop{\rm
Hom}(p_{\Omega}\mathop{\rm\daleth_{\mathcal{F}}},p_{\Lambda}\mathop{\rm\daleth_{\mathcal{F}}})\neq\emptyset$.
Thus, by Lemma 4.18(ii), $\Lambda\subseteq\Omega$.

We show that (a) follows from (b). So let $\Omega$ be as in (b). Set
$\mathop{\rm\gimel_{\mathcal{F}}^{m}}=\mathop{\rm\gimel_{\mathcal{F}}}/\Omega^{\perp}$
and
$\mathop{\rm\daleth_{\mathcal{F}}^{m}}=p_{\Omega}\mathop{\rm\daleth_{\mathcal{F}}}$;
we show that
$(\mathop{\rm\gimel_{\mathcal{F}}^{m}},\mathop{\rm\daleth_{\mathcal{F}}^{m}})$
is the free minimal extension of $\mathcal{F}$. First, the extension
$\mathop{\rm\daleth_{\mathcal{F}}^{m}}$ is minimal by Lemma 4.20. Further, fix
$G\in\mathsf{CAGp}$ and a minimal extension $\mathcal{C}\in\mathop{\rm{\bf
Z}_{\mathcal{F}}}(G)$; we need to find a morphism
$q\colon\mathop{\rm\gimel_{\mathcal{F}}^{m}}\to G$ with
$q\mathop{\rm\daleth_{\mathcal{F}}^{m}}=\mathcal{C}$. By Lemma 4.17(2), we may
assume, without loss of generality, that
$G=\mathop{\rm\gimel_{\mathcal{F}}}/\Lambda^{\perp}$ and
$\mathcal{C}=p_{\Lambda}\mathop{\rm\daleth_{\mathcal{F}}}$ for an appropriate
subgroup $\Lambda$ of $\mathop{\rm{\bf Z}_{\mathcal{F}}}$. By virtue of Lemma
4.20, $\Lambda\cap\mathop{\rm{\bf B}_{\mathcal{F}}}=1$ and hence
$\Lambda\subseteq\Omega$ by our choice of $\Omega$. Lemma 4.18(ii) then yields
a desired morphism $\mathop{\rm\daleth_{\mathcal{F}}^{m}}\to
p_{\Lambda}\mathop{\rm\daleth_{\mathcal{F}}}$. ∎

######  Theorem 4.23.

Let $\mathop{\rm\mathcal{F}\colon\Gamma\curvearrowright X}$ be a minimal flow
with both $\mathop{\rm{\bf B}_{\mathcal{F}}}$ and $\mathop{\rm{\bf
MinZ}_{\mathcal{F}}}$ non-trivial. Then the free minimal extension of
$\mathcal{F}$ does not exist.

######  Remark 4.24.

The assumptions of the theorem are satisfied, provided the flow $\mathcal{F}$
fulfills one of the three assumptions listed at the beginning of Section 6.1
in Chapter 6. First, in all the three situations $X$ is a non-trivial compact
space and so the group $\mathop{\rm{\bf B}_{\mathcal{F}}}\cong
C_{z}(X,\mathbb{T}^{1})$ is non-trivial. Second, it follows from Theorems 6.3,
6.5 and 6.6 that the groupoid $\mathop{\rm{\bf MinZ}_{\mathcal{F}}}$ is also
non-trivial.

###### Proof of Theorem 4.23.

We proceed by contradiction, assuming that $\mathcal{F}$ does possess the free
minimal extension. By virtue of Lemma 4.21, there is a largest subgroup
$\Omega$ of $\mathop{\rm{\bf Z}_{\mathcal{F}}}$ with the property
$\Omega\cap\mathop{\rm{\bf B}_{\mathcal{F}}}=1$. Fix
$\mathcal{C}\in\mathop{\rm{\bf MinZ}_{\mathcal{F}}}\setminus 1$ and
$\mathcal{D}\in\mathop{\rm{\bf B}_{\mathcal{F}}}\setminus 1$. Then the
subgroups $\langle\mathcal{C}\rangle$ and
$\langle\mathcal{C}\mathcal{D}\rangle$ of $\mathop{\rm{\bf Z}_{\mathcal{F}}}$
generated by $\mathcal{C}$ and $\mathcal{C}\mathcal{D}$, respectively, are
contained in the groupoid $\mathop{\rm{\bf MinZ}_{\mathcal{F}}}$, and hence
they intersect the group $\mathop{\rm{\bf B}_{\mathcal{F}}}$ only at the
identity. Thus, by definition of $\Omega$, both $\langle\mathcal{C}\rangle$
and $\langle\mathcal{C}\mathcal{D}\rangle$ are contained in $\Omega$.
Consequently,
$\mathcal{D}=(\mathcal{C}\mathcal{D})\mathcal{C}^{-1}\in\Omega\cap(\mathop{\rm{\bf
B}_{\mathcal{F}}}\setminus 1)$, which contradicts the definition of $\Omega$.
∎

######  Lemma 4.25.

Let $\mathop{\rm\mathcal{F}\colon\Gamma\curvearrowright X}$ be a minimal flow,
$\beth$ be a minimal extension from $\mathop{\rm\mathsf{CAGpZ}_{\mathcal{F}}}$
and $\Lambda$ be the subgroup of $\mathop{\rm{\bf Z}_{\mathcal{F}}}$ with
$p_{\Lambda}\mathop{\rm\daleth_{\mathcal{F}}}$ isomorphic to $\beth$. Then the
following conditions are equivalent:

  1. ($\alpha$)

every minimal extension $\beth^{\prime}$ from
$\mathop{\rm\mathsf{CAGpZ}_{\mathcal{F}}}$ with $\mathop{\rm
Hom}(\beth^{\prime},\beth)\neq\emptyset$ is isomorphic to $\beth$,

  2. ($\beta$)

$\Lambda$ is maximal (with respect to inclusion) among the subgroups of
$\mathop{\rm{\bf Z}_{\mathcal{F}}}$ intersecting $\mathop{\rm{\bf
B}_{\mathcal{F}}}$ only at the identity.

######  Remark 4.26.

We wish to mention the following facts.

  * •

Condition ($\alpha$) can be reformulated by saying that $\beth$ is a maximal
object in the category of the minimal extensions from
$\mathop{\rm\mathsf{CAGpZ}_{\mathcal{F}}}$.

  * •

By Lemma 4.17(2), the isomorphism classes of the extensions $\beth$ satisfying
($\alpha$) are in a one-to-one correspondence with the subgroups $\Lambda$ of
$\mathop{\rm{\bf Z}_{\mathcal{F}}}$ satisfying ($\beta$). A group $\Lambda$
satisfying ($\beta$) is not unique in general and so two extensions $\beth$
satisfying ($\alpha$) need not be isomorphic; see Example 4.29 below.

  * •

Let $\Lambda_{1},\Lambda_{2}$ be subgroups of $\mathop{\rm{\bf
Z}_{\mathcal{F}}}$ satisfying ($\beta$) and let
$\beth_{i}=p_{\Lambda_{i}}\mathop{\rm\daleth_{\mathcal{F}}}\in\mathop{\rm{\bf
Z}_{\mathcal{F}}}(\mathop{\rm\gimel_{\mathcal{F}}}/\Lambda_{i}^{\perp})$ for
$i=1,2$. As mentioned above, $\beth_{1},\beth_{2}$ need not be isomorphic in
$\mathop{\rm\mathsf{CAGpZ}_{\mathcal{F}}}$. As a matter of fact, it is
possible that the groups
$\mathop{\rm\gimel_{\mathcal{F}}}/\Lambda_{1}^{\perp}$,
$\mathop{\rm\gimel_{\mathcal{F}}}/\Lambda_{2}^{\perp}$ are not isomorphic in
$\mathsf{CAGp}$; see Example 4.30 below.

###### Proof of Lemma 4.25.

Before turning to the proof oberve that by Lemma 4.17(2), there is indeed a
unique subgroup $\Lambda$ of $\mathop{\rm{\bf Z}_{\mathcal{F}}}$ such that
$\beth$ is isomorphic to $p_{\Lambda}\mathop{\rm\daleth_{\mathcal{F}}}$. Also,
since $\beth$ is minimal, $\Lambda\cap\mathop{\rm{\bf B}_{\mathcal{F}}}=1$ by
virtue of Lemma 4.20.

We show that ($\beta$) follows from ($\alpha$). So assume that ($\alpha$)
holds and fix a subgroup $\Xi$ of $\mathop{\rm{\bf Z}_{\mathcal{F}}}$ with
$\Xi\cap\mathop{\rm{\bf B}_{\mathcal{F}}}=1$ and $\Lambda\subseteq\Xi$. Then
the extension $\beth^{\prime}=p_{\Xi}\mathop{\rm\daleth_{\mathcal{F}}}$ is
minimal by Lemma 4.20. Furthermore, Lemma 4.18(ii) yields $\mathop{\rm
Hom}(\beth^{\prime},\beth)\neq\emptyset$. Consequently, $\beth^{\prime}$ and
$\beth$ are isomorphic in $\mathop{\rm\mathsf{CAGpZ}_{\mathcal{F}}}$ by virtue
of ($\alpha$), and Lemma 4.18(ii) thus yields $\Lambda=\Xi$. This verifies
($\beta$).

We show that ($\alpha$) follows from ($\beta$). So assume that ($\beta$) holds
and fix a minimal extension $\beth^{\prime}$ from
$\mathop{\rm\mathsf{CAGpZ}_{\mathcal{F}}}$ with $\mathop{\rm
Hom}(\beth^{\prime},\beth)\neq\emptyset$. By Lemma 4.17(2), $\beth^{\prime}$
is isomorphic to $p_{\Xi}\mathop{\rm\daleth_{\mathcal{F}}}$ for an appropriate
subgroup $\Xi$ of $\mathop{\rm{\bf Z}_{\mathcal{F}}}$. Moreover, Lemmas 4.20
and 4.18 yield $\Xi\cap\mathop{\rm{\bf B}_{\mathcal{F}}}=1$ and
$\Lambda\subseteq\Xi$. Thus, by applying ($\beta$), we obtain $\Xi=\Lambda$.
Consequently, we have isomorphisms of extensions $\beth^{\prime}\cong
p_{\Xi}\mathop{\rm\daleth_{\mathcal{F}}}=p_{\Lambda}\mathop{\rm\daleth_{\mathcal{F}}}\cong\beth$.
This verifies ($\alpha$). ∎

######  Theorem 4.27.

Let $\mathop{\rm\mathcal{F}\colon\Gamma\curvearrowright X}$ be a minimal flow
and $\mathcal{C}$ be a minimal extension from
$\mathop{\rm\mathsf{CAGpZ}_{\mathcal{F}}}$. Then there is a minimal extension
$\beth$ in $\mathop{\rm\mathsf{CAGpZ}_{\mathcal{F}}}$ with the following
properties:

  1. (a)

there is an epimorphism $\beth\to\mathcal{C}$,

  2. (b)

every minimal extension $\beth^{\prime}$ from
$\mathop{\rm\mathsf{CAGpZ}_{\mathcal{F}}}$ with $\mathop{\rm
Hom}(\beth^{\prime},\beth)\neq\emptyset$ is isomorphic to $\beth$.

######  Remark 4.28.

In view of our discussion from Remark 4.26 we may reformulate the statement of
the theorem by saying that in the category of the minimal extensions from
$\mathop{\rm\mathsf{CAGpZ}_{\mathcal{F}}}$, every object $\mathcal{C}$ is an
epimorphic image of a maximal object $\beth$.

###### Proof of Theorem 4.27.

By Lemma 4.17(2), $\mathcal{C}$ is isomorphic in
$\mathop{\rm\mathsf{CAGpZ}_{\mathcal{F}}}$ to
$p_{\Xi}\mathop{\rm\daleth_{\mathcal{F}}}$ for an appropriate subgroup $\Xi$
of $\mathop{\rm{\bf Z}_{\mathcal{F}}}$. Moreover, Lemma 4.20 yields
$\Xi\cap\mathop{\rm{\bf B}_{\mathcal{F}}}=1$. By an elementary argument
involving Zorn’s lemma one finds a subgroup $\Lambda\supseteq\Xi$ of
$\mathop{\rm{\bf Z}_{\mathcal{F}}}$, which is maximal (with respect to
inclusion) among the subgroups of $\mathop{\rm{\bf Z}_{\mathcal{F}}}$
intersecting $\mathop{\rm{\bf B}_{\mathcal{F}}}$ only at the identity. Now set
$\beth=p_{\Lambda}\mathop{\rm\daleth_{\mathcal{F}}}$. Then $\beth$ is a
minimal extension of $\mathcal{F}$ due to Lemma 4.20. Further, since
$\Xi\subseteq\Lambda$, condition (a) holds by virtue of Lemma 4.18. Finally,
condition (b) follows from Lemma 4.25 by our choice of $\Lambda$. ∎

######  Example 4.29.

Let $\mathop{\rm\mathcal{F}\colon\Gamma\curvearrowright X}$ be a minimal flow
with $X$ non-degenerate locally compact and assume that $\mathop{\rm{\bf
MinZ}_{\mathcal{F}}}\neq 1$. Let $\mathfrak{A}$ be the family of all subgroups
$\Lambda$ of $\mathop{\rm{\bf Z}_{\mathcal{F}}}$, which are maximal with
respect to the condition $\Lambda\cap\mathop{\rm{\bf B}_{\mathcal{F}}}=1$. By
the second part of Remark 4.26, $\mathfrak{A}$ can be identified with the set
of all isomorphism classes of the maximal objects in the category of the
minimal extensions from $\mathop{\rm\mathsf{CAGpZ}_{\mathcal{F}}}$. To
demonstrate that there are many such isomorphism classes in general, we show
that $\mathop{\rm card}(\mathfrak{A})\geq\mathfrak{c}$ under our current
assumptions on $\mathcal{F}$.

Fix $\mathcal{C}\in\mathop{\rm{\bf MinZ}_{\mathcal{F}}}\setminus 1$. For
$\mathcal{D}\in\mathop{\rm{\bf B}_{\mathcal{F}}}$ let $\Lambda_{\mathcal{D}}$
be an element of $\mathfrak{A}$ with
$\mathcal{C}\mathcal{D}\in\Lambda_{\mathcal{D}}$. (Such a group
$\Lambda_{\mathcal{D}}$ of course exists by Zorn’s lemma, since
$\langle\mathcal{C}\mathcal{D}\rangle\cap\mathop{\rm{\bf
B}_{\mathcal{F}}}=1$.) We claim that distinct coboundaries $\mathcal{D}$
correspond to distinct groups $\Lambda_{\mathcal{D}}$. To see this, fix
$\mathcal{D}_{1},\mathcal{D}_{2}\in\mathop{\rm{\bf B}_{\mathcal{F}}}$ and
assume that $\Lambda_{\mathcal{D}_{1}}=\Lambda_{\mathcal{D}_{2}}$. Then

$\mathop{\rm{\bf
B}_{\mathcal{F}}}\ni\mathcal{D}_{1}\mathcal{D}_{2}^{-1}=(\mathcal{C}\mathcal{D}_{1})(\mathcal{C}\mathcal{D}_{2})^{-1}\in\Lambda_{\mathcal{D}_{1}}\Lambda_{\mathcal{D}_{2}}^{-1}=\Lambda_{\mathcal{D}_{1}}.$

Since $\Lambda_{\mathcal{D}_{1}}\cap\mathop{\rm{\bf B}_{\mathcal{F}}}=1$, it
follows that $\mathcal{D}_{1}=\mathcal{D}_{2}$. Consequently,

$\mathop{\rm card}(\mathfrak{A})\geq\mathop{\rm card}(\mathop{\rm{\bf
B}_{\mathcal{F}}})=\mathop{\rm
card}(C_{z}(X,\mathbb{T}^{1}))\geq\mathfrak{c},$

as was to be shown.

######  Example 4.30.

In the previous example we have seen that two extensions $\beth$ satisfying
condition ($\alpha$) from Lemma 4.25 need not be isomorphic in
$\mathop{\rm\mathsf{CAGpZ}_{\mathcal{F}}}$. In this example we show that it is
in fact possible for their sections $F(\beth)=G_{\beth}$ to be non-isomorphic
in $\mathsf{CAGp}$. To this end, we shall use a result from Chapter 6.

Let
$\mathcal{F}\colon\mathbb{R}\times\mathbb{T}^{1}\curvearrowright\mathbb{T}^{2}$
be defined as the product action of a minimal continuous flow on
$\mathbb{T}^{1}$ with the natural action of $\mathbb{T}^{1}$ on itself.
Clearly, $\mathcal{F}$ thus defined is a minimal flow. Choose $z=(1,1)$ as the
base point for $\mathbb{T}^{2}$ and consider the corresponding motion map
$\mathcal{F}_{z}\colon\mathbb{R}\times\mathbb{T}^{1}\to\mathbb{T}^{2}$. The
induced morphism $\mathcal{F}_{z}^{\sharp}\colon
H_{1}^{w}(\mathbb{R}\times\mathbb{T}^{1})\to H_{1}^{w}(\mathbb{T}^{2})$ is a
monomorphism and it takes the form of an inclusion of groups
$0\oplus\mathbb{Z}\subseteq\mathbb{Z}\oplus\mathbb{Z}$. It follows that the
flow $\mathcal{F}$ is topologically free and possesses a free cycle. By virtue
of (6.9) from Theorem 6.14, the inclusion $\mathop{\rm{\bf
B}_{\mathcal{F}}}\subseteq\mathop{\rm{\bf Z}_{\mathcal{F}}}$ takes the form of
an inclusion of abelian groups

$\begin{array}[]{ccccccccccc}\mathop{\rm{\bf
B}_{\mathcal{F}}}&\cong&0&\oplus&\mathbb{R}&\oplus&\mathbb{Z}&\oplus&\mathbb{Z}&\subseteq&{}\hfil\\\
{}\hfil&\subseteq&\mathbb{R}&\oplus&\mathbb{R}&\oplus&\mathbb{Q}&\oplus&\mathbb{Z}&\cong&\mathop{\rm{\bf
Z}_{\mathcal{F}}}.\end{array}$ (4.4)

By writing the inclusion $0\subseteq\mathbb{R}$ in the form $0\oplus
0\subseteq\mathbb{R}\oplus\mathbb{Q}$ and by reorganizing the direct summands
from the isomorphism above, we put (4.4) into the following form:

$\begin{array}[]{ccccccccccc}\mathop{\rm{\bf
B}_{\mathcal{F}}}&\cong&0&\oplus&\mathbb{R}&\oplus&\mathbb{Z}&\oplus&(0\oplus\mathbb{Z})&\subseteq&{}\hfil\\\
{}\hfil&\subseteq&\mathbb{R}&\oplus&\mathbb{R}&\oplus&\mathbb{Q}&\oplus&(\mathbb{Q}\oplus\mathbb{Z})&\cong&\mathop{\rm{\bf
Z}_{\mathcal{F}}}.\end{array}$

Set $\Lambda_{1}=\mathbb{R}\oplus 0\oplus 0\oplus A_{1}$ and
$\Lambda_{2}=\mathbb{R}\oplus 0\oplus 0\oplus A_{2}$, where
$A_{1}=\mathbb{Q}\oplus 0\cong\mathbb{Q}$ and
$A_{2}=\\{(k,k):k\in\mathbb{Z}\\}\cong\mathbb{Z}$. For $i=1,2$ set
$\beth_{i}=p_{\Lambda_{i}}\mathop{\rm\daleth_{\mathcal{F}}}$. Elementary
arguments show that both $\Lambda_{1}$ and $\Lambda_{2}$ are maximal among the
subgroups of $\mathop{\rm{\bf Z}_{\mathcal{F}}}$ intersecting $\mathop{\rm{\bf
B}_{\mathcal{F}}}$ only at the identity, and hence both $\beth_{1}$ and
$\beth_{2}$ are minimal and satisfy condition ($\alpha$) from Lemma 4.25. We
claim that the groups $G_{\beth_{1}}$ and $G_{\beth_{2}}$ are not isomorphic
in $\mathsf{CAGp}$. Indeed, there are topological isomorphisms

$G_{\beth_{1}}=F(\beth_{1})=\mathop{\rm\gimel_{\mathcal{F}}}/\Lambda_{1}^{\perp}\cong\Lambda_{1}^{*}\cong
b\mathbb{R}\times\mathbb{Q}^{*}$

and

$G_{\beth_{2}}=F(\beth_{2})=\mathop{\rm\gimel_{\mathcal{F}}}/\Lambda_{2}^{\perp}\cong\Lambda_{2}^{*}\cong
b\mathbb{R}\times\mathbb{T}^{1}.$

These two groups are not topologically isomorphic, for the first one is
torsion-free, while the other one is not.

######  Remark 4.31.

Let $\mathop{\rm\mathcal{F}\colon\Gamma\curvearrowright X}$ be a minimal flow
with $\mathop{\rm{\bf Z}_{\mathcal{F}}}$ torsion-free. As has been shown in
the preceding example, two distinct extensions $\beth$ satisfying condition
($\alpha$) from Lemma 4.25 may have non-isomorphic sections
$F(\beth)=G_{\beth}$. We would like to show now that, after all, there does
exist an invariant for such extensions $\beth$, namely the topological
dimension of the section $\dim(G_{\beth})$. So fix $\beth$ satisfying
($\alpha$) from Lemma 4.25 and let $\Lambda$ be the subgroup of
$\mathop{\rm{\bf Z}_{\mathcal{F}}}$ with $\beth\cong
p_{\Lambda}\mathop{\rm\daleth_{\mathcal{F}}}$ in
$\mathop{\rm\mathsf{CAGpZ}_{\mathcal{F}}}$. By Lemma 4.25, $\Lambda$ is
maximal among the subgroups of $\mathop{\rm{\bf Z}_{\mathcal{F}}}$
intersecting $\mathop{\rm{\bf B}_{\mathcal{F}}}$ only at the identity.
Consequently, the divisible hull $\mathbb{Q}\otimes\mathop{\rm{\bf
Z}_{\mathcal{F}}}$ of $\mathop{\rm{\bf Z}_{\mathcal{F}}}$ takes the form of a
direct sum of abelian groups (and hence also of rational linear spaces)
$\mathbb{Q}\otimes\mathop{\rm{\bf
Z}_{\mathcal{F}}}=\left(\mathbb{Q}\otimes\mathop{\rm{\bf
B}_{\mathcal{F}}}\right)\oplus\left(\mathbb{Q}\otimes\Lambda\right)$. Thus,

$\begin{split}\dim\left(G_{\beth}\right)&=\dim\left(\mathop{\rm\gimel_{\mathcal{F}}}/\Lambda^{\perp}\right)=\dim\left(\Lambda^{*}\right)=\mathop{\rm
rank}(\Lambda)=\mathop{\rm dim_{\mathbb{Q}LS}}(\mathbb{Q}\otimes\Lambda)\\\
&=\mathop{\rm dim_{\mathbb{Q}LS}}\left((\mathbb{Q}\otimes\mathop{\rm{\bf
Z}_{\mathcal{F}}})/(\mathbb{Q}\otimes\mathop{\rm{\bf
B}_{\mathcal{F}}})\right)\end{split}$

does not depend on the choice of $\beth$.

###  4.5 Cohomology classes as group morphisms

Let $\mathop{\rm\mathcal{F}\colon\Gamma\curvearrowright X}$ be a minimal flow
and assume that the group $\mathop{\rm{\bf Z}_{\mathcal{F}}}$ is divisible. In
Theorem 3.1 from Section 3.1 we have seen that for every $G\in\mathsf{CAGp}$,
the group $\mathop{\rm{\bf Z}_{\mathcal{F}}}(G)$ can be viewed as the group of
morphisms $G^{*}\to\mathop{\rm{\bf Z}_{\mathcal{F}}}$. For cohomology groups
such an isomorphism fails to hold. Though the group $\mathop{\rm{\bf
H}_{\mathcal{F}}}(G)$ can be viewed as a subgroup of $\mathop{\rm
Hom}(G^{*},\mathop{\rm{\bf H}_{\mathcal{F}}})$, the extent to which these two
groups differ is represented by the group of extensions $\mathop{\rm
Ext}(G^{*},\pi^{1}(X))$; this is proved in Theorems 4.32 and 4.36. In fact, as
we show in Theorem 4.39, the obstruction to an isomorphism $\mathop{\rm{\bf
H}_{\mathcal{F}}}(G)\cong\mathop{\rm Hom}(G^{*},\mathop{\rm{\bf
H}_{\mathcal{F}}})$ lies in the existence of extensions
$\mathcal{C}\in\mathop{\rm{\bf Z}_{\mathcal{F}}}$ of $\mathcal{F}$ with
totally disconnected sections $F(\mathcal{C})$. This also suggests that there
might be a connection between the existence of such extensions $\mathcal{C}$
on one side and the first cohomotopy group $\pi^{1}(X)$ of $X$ on the other
side. This is indeed the case, as we shall demonstrate later in Chapter 5.

######  Theorem 4.32.

Let $\mathop{\rm\mathcal{F}\colon\Gamma\curvearrowright X}$ be a minimal flow,
$G\in\mathsf{CAGp}$ and $\pi$ be the canonical quotient morphism
$\mathop{\rm{\bf Z}_{\mathcal{F}}}\to\mathop{\rm{\bf H}_{\mathcal{F}}}$. Then
the map

$\Psi_{G}\colon\mathop{\rm{\bf
H}_{\mathcal{F}}}(G)\ni\mathcal{C}\mapsto\pi\mathcal{C}^{*}\in\mathop{\rm
Hom}(G^{*},\mathop{\rm{\bf H}_{\mathcal{F}}})$ (4.5)

is a topological isomorphism onto its image, provided the groups
$\mathop{\rm{\bf H}_{\mathcal{F}}}$, $\mathop{\rm{\bf H}_{\mathcal{F}}}(G)$
are equipped with the ext-topology and the group $\mathop{\rm
Hom}(G^{*},\mathop{\rm{\bf H}_{\mathcal{F}}})$ carries the topology of point-
wise convergence.

######  Remark 4.33.

We wish to add the following remarks.

  * •

In order to simplify notation, we are identifying an extension
$\mathcal{C}\in\mathop{\rm{\bf Z}_{\mathcal{F}}}(G)$ with its cohomology class
in the definition of $\Psi_{G}$. We shall do so also in the sequel in
situations when no misunderstanding can arise.

  * •

We shall see in the proof of the theorem that the morphism $\Psi_{G}$ can also
be defined formally as a unique morphism $\mathop{\rm{\bf
H}_{\mathcal{F}}}(G)\to\mathop{\rm Hom}(G^{*},\mathop{\rm{\bf
H}_{\mathcal{F}}})$ with $\Psi_{G}\pi_{G}=\mathop{\rm
Hom}(G^{*},\pi)\Phi_{G}$.

  * •

The theorem provides us with the following inclusions of topological groups
for every $G\in\mathsf{CAGp}$:

$\mathop{\rm{\bf H}_{\mathcal{F}}}(G)\subseteq\mathop{\rm
Hom}(G^{*},\mathop{\rm{\bf H}_{\mathcal{F}}})\subseteq\left(\mathop{\rm{\bf
H}_{\mathcal{F}}}\right)^{G^{*}}.$

Since $\mathop{\rm{\bf H}_{\mathcal{F}}}$ is a discrete group by Remark 3.35,
it follows that $\mathop{\rm{\bf H}_{\mathcal{F}}}(G)$ is always totally
disconnected.

  * •

The image $\mathop{\rm im}(\Psi_{G})$ of $\Psi_{G}$ may or may not be closed
in $\mathop{\rm Hom}(G^{*},\mathop{\rm{\bf H}_{\mathcal{F}}})$, depending on a
particular situation. See, for instance, Theorem 6.28 in Chapter 6.

###### Proof of Theorem 4.32.

Consider the isomorphism $\Phi_{G}\colon\mathop{\rm{\bf
Z}_{\mathcal{F}}}(G)\to\mathop{\rm Hom}(G^{*},\mathop{\rm{\bf
Z}_{\mathcal{F}}})$ defined in Theorem 3.1 and the morphism

$\mathop{\rm Hom}(G^{*},\pi)\colon\mathop{\rm Hom}(G^{*},\mathop{\rm{\bf
Z}_{\mathcal{F}}})\to\mathop{\rm Hom}(G^{*},\mathop{\rm{\bf
H}_{\mathcal{F}}})$

derived from $\pi$. We claim that

$\ker(\pi_{G})=\mathop{\rm{\bf B}_{\mathcal{F}}}(G)=\ker(\mathop{\rm
Hom}(G^{*},\pi)\Phi_{G}).$

Indeed, the first equality is an immediate consequence of the definition of
$\pi_{G}$, whereas the second equality follows from the equivalence of the
following statements for every $\mathcal{C}\in\mathop{\rm{\bf
Z}_{\mathcal{F}}}(G)$:

  * •

$\mathcal{C}\in\ker(\mathop{\rm Hom}(G^{*},\pi)\Phi_{G})$,

  * •

$\pi\mathcal{C}^{*}=1$,

  * •

$\chi\mathcal{C}\in\mathop{\rm{\bf B}_{\mathcal{F}}}$ for every $\chi\in
G^{*}$,

  * •

$\mathcal{C}\in\mathop{\rm{\bf B}_{\mathcal{F}}}(G)$.

(For the equivalence of the third and the fourth statement, see Corollary
3.8.) It follows that there exists a unique morphism of groups

$\Psi_{G}\colon\mathop{\rm{\bf H}_{\mathcal{F}}}(G)\to\mathop{\rm
Hom}(G^{*},\mathop{\rm{\bf H}_{\mathcal{F}}})$

with $\Psi_{G}\pi_{G}=\mathop{\rm Hom}(G^{*},\pi_{G})\Phi_{G}$ and that
$\Psi_{G}$ is in fact a monomorphism. One verifies at once that
$\Psi_{G}(\mathcal{C})=\pi\mathcal{C}^{*}$ for every
$\mathcal{C}\in\mathop{\rm{\bf H}_{\mathcal{F}}}(G)$.

Now we show that $\Psi_{G}$ is a topological isomorphism onto its image. First
we verify the continuity of $\Psi_{G}$. So assume that a convergent net
$\mathcal{C}_{i}\to e$ is given in $\mathop{\rm{\bf H}_{\mathcal{F}}}(G)$.
Then $F(\mathcal{C}_{i})\to e$ in $2^{G}$ and hence
$F(\mathcal{C}^{*}_{i}(\chi))=F(\chi\mathcal{C}_{i})=\chi
F(\mathcal{C}_{i})\to 1$ in $2^{\mathbb{T}^{1}}$ for every $\chi\in G^{*}$.
This yields convergence $\pi\mathcal{C}^{*}_{i}\to e$ in $\mathop{\rm
Hom}(G^{*},\mathop{\rm{\bf H}_{\mathcal{F}}})$ and verifies the continuity of
$\Psi_{G}$.

Conversely, assume that for a given net $(\mathcal{C}_{i})_{i\in J}$ in
$\mathop{\rm{\bf H}_{\mathcal{F}}}(G)$, we have convergence
$\pi\mathcal{C}^{*}_{i}\to e$ in $\mathop{\rm Hom}(G^{*},\mathop{\rm{\bf
H}_{\mathcal{F}}})$. Fix a neighbourhood $V$ of $e$ in $G$. There exist
$n\in\mathbb{N}$, $\chi_{1},\dots,\chi_{n}\in G^{*}$ and neighbourhoods
$W_{1}\dots,W_{n}$ of $1$ in $\mathbb{T}^{1}$ such that
$V\supseteq\bigcap_{l=1}^{n}\chi_{l}^{-1}(W_{l})$. From the convergence
$\pi\mathcal{C}^{*}_{i}\to e$ in $\mathop{\rm Hom}(G^{*},\mathop{\rm{\bf
H}_{\mathcal{F}}})$ it follows that there is $i_{0}\in J$ with
$\chi_{l}F(\mathcal{C}_{i})=F(\chi_{l}\mathcal{C}_{i})=F(\mathcal{C}^{*}_{i}(\chi_{l}))\subseteq
W_{l}$ for all $i\geq i_{0}$ and $l=1,\dots,n$. Consequently,
$F(\mathcal{C}_{i})\subseteq\bigcap_{l=1}^{n}\chi_{l}^{-1}(W_{l})\subseteq V$
for every $i\geq i_{0}$. This shows that $F(\mathcal{C}_{i})\to e$ in $2^{G}$
and hence $\mathcal{C}_{i}\to e$ in $\mathop{\rm{\bf H}_{\mathcal{F}}}(G)$. ∎

######  Corollary 4.34.

Let $\mathop{\rm\mathcal{F}\colon\Gamma\curvearrowright X}$ be a minimal flow
and $G\in\mathsf{CAGp}$. Given $\mathcal{C}\in\mathop{\rm{\bf
Z}_{\mathcal{F}}}(G)$, let $h_{\mathcal{C}}$ be the morphism
$\mathop{\rm\gimel_{\mathcal{F}}}\to G$ with
$h_{\mathcal{C}}\mathop{\rm\daleth_{\mathcal{F}}}=\mathcal{C}$. Then the map

$\widetilde{\Psi}_{G}\colon\mathop{\rm{\bf
H}_{\mathcal{F}}}(G)\ni\mathcal{C}\mapsto\left(\omega_{G}\right)^{-1}\Psi_{G}(\mathcal{C})^{*}(\pi^{*})^{-1}=h_{\mathcal{C}}\in\mathop{\rm
Hom}(F(\mathop{\rm\daleth_{\mathcal{F}}}),G)$

is a topological isomorphism onto its image, provided $\mathop{\rm{\bf
H}_{\mathcal{F}}}(G)$ is equipped with the ext-topology and $\mathop{\rm
Hom}(F(\mathop{\rm\daleth_{\mathcal{F}}}),G)$ carries the topology of uniform
convergence.

######  Remark 4.35.

Similarly to Theorem 4.32, in our definition of $\widetilde{\Psi}_{G}$ we are
identifying every extension $\mathcal{C}\in\mathop{\rm{\bf
Z}_{\mathcal{F}}}(G)$ with its cohomology class. This time, the abuse of
notation is excused by the fact that for equivalent extensions
$\mathcal{C},\mathcal{D}\in\mathop{\rm{\bf Z}_{\mathcal{F}}}(G)$, the
corresponding morphisms $h_{\mathcal{C}}$ and $h_{\mathcal{D}}$ agree on
$F(\mathop{\rm\daleth_{\mathcal{F}}})$. (Indeed, if
$\mathcal{C}=\mathcal{D}\mathop{\rm co}(\xi)$ then
$h_{\mathcal{C}}(h_{\mathcal{D}})^{-1}=h_{\mathop{\rm co}(\xi)}$ and
$h_{\mathop{\rm co}(\xi)}F(\mathop{\rm\daleth_{\mathcal{F}}})=F(h_{\mathop{\rm
co}(\xi)}\mathop{\rm\daleth_{\mathcal{F}}})=F(\mathop{\rm co}(\xi))=e$.) We
shall use this identification also in the proof of Corollary 4.34 below.

###### Proof of Corollary 4.34.

Since both $G^{*}$ and $\mathop{\rm{\bf H}_{\mathcal{F}}}$ carry the discrete
topology, there is a topological isomorphism $\mathop{\rm
Hom}(G^{*},\mathop{\rm{\bf H}_{\mathcal{F}}})\to\mathop{\rm
Hom}((\mathop{\rm{\bf H}_{\mathcal{F}}})^{*},G^{**})$, which associates with
every morphism $k\colon G^{*}\to\mathop{\rm{\bf H}_{\mathcal{F}}}$ its
Pontryagin dual $k^{*}\colon G^{**}\to(\mathop{\rm{\bf
H}_{\mathcal{F}}})^{*}$. Moreover, the Pontryagin isomorphism $\omega_{G}$
corresponding to $G$ is a topological isomorphism $G\to G^{**}$ and the dual
morphism $\pi^{*}$ of $\pi$ is a topological isomorphism $(\mathop{\rm{\bf
H}_{\mathcal{F}}})^{*}\to(\mathop{\rm{\bf B}_{\mathcal{F}}})^{\perp}$.
Finally, since $(\mathop{\rm{\bf
B}_{\mathcal{F}}})^{\perp}=F(\mathop{\rm\daleth_{\mathcal{F}}})$ by the first
part of Remark 4.8, we have a topological isomorphism $\mathop{\rm
Hom}(G^{*},\mathop{\rm{\bf H}_{\mathcal{F}}})\to\mathop{\rm
Hom}(F(\mathop{\rm\daleth_{\mathcal{F}}}),G)$, which associates with every
morphism $k\colon G^{*}\to\mathop{\rm{\bf H}_{\mathcal{F}}}$ the morphism
$(\omega_{G})^{-1}k^{*}(\pi^{*})^{-1}\colon
F(\mathop{\rm\daleth_{\mathcal{F}}})\to G$. Thus, since
$\Psi_{G}\colon\mathop{\rm{\bf H}_{\mathcal{F}}}(G)\to\mathop{\rm
Hom}(G^{*},\mathop{\rm{\bf H}_{\mathcal{F}}})$ is a topological isomorphism
onto its image by virtue of Theorem 4.32, it follows that
$\widetilde{\Psi}_{G}$ is also a topological isomorphism onto its image. To
finish the proof, we need to show that
$(\omega_{G})^{-1}\Psi_{G}(\mathcal{C})^{*}(\pi^{*})^{-1}=h_{\mathcal{C}}$ for
every $\mathcal{C}\in\mathop{\rm{\bf Z}_{\mathcal{F}}}(G)$.

Fix $\mathcal{C}\in\mathop{\rm{\bf Z}_{\mathcal{F}}}(G)$. Given $\chi\in
G^{*}$, consider the set

$\mathcal{A}_{\chi,\mathcal{C}}=\left\\{\Lambda\in\mathop{\rm\gimel_{\mathcal{F}}}=(\mathop{\rm{\bf
Z}_{\mathcal{F}}})^{*}:\Lambda(\chi\mathcal{C})=(\chi
h_{\mathcal{C}})(\Lambda)\right\\}.$

Clearly, $\mathcal{A}_{\chi,\mathcal{C}}$ is a closed subgroup of
$\mathop{\rm\gimel_{\mathcal{F}}}$. Further, $\mathcal{A}_{\chi,\mathcal{C}}$
contains all the values of $\mathop{\rm\daleth_{\mathcal{F}}}$; indeed, for
all $\gamma\in\Gamma$ and $x\in X$,

$\mathop{\rm\daleth_{\mathcal{F}}}(\gamma,x)(\chi\mathcal{C})=(\chi\mathcal{C})(\gamma,x)=\chi(\mathcal{C}(\gamma,x))=\chi(h_{\mathcal{C}}\mathop{\rm\daleth_{\mathcal{F}}}(\gamma,x))=(\chi
h_{\mathcal{C}})(\mathop{\rm\daleth_{\mathcal{F}}}(\gamma,x)).$

These two observations lead to the inclusion
$F(\mathop{\rm\daleth_{\mathcal{F}}})\subseteq\mathcal{A}_{\chi,\mathcal{C}}$.

We verify the desired identity
$(\omega_{G})^{-1}\Psi_{G}(\mathcal{C})^{*}(\pi^{*})^{-1}=h_{\mathcal{C}}$. We
accomplish this by showing that
$\Psi_{G}(\mathcal{C})^{*}=\omega_{G}h_{\mathcal{C}}\pi^{*}$. For all
$\Upsilon\in(\mathop{\rm{\bf H}_{\mathcal{F}}})^{*}$ and $\chi\in G^{*}$ we
have

$\Psi_{G}(\mathcal{C})^{*}(\Upsilon)=\Upsilon\Psi_{G}(\mathcal{C})\colon\chi\mapsto\Upsilon\left(\Psi_{G}(\mathcal{C})(\chi)\right)=\Upsilon(\pi(\chi\mathcal{C}))=(\Upsilon\pi)(\chi\mathcal{C}),$
(4.6)

and

$\left(\omega_{G}h_{\mathcal{C}}\pi^{*}\right)(\Upsilon)=\omega_{G}\left(h_{\mathcal{C}}(\Upsilon\pi)\right)\colon\chi\mapsto\chi\left(h_{\mathcal{C}}(\Upsilon\pi)\right)=(\chi
h_{\mathcal{C}})(\Upsilon\pi).$ (4.7)

Since $\Upsilon\pi=\pi^{*}(\Upsilon)\in\mathop{\rm
im}(\pi^{*})=F(\mathop{\rm\daleth_{\mathcal{F}}})\subseteq\mathcal{A}_{\chi,\mathcal{C}}$
for all $\Upsilon$ and $\chi$ by our discussion above, the right-hand sides of
(4.6) and (4.7) are equal. This verifies the identity
$\Psi_{G}(\mathcal{C})^{*}=\omega_{G}h_{\mathcal{C}}\pi^{*}$. ∎

######  Theorem 4.36.

Let $\mathop{\rm\mathcal{F}\colon\Gamma\curvearrowright X}$ be a minimal flow
on a compact connected space $X$ and let $G\in\mathsf{CAGp}$. Then there exist

  * •

a morphism $\Omega\colon\mathop{\rm Hom}(G^{*},\mathop{\rm{\bf
H}_{\mathcal{F}}})\to\mathop{\rm Ext}(G^{*},\pi^{1}(X))$, and

  * •

a monomorphism $\nu\colon\pi^{1}(X)\to\mathop{\rm{\bf Z}_{\mathcal{F}}}$,

such that the following sequence is exact:

$\begin{split}0&\longrightarrow\mathop{\rm{\bf
B}_{\mathcal{F}}}(G)\stackrel{{\scriptstyle\mu_{G}}}{{\longrightarrow}}\mathop{\rm{\bf
Z}_{\mathcal{F}}}(G)\stackrel{{\scriptstyle\kappa_{G}}}{{\longrightarrow}}\mathop{\rm
Hom}(G^{*},\mathop{\rm{\bf
H}_{\mathcal{F}}})\stackrel{{\scriptstyle\Omega}}{{\longrightarrow}}\\\
&\stackrel{{\scriptstyle\Omega}}{{\longrightarrow}}\mathop{\rm
Ext}(G^{*},\pi^{1}(X))\stackrel{{\scriptstyle\nu_{e}}}{{\longrightarrow}}\mathop{\rm
Ext}(G^{*},\mathop{\rm{\bf
Z}_{\mathcal{F}}})\stackrel{{\scriptstyle\pi_{e}}}{{\longrightarrow}}\mathop{\rm
Ext}(G^{*},\mathop{\rm{\bf H}_{\mathcal{F}}})\longrightarrow 0,\end{split}$
(4.8)

where $\kappa_{G}=\Psi_{G}\pi_{G}$, $\nu_{e}=\mathop{\rm Ext}(G^{*},\nu)$ and
$\pi_{e}=\mathop{\rm Ext}(G^{*},\pi)$. If, in addition, the group
$\mathop{\rm{\bf Z}_{\mathcal{F}}}$ is divisible then the following sequence
is exact:

$\begin{split}0&\longrightarrow\mathop{\rm{\bf
B}_{\mathcal{F}}}(G)\stackrel{{\scriptstyle\mu_{G}}}{{\longrightarrow}}\mathop{\rm{\bf
Z}_{\mathcal{F}}}(G)\stackrel{{\scriptstyle\kappa_{G}}}{{\longrightarrow}}\mathop{\rm
Hom}(G^{*},\mathop{\rm{\bf
H}_{\mathcal{F}}})\stackrel{{\scriptstyle\Omega}}{{\longrightarrow}}\mathop{\rm
Ext}(G^{*},\pi^{1}(X))\longrightarrow 0.\end{split}$ (4.9)

Finally, if $\mathop{\rm{\bf Z}_{\mathcal{F}}}$ is divisible and $G$ is
connected then, under the identification $\mathop{\rm
im}(\Psi_{G})\cong\mathop{\rm{\bf H}_{\mathcal{F}}}(G)$, the latter group is a
direct summand in $\mathop{\rm Hom}(G^{*},\mathop{\rm{\bf H}_{\mathcal{F}}})$
with the complementary summand isomorphic to $\mathop{\rm
Ext}(G^{*},\pi^{1}(X))$. Thus, there is a direct sum

$\mathop{\rm Hom}\left(G^{*},\mathop{\rm{\bf
H}_{\mathcal{F}}}\right)=\mathop{\rm{\bf H}_{\mathcal{F}}}(G)\oplus\mathop{\rm
Ext}\left(G^{*},\pi^{1}(X)\right).$ (4.10)

######  Remark 4.37.

It follows from the exactness of (4.8) and from the definition of $\kappa_{G}$
that $\Psi_{G}$ is an isomorphism, provided $\mathop{\rm
Ext}(G^{*},\pi^{1}(X))=0$. This is the case in the following situations:

  * •

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

  * •

the group $G^{*}$ is free abelian, that is, $G$ is a torus,

  * •

$\pi^{1}(X)$ is finitely generated (in particular, $X$ is a manifold) and
$G^{*}$ is a Whitehead group (that is, $G$ is arc-wise connected).

###### Proof of Theorem 4.36.

Consider the covariant Hom-Ext sequence derived from the sequence (2.27) and
corresponding to the group $G^{*}$:

$\begin{split}0&\longrightarrow\mathop{\rm Hom}(G^{*},\mathop{\rm{\bf
B}_{\mathcal{F}}})\stackrel{{\scriptstyle\mu_{h}}}{{\longrightarrow}}\mathop{\rm
Hom}(G^{*},\mathop{\rm{\bf
Z}_{\mathcal{F}}})\stackrel{{\scriptstyle\pi_{h}}}{{\longrightarrow}}\mathop{\rm
Hom}(G^{*},\mathop{\rm{\bf H}_{\mathcal{F}}})\\\
&\stackrel{{\scriptstyle\epsilon}}{{\longrightarrow}}\mathop{\rm
Ext}(G^{*},\mathop{\rm{\bf
B}_{\mathcal{F}}})\stackrel{{\scriptstyle\mu_{e}}}{{\longrightarrow}}\mathop{\rm
Ext}(G^{*},\mathop{\rm{\bf
Z}_{\mathcal{F}}})\stackrel{{\scriptstyle\pi_{e}}}{{\longrightarrow}}\mathop{\rm
Ext}(G^{*},\mathop{\rm{\bf H}_{\mathcal{F}}})\longrightarrow 0.\end{split}$
(4.11)

Recall that $\mu_{h}=\mathop{\rm Hom}(G^{*},\mu)$, $\pi_{h}=\mathop{\rm
Hom}(G^{*},\pi)$, $\mu_{e}=\mathop{\rm Ext}(G^{*},\mu)$, $\pi_{e}=\mathop{\rm
Ext}(G^{*},\pi)$ and $\epsilon$ is the connecting morphism. Our aim now is to
show that the sequence (4.11) is equivalent to the sequence (4.8). We shall
proceed by replacing all the arrows in (4.11) step by step. The result of this
construction will be summarized in the diagrams in Figures 4.11 and 4.12
below.

By virtue of Corollary 3.8, the isomorphism $\Phi_{G}$ from Theorem 3.1
restricts to an isomorphism $\Phi_{G}^{\prime}\colon\mathop{\rm{\bf
B}_{\mathcal{F}}}(G)\to\mathop{\rm Hom}(G^{*},\mathop{\rm{\bf
B}_{\mathcal{F}}})$. Moreover, $\Phi_{G}\mu_{G}=\mathop{\rm
Hom}(G^{*},\mu)\Phi_{G}^{\prime}=\mu_{h}\Phi_{G}^{\prime}$. It follows that
the arrow

$\mu_{h}=\mathop{\rm Hom}(G^{*},\mu)\colon\mathop{\rm
Hom}(G^{*},\mathop{\rm{\bf B}_{\mathcal{F}}})\longrightarrow\mathop{\rm
Hom}(G^{*},\mathop{\rm{\bf Z}_{\mathcal{F}}})$

in (4.11) can be replaced by the arrow $\mu_{G}\colon\mathop{\rm{\bf
B}_{\mathcal{F}}}(G)\longrightarrow\mathop{\rm{\bf Z}_{\mathcal{F}}}(G)$.

Further, the arrow

$\pi_{h}=\mathop{\rm Hom}(G^{*},\pi)\colon\mathop{\rm
Hom}(G^{*},\mathop{\rm{\bf Z}_{\mathcal{F}}})\to\mathop{\rm
Hom}(G^{*},\mathop{\rm{\bf H}_{\mathcal{F}}})$

in (4.11) can be replaced by the arrow $\pi_{h}\Phi_{G}\colon\mathop{\rm{\bf
Z}_{\mathcal{F}}}(G)\to\mathop{\rm Hom}(G^{*},\mathop{\rm{\bf
H}_{\mathcal{F}}})$, this is clear. Since
$\pi_{h}\Phi_{G}=\Psi_{G}\pi_{G}=\kappa_{G}$ by definition of $\Psi_{G}$ and
$\kappa_{G}$, this yields the third arrow in (4.8).

Fix a base point $z$ for $X$. There are isomorphisms of groups
$\mathop{\rm{\bf B}_{\mathcal{F}}}\cong C_{z}(X,\mathbb{T}^{1})\cong
C_{z}(X,\mathbb{R})\oplus\pi^{1}(X)$; denote by $j$ the underlying inclusion
morphism $\pi^{1}(X)\to\mathop{\rm{\bf B}_{\mathcal{F}}}$. We claim that $j$
induces an isomorphism of groups

$j_{e}=\mathop{\rm Ext}(G^{*},j)\colon\mathop{\rm
Ext}(G^{*},\pi^{1}(X))\to\mathop{\rm Ext}(G^{*},\mathop{\rm{\bf
B}_{\mathcal{F}}}).$

To see this, let $\mathop{\rm pr}_{1}$, $\mathop{\rm pr}_{2}$ be the
projection morphisms from $\mathop{\rm{\bf B}_{\mathcal{F}}}$ onto
$C_{z}(X,\mathbb{R})$, $\pi^{1}(X)$, respectively, and write
$q_{1}=\mathop{\rm Ext}(G^{*},\mathop{\rm pr}_{1})$, $q_{2}=\mathop{\rm
Ext}(G^{*},\mathop{\rm pr}_{2})$. Since $\mathop{\rm pr}_{2}j=\mathop{\rm
id}_{\pi^{1}(X)}$, it follows that $q_{2}j_{e}=\mathop{\rm id}_{\mathop{\rm
Ext}(G^{*},\pi^{1}(X))}$ and hence $j_{e}$ is a monomorphism. Further, from
the exactness of the sequence

$0\longrightarrow\pi^{1}(X)\stackrel{{\scriptstyle
j}}{{\longrightarrow}}\mathop{\rm{\bf
B}_{\mathcal{F}}}\stackrel{{\scriptstyle\mathop{\rm
pr}_{1}}}{{\longrightarrow}}C_{z}(X,\mathbb{R})\longrightarrow 0$

we obtain an exact sequence

$\mathop{\rm Ext}(G^{*},\pi^{1}(X))\stackrel{{\scriptstyle
j_{e}}}{{\longrightarrow}}\mathop{\rm Ext}(G^{*},\mathop{\rm{\bf
B}_{\mathcal{F}}})\stackrel{{\scriptstyle
q_{1}}}{{\longrightarrow}}\mathop{\rm
Ext}(G^{*},C_{z}(X,\mathbb{R}))\longrightarrow 0.$ (4.12)

By divisibility of $C_{z}(X,\mathbb{R})$, we have $\mathop{\rm
Ext}(G^{*},C_{z}(X,\mathbb{R}))=0$ and so $j_{e}$ is an epimorphism due to the
exactness of (4.12). Thus, indeed, $j_{e}$ is an isomorphism of groups.
Consequently, the arrow $\epsilon\colon\mathop{\rm Hom}(G^{*},\mathop{\rm{\bf
H}_{\mathcal{F}}})\to\mathop{\rm Ext}(G^{*},\mathop{\rm{\bf
B}_{\mathcal{F}}})$ in (4.11) can be replaced by the arrow
$\Omega\colon\mathop{\rm Hom}(G^{*},\mathop{\rm{\bf
H}_{\mathcal{F}}})\to\mathop{\rm Ext}(G^{*},\pi^{1}(X))$, where
$\Omega=j_{e}^{-1}\epsilon$.

Finally, set $\nu=\mu j$ and $\nu_{e}=\mathop{\rm Ext}(G^{*},\nu)$. Then
$\nu_{e}=\mu_{e}j_{e}$ and the arrow

$\mu_{e}=\mathop{\rm Ext}(G^{*},\mu)\colon\mathop{\rm
Ext}(G^{*},\mathop{\rm{\bf B}_{\mathcal{F}}})\to\mathop{\rm
Ext}(G^{*},\mathop{\rm{\bf Z}_{\mathcal{F}}})$

in (4.11) thus turns into

$\nu_{e}=\mathop{\rm Ext}(G^{*},\nu)\colon\mathop{\rm
Ext}(G^{*},\pi^{1}(X))\to\mathop{\rm Ext}(G^{*},\mathop{\rm{\bf
Z}_{\mathcal{F}}}),$

which is precisely the fifth arrow in (4.8). The observations made so far can
be summarized in the form of the commutative diagrams from Figures 4.11 and
4.12.

$\begin{CD}\mathop{\rm Hom}(G^{*},\mathop{\rm{\bf
B}_{\mathcal{F}}})@>{\mu_{h}}>{}>\mathop{\rm Hom}(G^{*},\mathop{\rm{\bf
Z}_{\mathcal{F}}})@>{\pi_{h}}>{}>\mathop{\rm Hom}(G^{*},\mathop{\rm{\bf
H}_{\mathcal{F}}})\\\ @A{\Phi_{G}^{\prime}}A{}A@A{\Phi_{G}}A{}A\Big{\|}\\\
\mathop{\rm{\bf B}_{\mathcal{F}}}(G)@>{}>{\mu_{G}}>\mathop{\rm{\bf
Z}_{\mathcal{F}}}(G)@>{}>{\kappa_{G}}>\mathop{\rm Hom}(G^{*},\mathop{\rm{\bf
H}_{\mathcal{F}}})\end{CD}$ Figure 4.11: Obtaining (4.8) from (4.11), part 1.
$\begin{CD}\mathop{\rm Hom}(G^{*},\mathop{\rm{\bf
H}_{\mathcal{F}}})@>{\epsilon}>{}>\mathop{\rm Ext}(G^{*},\mathop{\rm{\bf
B}_{\mathcal{F}}})@>{\mu_{e}}>{}>\mathop{\rm Ext}(G^{*},\mathop{\rm{\bf
Z}_{\mathcal{F}}})@>{\pi_{e}}>{}>\mathop{\rm Ext}(G^{*},\mathop{\rm{\bf
H}_{\mathcal{F}}})\\\ \Big{\|}@A{j_{e}}A{}A\Big{\|}\Big{\|}\\\ \mathop{\rm
Hom}(G^{*},\mathop{\rm{\bf H}_{\mathcal{F}}})@>{}>{\Omega}>\mathop{\rm
Ext}(G^{*},\pi^{1}(X))@>{}>{\nu_{e}}>\mathop{\rm Ext}(G^{*},\mathop{\rm{\bf
Z}_{\mathcal{F}}})@>{}>{\pi_{e}}>\mathop{\rm Ext}(G^{*},\mathop{\rm{\bf
H}_{\mathcal{F}}})\end{CD}$ Figure 4.12: Obtaining (4.8) from (4.11), part 2.

Since all the vertical arrows in Figures 4.11 and 4.12 are isomorphisms, the
exactness of (4.8) follows from the exactness of (4.11). The exactness of
(4.9) follows at once from the exactness of (4.8), for $\mathop{\rm
Ext}(G^{*},\mathop{\rm{\bf Z}_{\mathcal{F}}})=0$ as soon as the group
$\mathop{\rm{\bf Z}_{\mathcal{F}}}$ is divisible.

We finish the proof of the theorem by verifying (4.10). First, since
$\pi_{G}\colon\mathop{\rm{\bf Z}_{\mathcal{F}}}(G)\to\mathop{\rm{\bf
H}_{\mathcal{F}}}(G)$ is an epimorphism and $\kappa_{G}=\Psi_{G}\pi_{G}$ by
definition, we have $\mathop{\rm im}(\Psi_{G})=\mathop{\rm im}(\kappa_{G})$.
Further, by the exactness of (4.9), $\mathop{\rm im}(\kappa_{G})=\ker(\Omega)$
and

$\mathop{\rm Hom}(G^{*},\mathop{\rm{\bf
H}_{\mathcal{F}}})/\ker(\Omega)\cong\mathop{\rm Ext}(G^{*},\pi^{1}(X)).$

Consequently,

$\begin{split}\mathop{\rm Hom}(G^{*},\mathop{\rm{\bf
H}_{\mathcal{F}}})/\mathop{\rm{\bf H}_{\mathcal{F}}}(G)&=\mathop{\rm
Hom}(G^{*},\mathop{\rm{\bf H}_{\mathcal{F}}})/\mathop{\rm im}(\Psi_{G})\\\
&=\mathop{\rm Hom}(G^{*},\mathop{\rm{\bf H}_{\mathcal{F}}})/\mathop{\rm
im}(\kappa_{G})\\\ &=\mathop{\rm Hom}(G^{*},\mathop{\rm{\bf
H}_{\mathcal{F}}})/\ker(\Omega)\\\ &\cong\mathop{\rm
Ext}\left(G^{*},\pi^{1}(X)\right).\end{split}$ (4.13)

By virtue of Theorem 4.12, the group $\mathop{\rm{\bf Z}_{\mathcal{F}}}(G)$ is
divisible due to divisibility of $\mathop{\rm{\bf Z}_{\mathcal{F}}}$ and
connectedness of $G$, and hence its quotient group $\mathop{\rm{\bf
H}_{\mathcal{F}}}(G)$ is also divisible. It follows that $\mathop{\rm{\bf
H}_{\mathcal{F}}}(G)$ is a direct summand in $\mathop{\rm
Hom}(G^{*},\mathop{\rm{\bf H}_{\mathcal{F}}})$ and, by virtue of (4.13), its
complementary summand is isomorphic to $\mathop{\rm
Ext}\left(G^{*},\pi^{1}(X)\right)$. This verifies (4.10). ∎

In order to formulate our next result we fix some notation. Given a minimal
flow $\mathcal{F}$ and a group $G\in\mathsf{CAGp}$, we use the symbol
$\mathop{\rm{\bf Z}^{td}_{\mathcal{F}}}(G)$ to denote the set of all
extensions $\mathcal{C}\in\mathop{\rm{\bf Z}_{\mathcal{F}}}(G)$ with a totally
disconnected section $F(\mathcal{C})\subseteq G$. Also, we write
$\mathop{\rm{\bf Z}^{td}_{\mathcal{F}}}$ instead of $\mathop{\rm{\bf
Z}^{td}_{\mathcal{F}}}(\mathbb{T}^{1})$.

######  Lemma 4.38.

Let $\mathop{\rm\mathcal{F}\colon\Gamma\curvearrowright X}$ be a minimal flow.
Then for every $G\in\mathsf{CAGp}$, $\mathop{\rm{\bf
Z}^{td}_{\mathcal{F}}}(G)$ is a subgroup of $\mathop{\rm{\bf
Z}_{\mathcal{F}}}(G)$. Moreover, if $\mathop{\rm{\bf Z}_{\mathcal{F}}}$ is
divisible and $G\in\mathsf{CAGp}$ is connected then $\mathop{\rm{\bf
Z}^{td}_{\mathcal{F}}}(G)$ is also divisible and hence it is a direct summand
in $\mathop{\rm{\bf Z}_{\mathcal{F}}}(G)$.

###### Proof.

We divide the proof into four steps.

_1st step._ Let $G\in\mathsf{CAGp}$ and $H\mathrel{\lhd}G$. We recall the
equivalence of the following statements:

  1. ($\alpha$)

$H$ is totally disconnected,

  2. ($\beta$)

$\bigcap_{k\in\mathbb{N}}\kappa_{k}(H)=e$,

  3. ($\gamma$)

for every identity neighbourhood $U$ in $G$ there is $k\in\mathbb{N}$ such
that $\kappa_{l}(H)\subseteq U$ for every $l\in\mathbb{N}$ divisible by $k$.

First, ($\alpha$) follows from ($\beta$), for if $H_{0}\neq e$ then, for every
$k\in\mathbb{N}$, $\kappa_{k}(H)\supseteq\kappa_{k}(H_{0})=H_{0}\neq e$. To
show that ($\beta$) follows from ($\alpha$), assume that $H$ is totally
disconnected. Then $H$ is isomorphic to a subgroup of the product of a family
of finite abelian groups. Since every finite abelian group is annihilated by
$\kappa_{k}$ for some $k\in\mathbb{N}$, we conclude that for every identity
neighbourhood $V$ in $H$ there is $k$ with $\kappa_{k}(H)\subseteq V$. This
verifies ($\beta$). The equivalence of ($\beta$) and ($\gamma$) follows from
the fact that $\kappa_{l}(H)\subseteq\kappa_{k}(H)$ as soon as $k$ divides
$l$.

_2nd step._ We show that if $H,K$ are closed totally disconnected subgroups of
$G$ and $h\in\mathbb{N}$, then both $HK$ and $\kappa_{h}^{-1}(H)$ are (closed
and) totally disconnected. (Here, $\kappa_{h}$ is the $h$-endomorphism of the
whole group $G$.)

Fix an identity neighbourhood $U$ in $G$. Further, choose an identity
neighbourhood $V$ in $G$ with $VV\subseteq U$. By virtue of the equivalence
($\alpha$)$\Leftrightarrow$($\gamma$) from the first step of the proof, there
are $k,l\in\mathbb{N}$ with $\kappa_{k}(H)\subseteq V$ and
$\kappa_{l}(K)\subseteq V$. Then

$\kappa_{kl}(HK)=\kappa_{kl}(H)\kappa_{kl}(K)\subseteq\kappa_{k}(H)\kappa_{l}(K)\subseteq
VV\subseteq U.$

Since $U$ was an arbitrary identity neighbourhood in $G$, it follows by the
first step of the proof that $HK$ is indeed totally disconnected. Finally, the
group $\kappa_{h}^{-1}(H)$ is totally disconnected by virtue of the
equivalence ($\alpha$)$\Leftrightarrow$($\beta$), for

$\begin{split}\bigcap_{k\in\mathbb{N}}\kappa_{k}\left(\kappa_{h}^{-1}(H)\right)&\subseteq\bigcap_{k\in\mathbb{N}}\kappa_{kh}\left(\kappa_{h}^{-1}(H)\right)=\bigcap_{k\in\mathbb{N}}\kappa_{k}\left(\kappa_{h}\left(\kappa_{h}^{-1}(H)\right)\right)\\\
&\subseteq\bigcap_{k\in\mathbb{N}}\kappa_{k}(H)=e.\end{split}$

_3rd step._ Let $G\in\mathsf{CAGp}$. We show that $\mathop{\rm{\bf
Z}^{td}_{\mathcal{F}}}(G)$ is a subgroup of $\mathop{\rm{\bf
Z}_{\mathcal{F}}}(G)$.

Fix $\mathcal{C},\mathcal{D}\in\mathop{\rm{\bf Z}^{td}_{\mathcal{F}}}(G)$.
Then $F(\mathcal{C}\mathcal{D})\subseteq F(\mathcal{C})F(\mathcal{D})$ by
Lemma 3.31(c). Since $F(\mathcal{C})F(\mathcal{D})$ is a totally disconnected
subgroup of $G$ by the second step of the proof, it follows that the group
$F(\mathcal{C}\mathcal{D})$ is also totally disconnected and hence
$\mathcal{C}\mathcal{D}\in\mathop{\rm{\bf Z}^{td}_{\mathcal{F}}}(G)$. Also,
$F(\mathcal{C}^{-1})=F(\mathcal{C})$ by Lemma 3.31(b) and so
$\mathcal{C}^{-1}\in\mathop{\rm{\bf Z}^{td}_{\mathcal{F}}}(G)$. This shows
that $\mathop{\rm{\bf Z}^{td}_{\mathcal{F}}}(G)$ is indeed a subgroup of
$\mathop{\rm{\bf Z}_{\mathcal{F}}}(G)$.

_4th step._ Assume that the group $\mathop{\rm{\bf Z}_{\mathcal{F}}}$ is
divisible and fix $G\in\mathsf{CAGp}$ connected. We show that the group
$\mathop{\rm{\bf Z}^{td}_{\mathcal{F}}}(G)$ is divisible.

Fix $\mathcal{C}\in\mathop{\rm{\bf Z}^{td}_{\mathcal{F}}}(G)$ and
$h\in\mathbb{N}$. By the assumptions of the theorem and by virtue of Theorem
4.12, the group $\mathop{\rm{\bf Z}_{\mathcal{F}}}(G)$ is divisible.
Consequently, there is $\mathcal{D}\in\mathop{\rm{\bf Z}_{\mathcal{F}}}(G)$
with $\mathcal{D}^{h}=\mathcal{C}$; we show that
$\mathcal{D}\in\mathop{\rm{\bf Z}^{td}_{\mathcal{F}}}(G)$. We have
$\kappa_{h}(F(\mathcal{D}))=F(\kappa_{h}\mathcal{D})=F(\mathcal{D}^{h})=F(\mathcal{C})$
and so $F(\mathcal{D})\subseteq\kappa_{h}^{-1}(F(\mathcal{C}))$. By the second
step of the proof, $\kappa_{h}^{-1}(F(\mathcal{C}))$ is totally disconnected
and hence so is $F(\mathcal{D})$. Thus, $\mathcal{D}\in\mathop{\rm{\bf
Z}^{td}_{\mathcal{F}}}(G)$, as was to be shown. ∎

######  Theorem 4.39.

Let $\mathop{\rm\mathcal{F}\colon\Gamma\curvearrowright X}$ be a minimal flow
on a compact connected space $X$. Assume that the group $\mathop{\rm{\bf
Z}_{\mathcal{F}}}$ is divisible and fix a complementary direct summand
$\mathop{\rm{\bf Z}^{cn}_{\mathcal{F}}}$ to $\mathop{\rm{\bf
Z}^{td}_{\mathcal{F}}}$ in $\mathop{\rm{\bf Z}_{\mathcal{F}}}$. Given
$G\in\mathsf{CAGp}$ connected, set

  * •

$\mathop{\rm{\bf H}^{td}_{\mathcal{F}}}(G)=\pi_{G}(\mathop{\rm{\bf
Z}^{td}_{\mathcal{F}}}(G))=\mathop{\rm{\bf
Z}^{td}_{\mathcal{F}}}(G)/\mathop{\rm{\bf B}_{\mathcal{F}}}(G)$,

  * •

$\mathop{\rm{\bf Z}^{cn}_{\mathcal{F}}}(G)=\Phi_{G}^{-1}\left(\mathop{\rm
Hom}(G^{*},\mathop{\rm{\bf Z}^{cn}_{\mathcal{F}}})\right)$,

  * •

$\mathop{\rm{\bf H}^{cn}_{\mathcal{F}}}(G)=\pi_{G}\left(\mathop{\rm{\bf
Z}^{cn}_{\mathcal{F}}}(G)\right)$,

  * •

$\mathop{\rm{\bf H}^{td}_{\mathcal{F}}}=\mathop{\rm{\bf
H}^{td}_{\mathcal{F}}}(\mathbb{T}^{1})$ and $\mathop{\rm{\bf
H}^{cn}_{\mathcal{F}}}=\mathop{\rm{\bf
H}^{cn}_{\mathcal{F}}}(\mathbb{T}^{1})$.

Then the following statements hold:

  1. (1)

$F(\mathcal{C})\subseteq G$ is connected for every
$\mathcal{C}\in\mathop{\rm{\bf Z}^{cn}_{\mathcal{F}}}(G)$,

  2. (2)

there is a direct sum

$\mathop{\rm{\bf Z}_{\mathcal{F}}}(G)=\mathop{\rm{\bf
Z}^{td}_{\mathcal{F}}}(G)\oplus\mathop{\rm{\bf Z}^{cn}_{\mathcal{F}}}(G),$

  3. (3)

there is a direct sum

$\mathop{\rm{\bf H}_{\mathcal{F}}}(G)=\mathop{\rm{\bf
H}^{td}_{\mathcal{F}}}(G)\oplus\mathop{\rm{\bf H}^{cn}_{\mathcal{F}}}(G),$

which is topological, provided $\mathop{\rm{\bf H}_{\mathcal{F}}}(G)$ carries
the ext-topology,

  4. (4)

the restriction $\Psi_{G}\colon\mathop{\rm{\bf
H}^{cn}_{\mathcal{F}}}(G)\to\mathop{\rm Hom}(G^{*},\mathop{\rm{\bf
H}^{cn}_{\mathcal{F}}})$ is an isomorphism of groups,

  5. (5)

under the identification $\Psi_{G}(\mathop{\rm{\bf
H}^{td}_{\mathcal{F}}}(G))\cong\mathop{\rm{\bf H}^{td}_{\mathcal{F}}}(G)$, the
latter group is a direct summand in $\mathop{\rm Hom}(G^{*},\mathop{\rm{\bf
H}^{td}_{\mathcal{F}}})$ and its complementary summand is isomorphic to the
group $\mathop{\rm Ext}(G^{*},\pi^{1}(X))$; consequently, there is a direct
sum

$\mathop{\rm Hom}(G^{*},\mathop{\rm{\bf
H}^{td}_{\mathcal{F}}})=\mathop{\rm{\bf
H}^{td}_{\mathcal{F}}}(G)\oplus\mathop{\rm Ext}(G^{*},\pi^{1}(X)).$ (4.14)

######  Remark 4.40.

Notice the following facts.

  * •

Assume that $G$ has no proper non-trivial closed connected subgroups. Then,
since $\mathop{\rm{\bf B}_{\mathcal{F}}}(G)\subseteq\mathop{\rm{\bf
Z}^{td}_{\mathcal{F}}}(G)$ by definition of $\mathop{\rm{\bf
Z}^{td}_{\mathcal{F}}}(G)$, it follows from statement (1) that
$\mathop{\rm{\bf Z}^{cn}_{\mathcal{F}}}(G)$ is a subgroup of the groupoid
$\mathop{\rm{\bf MinZ}_{\mathcal{F}}}(G)$. Recall also that for a non-trivial
connected group $G\in\mathsf{CAGp}$ the following conditions are equivalent:

    1. $\circ$

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

    2. $\circ$

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

These conditions are satisfied, for instance, by the circle group
$G=\mathbb{T}^{1}$ and by the solenoids.

  * •

By virtue of (3) applied to the group $\mathbb{T}^{1}$, there is a direct sum

$\mathop{\rm Hom}(G^{*},\mathop{\rm{\bf H}_{\mathcal{F}}})=\mathop{\rm
Hom}(G^{*},\mathop{\rm{\bf H}^{td}_{\mathcal{F}}})\oplus\mathop{\rm
Hom}(G^{*},\mathop{\rm{\bf H}^{cn}_{\mathcal{F}}}).$

Consequently, by virtue of (4) and (5), it is the groups $\mathop{\rm{\bf
H}^{td}_{\mathcal{F}}}$ and $\mathop{\rm{\bf H}^{td}_{\mathcal{F}}}(G)$ that
are “responsible” for the non-surjectivity of the monomorphism $\Psi_{G}$.
That is, $\Psi_{G}$ is an isomorphism if and only if its restriction
$\Psi_{G}\colon\mathop{\rm{\bf H}^{td}_{\mathcal{F}}}(G)\to\mathop{\rm
Hom}(G^{*},\mathop{\rm{\bf H}^{td}_{\mathcal{F}}})$ is an isomorphism.

  * •

The splitting in statement (2), unlike the one in statement (3), need not be
topological if $\mathop{\rm{\bf Z}_{\mathcal{F}}}(G)$ carries the topology of
u.c.s. convergence. For examples of such a situation we refer to Proposition
5.45 and Remark 5.46 in Chapter 5.

###### Proof of Theorem 4.39.

Before turning to the proof of the theorem notice the following facts. First,
by virtue of Lemma 4.38, $\mathop{\rm{\bf Z}^{td}_{\mathcal{F}}}$ is indeed a
direct summand in $\mathop{\rm{\bf Z}_{\mathcal{F}}}$. Second, though unique
only up to an isomorphism, $\mathop{\rm{\bf Z}^{cn}_{\mathcal{F}}}$ is always
a subgroup of the groupoid $\mathop{\rm{\bf MinZ}_{\mathcal{F}}}$. Now we turn
to the proof of the theorem, proceeding in five steps.

_1st step._ We verify statement (1).

Fix $\mathcal{C}\in\mathop{\rm{\bf Z}^{cn}_{\mathcal{F}}}(G)$. By definition
of $\mathop{\rm{\bf Z}^{cn}_{\mathcal{F}}}(G)$,
$\chi\mathcal{C}=\mathcal{C}^{*}(\chi)\in\mathop{\rm{\bf
Z}^{cn}_{\mathcal{F}}}$ for every $\chi\in G^{*}$. It follows that for every
$\chi\in G^{*}$, $\chi F(\mathcal{C})=F(\chi\mathcal{C})$ is either $1$ or
$\mathbb{T}^{1}$. Consequently, $F(\mathcal{C})$ is a connected subgroup of
$G$ and statement (1) thus follows.

_2nd step._ We verify statement (2).

By definition of $\mathop{\rm{\bf Z}^{cn}_{\mathcal{F}}}$, there is a direct
sum

$\mathop{\rm Hom}(G^{*},\mathop{\rm{\bf Z}_{\mathcal{F}}})=\mathop{\rm
Hom}(G^{*},\mathop{\rm{\bf Z}^{td}_{\mathcal{F}}})\oplus\mathop{\rm
Hom}(G^{*},\mathop{\rm{\bf Z}^{cn}_{\mathcal{F}}}).$ (4.15)

Under the isomorphism $\Phi_{G}$ from Theorem 3.1, the following
correspondences hold:

  1. (i)

$\mathop{\rm{\bf Z}^{td}_{\mathcal{F}}}(G)$ corresponds to $\mathop{\rm
Hom}(G^{*},\mathop{\rm{\bf Z}^{td}_{\mathcal{F}}})$; indeed, given
$\mathcal{C}\in\mathop{\rm{\bf Z}_{\mathcal{F}}}(G)$, the following conditions
are equivalent:

     * •

$\mathcal{C}\in\mathop{\rm{\bf Z}^{td}_{\mathcal{F}}}(G)$,

     * •

$F(\mathcal{C})\subseteq G$ is totally disconnected,

     * •

$F(\chi\mathcal{C})=\chi F(\mathcal{C})\subseteq\mathbb{T}^{1}$ is totally
disconnected for every $\chi\in G^{*}$,

     * •

$\mathcal{C}^{*}(\chi)=\chi\mathcal{C}\in\mathop{\rm{\bf
Z}^{td}_{\mathcal{F}}}$ for every $\chi\in G^{*}$,

     * •

$\Phi_{G}(\mathcal{C})=\mathcal{C}^{*}\in\mathop{\rm
Hom}(G^{*},\mathop{\rm{\bf Z}^{td}_{\mathcal{F}}})$,

  2. (ii)

$\mathop{\rm{\bf Z}^{cn}_{\mathcal{F}}}(G)$ corresponds to $\mathop{\rm
Hom}(G^{*},\mathop{\rm{\bf Z}^{cn}_{\mathcal{F}}})$; this holds by definition
of $\mathop{\rm{\bf Z}^{cn}_{\mathcal{F}}}(G)$.

Thus, statement (2) follows from (4.15), (i) and (ii).

_3rd step._ We verify statement (3).

First we check that the direct sum in (3) holds algebraically. The fact that
$\mathop{\rm{\bf H}_{\mathcal{F}}}(G)$ is a sum of $\mathop{\rm{\bf
H}^{td}_{\mathcal{F}}}(G)$ and $\mathop{\rm{\bf H}^{cn}_{\mathcal{F}}}(G)$
follows by applying $\pi_{G}$ to the direct sum in (2). We need only to show
that $\mathop{\rm{\bf H}^{td}_{\mathcal{F}}}(G)\cap\mathop{\rm{\bf
H}^{cn}_{\mathcal{F}}}(G)=e$, but this is immediate. For if
$\mathcal{C}\in\mathop{\rm{\bf Z}_{\mathcal{F}}}(G)$ and $F(\mathcal{C})$ is
both connected and totally disconnected then $F(\mathcal{C})=e$ and hence
$\mathcal{C}\in\mathop{\rm{\bf B}_{\mathcal{F}}}$.

Now we show that the splitting in (3) is topological. To this end, fix a
directed set $J$ and nets $(\mathcal{C}_{i})_{i\in J}$ in $\mathop{\rm{\bf
Z}^{td}_{\mathcal{F}}}(G)$ and $(\mathcal{D}_{i})_{i\in J}$ in
$\mathop{\rm{\bf Z}^{cn}_{\mathcal{F}}}(G)$. We need to show that
$F(\mathcal{C}_{i}\mathcal{D}_{i})\to e$ in $2^{G}$ if and only if
$F(\mathcal{C}_{i}),F(\mathcal{D}_{i})\to e$ in $2^{G}$. Since all
$F(\mathcal{C}_{i})$ ($i\in J$) are totally disconnected and all
$F(\mathcal{D}_{i})$ ($i\in J$) are connected, Corollary 3.20 yields
$F(\mathcal{C}_{i}\mathcal{D}_{i})=F(\mathcal{C}_{i})F(\mathcal{D}_{i})$ for
every $i\in J$. From this identity the desired equivalence follows at once.

_4th step._ We verify statement (4).

Notice the following facts:

  * •

the restriction $\Phi_{G}\colon\mathop{\rm{\bf
Z}^{cn}_{\mathcal{F}}}(G)\to\mathop{\rm Hom}(G^{*},\mathop{\rm{\bf
Z}^{cn}_{\mathcal{F}}})$ is an isomorphism; this follows from the definition
of $\mathop{\rm{\bf Z}^{cn}_{\mathcal{F}}}(G)$ and from the fact that
$\Phi_{G}$ is an isomorphism,

  * •

the restriction $\pi_{G}\colon\mathop{\rm{\bf
Z}^{cn}_{\mathcal{F}}}(G)\to\mathop{\rm{\bf H}^{cn}_{\mathcal{F}}}(G)$ is an
isomorphism; this follows from the definition of $\mathop{\rm{\bf
H}^{cn}_{\mathcal{F}}}(G)$ and from the identity $\mathop{\rm{\bf
Z}^{cn}_{\mathcal{F}}}(G)\cap\mathop{\rm{\bf B}_{\mathcal{F}}}(G)=e$ (the
latter holds by virtue of (2)),

  * •

the restriction $\mathop{\rm Hom}(G^{*},\pi)\colon\mathop{\rm
Hom}(G^{*},\mathop{\rm{\bf Z}^{cn}_{\mathcal{F}}})\to\mathop{\rm
Hom}(G^{*},\mathop{\rm{\bf H}^{cn}_{\mathcal{F}}})$ is an isomorphism; this
follows from the fact that (by the preceding statement applied to the group
$\mathbb{T}^{1}$) the restriction $\pi\colon\mathop{\rm{\bf
Z}^{cn}_{\mathcal{F}}}\to\mathop{\rm{\bf H}^{cn}_{\mathcal{F}}}$ is an
isomorphism,

  * •

the morphism $\Psi_{G}\colon\mathop{\rm{\bf H}_{\mathcal{F}}}(G)\to\mathop{\rm
Hom}(G^{*},\mathop{\rm{\bf H}_{\mathcal{F}}})$ is defined by the identity
$\Psi_{G}\pi_{G}=\mathop{\rm Hom}(G^{*},\pi)\Phi_{G}$; this follows from
Remark 4.33.

The four facts listed above now yield statement (4).

_5th step._ We verify statement (5).

First notice that $\Psi_{G}$ maps $\mathop{\rm{\bf H}^{td}_{\mathcal{F}}}(G)$
into $\mathop{\rm Hom}(G^{*},\mathop{\rm{\bf H}^{td}_{\mathcal{F}}})$, since
$\Phi_{G}$ maps $\mathop{\rm{\bf Z}^{td}_{\mathcal{F}}}(G)$ into $\mathop{\rm
Hom}(G^{*},\mathop{\rm{\bf Z}^{td}_{\mathcal{F}}})$ by virtue of (i) from the
second step of the proof. We shall need the following facts:

  * •

under the isomorphism $\Psi_{G}(\mathop{\rm{\bf
H}_{\mathcal{F}}}(G))\cong\mathop{\rm{\bf H}_{\mathcal{F}}}(G)$, the latter
group is a direct summand in $\mathop{\rm Hom}(G^{*},\mathop{\rm{\bf
H}_{\mathcal{F}}})$ and its complementary summand is isomorphic to the group
$\mathop{\rm Ext}\left(G^{*},\pi^{1}(X)\right)$; this follows from (4.10) in
Theorem 4.36,

  * •

under the isomorphism $\Psi_{G}(\mathop{\rm{\bf
H}^{td}_{\mathcal{F}}}(G))\cong\mathop{\rm{\bf H}^{td}_{\mathcal{F}}}(G)$, the
latter group is a direct summand in $\mathop{\rm Hom}(G^{*},\mathop{\rm{\bf
H}^{td}_{\mathcal{F}}})$; indeed, by virtue of Lemma 4.38, $\mathop{\rm{\bf
Z}^{td}_{\mathcal{F}}}(G)$ is a divisible group and hence so is its quotient
group $\mathop{\rm{\bf H}^{td}_{\mathcal{F}}}(G)$,

  * •

there is a direct sum

$\mathop{\rm{\bf H}_{\mathcal{F}}}(G)\cong\mathop{\rm{\bf
H}^{td}_{\mathcal{F}}}(G)\oplus\mathop{\rm{\bf H}^{cn}_{\mathcal{F}}}(G);$

this follows from statement (3),

  * •

there is a direct sum

$\mathop{\rm Hom}(G^{*},\mathop{\rm{\bf H}_{\mathcal{F}}})=\mathop{\rm
Hom}(G^{*},\mathop{\rm{\bf H}^{td}_{\mathcal{F}}})\oplus\mathop{\rm
Hom}(G^{*},\mathop{\rm{\bf H}^{cn}_{\mathcal{F}}});$

this follows from statement (3) applied to the group $\mathbb{T}^{1}$,

  * •

$\Psi_{G}$ maps $\mathop{\rm{\bf H}^{cn}_{\mathcal{F}}}(G)$ isomorphically
onto $\mathop{\rm Hom}(G^{*},\mathop{\rm{\bf H}^{cn}_{\mathcal{F}}})$; this
follows from statement (4).

These five facts lead, on one hand, to a diagram in Figure 4.13 and, on the
other hand, to the isomorphisms

$\begin{split}\mathop{\rm
Ext}\left(G^{*},\pi^{1}(X)\right)&\cong\frac{\mathop{\rm
Hom}\left(G^{*},\mathop{\rm{\bf H}_{\mathcal{F}}}\right)}{\mathop{\rm{\bf
H}_{\mathcal{F}}}(G)}\\\ &\cong\frac{\mathop{\rm Hom}(G^{*},\mathop{\rm{\bf
H}^{td}_{\mathcal{F}}})}{\mathop{\rm{\bf
H}^{td}_{\mathcal{F}}}(G)}\oplus\frac{\mathop{\rm
Hom}\left(G^{*},\mathop{\rm{\bf H}^{cn}_{\mathcal{F}}}\right)}{\mathop{\rm{\bf
H}^{cn}_{\mathcal{F}}}(G)}\\\ &\cong\frac{\mathop{\rm
Hom}(G^{*},\mathop{\rm{\bf H}^{td}_{\mathcal{F}}})}{\mathop{\rm{\bf
H}^{td}_{\mathcal{F}}}(G)}.\end{split}$

$\setcounter{MaxMatrixCols}{11}\begin{CD}\mathop{\rm
Hom}(G^{*},\mathop{\rm{\bf H}_{\mathcal{F}}}){}={}\mathop{\rm
Hom}(G^{*},\mathop{\rm{\bf H}^{td}_{\mathcal{F}}}){}\oplus{}\mathop{\rm
Hom}(G^{*},\mathop{\rm{\bf H}^{cn}_{\mathcal{F}}})\\\
@A{\Psi_{G}}A{\text{mon}}A@A{\Psi_{G}}A{\text{mon}}A@A{\Psi_{G}}A{\text{iso}}A\\\
\mathop{\rm{\bf H}_{\mathcal{F}}}(G)=\mathop{\rm{\bf
H}^{td}_{\mathcal{F}}}(G)\oplus\mathop{\rm{\bf
H}^{cn}_{\mathcal{F}}}(G).\end{CD}$ Figure 4.13: Monomorphic and isomorphic
restrictions of $\Psi_{G}$.

This verifies statement (5). ∎

###  4.6 Torsions as non-minimal extensions

Let $\mathop{\rm\mathcal{F}\colon\Gamma\curvearrowright X}$ be a minimal flow.
In this section we study the torsion subgroup $\mathop{\rm
tor}(\mathop{\rm{\bf H}_{\mathcal{F}}})$ of $\mathop{\rm{\bf
H}_{\mathcal{F}}}$ and its subgroups $\mathop{\rm tor_{k}}(\mathop{\rm{\bf
H}_{\mathcal{F}}})$ ($k\in\mathbb{N}$). We are motivated by the fact that the
torsion elements of $\mathop{\rm{\bf H}_{\mathcal{F}}}$ correspond to the non-
minimal extensions from $\mathop{\rm{\bf Z}_{\mathcal{F}}}$, see Corollary
3.9. We begin our investigation with the situation when $\mathop{\rm{\bf
Z}_{\mathcal{F}}}$ is a divisible group, which includes the case when the
acting group $\Gamma$ of $\mathcal{F}$ is simply connected; see Theorem 4.41.
Then we embark on the case when $\Gamma$ is a connected Lie group and $X$ is a
compact (connected) manifold; see Theorem 4.44.

######  Theorem 4.41.

Let $\mathop{\rm\mathcal{F}\colon\Gamma\curvearrowright X}$ be a minimal flow
on a compact connected space $X$ and assume that the group $\mathop{\rm{\bf
Z}_{\mathcal{F}}}$ is divisible. Given $k\in\mathbb{N}$, there are short exact
sequences

$0\longrightarrow\mathop{\rm
Hom}(\Gamma,\mathbb{Z}_{k})\longrightarrow\mathop{\rm tor_{k}}(\mathop{\rm{\bf
H}_{\mathcal{F}}})\longrightarrow\pi^{1}(X)/k\pi^{1}(X)\longrightarrow 0$
(4.16)

and

$0\longrightarrow\bigcup_{l\in\mathbb{N}}\mathop{\rm
Hom}(\Gamma,\mathbb{Z}_{l})\longrightarrow\mathop{\rm tor}(\mathop{\rm{\bf
H}_{\mathcal{F}}})\longrightarrow(\mathbb{Q}/\mathbb{Z})\otimes\pi^{1}(X)\longrightarrow
0.$ (4.17)

Moreover, if $\Gamma$ has no non-trivial finite abelian quotient groups then
there are isomorphisms

$\mathop{\rm tor_{k}}(\mathop{\rm{\bf
H}_{\mathcal{F}}})\cong\pi^{1}(X)/k\pi^{1}(X)$ (4.18)

and

$\mathop{\rm tor}(\mathop{\rm{\bf
H}_{\mathcal{F}}})\cong(\mathbb{Q}/\mathbb{Z})\otimes\pi^{1}(X),$ (4.19)

and, finally, there exists a divisible subgroup $\mathfrak{D}$ of the groupoid
$\mathop{\rm{\bf MinH}_{\mathcal{F}}}$ such that

$\mathop{\rm{\bf H}_{\mathcal{F}}}=\mathfrak{D}\oplus\mathop{\rm
tor}(\mathop{\rm{\bf
H}_{\mathcal{F}}})\cong\mathfrak{D}\oplus\Big{(}(\mathbb{Q}/\mathbb{Z})\otimes\pi^{1}(X)\Big{)}.$
(4.20)

######  Remark 4.42.

Notice the following facts.

  * •

The union $\bigcup_{l\in\mathbb{N}}\mathop{\rm Hom}(\Gamma,\mathbb{Z}_{l})$
from (4.17) is to be interpreted as a subgroup of $\mathop{\rm
Hom}(\Gamma,(\mathbb{Q}/\mathbb{Z})_{d})\subseteq\mathop{\rm
Hom}(\Gamma,(\mathbb{T}^{1})_{d})$, under the usual identifications
$\mathbb{Z}_{l}\cong\mathop{\rm
tor}_{l}(\mathbb{T}^{1})\subseteq\mathbb{Q}/\mathbb{Z}\subseteq\mathbb{T}^{1}$.
An elementary argument shows that if the quotient group $\Gamma/\Gamma_{0}$ is
finitely generated then this union coincides with $\mathop{\rm
Hom}(\Gamma,(\mathbb{Q}/\mathbb{Z})_{d})$.

  * •

An extension $\mathcal{C}\in\mathop{\rm{\bf Z}_{\mathcal{F}}}$ corresponds to
an element of $\mathop{\rm tor_{k}}(\mathop{\rm{\bf H}_{\mathcal{F}}})$ if and
only if $\mathcal{C}^{k}\in\mathop{\rm{\bf B}_{\mathcal{F}}}$. This is
equivalent to the inclusion $F(\mathcal{C})\subseteq\mathbb{Z}_{k}$.

###### Proof of Theorem 4.41.

Consider the exact sequence for tensor and torsion products derived from
(2.27) and corresponding to the group $\mathbb{Z}_{k}$:

$\begin{split}0&\longrightarrow\mathop{\rm Tor}(\mathbb{Z}_{k},\mathop{\rm{\bf
B}_{\mathcal{F}}})\longrightarrow\mathop{\rm
Tor}(\mathbb{Z}_{k},\mathop{\rm{\bf
Z}_{\mathcal{F}}})\longrightarrow\mathop{\rm
Tor}(\mathbb{Z}_{k},\mathop{\rm{\bf H}_{\mathcal{F}}})\\\
&\longrightarrow\mathbb{Z}_{k}\otimes\mathop{\rm{\bf
B}_{\mathcal{F}}}\longrightarrow\mathbb{Z}_{k}\otimes\mathop{\rm{\bf
Z}_{\mathcal{F}}}\longrightarrow\mathbb{Z}_{k}\otimes\mathop{\rm{\bf
H}_{\mathcal{F}}}\longrightarrow 0.\end{split}$ (4.21)

Then (4.21) yields (4.16) with the help of the following observations:

  * •

$\mathop{\rm Tor}(\mathbb{Z}_{k},\mathop{\rm{\bf B}_{\mathcal{F}}})=0$; this
follows from the fact that for an arbitrary $z\in X$, $\mathop{\rm{\bf
B}_{\mathcal{F}}}\cong C_{z}(X,\mathbb{T}^{1})$ is a torsion-free group by
connectedness of $X$,

  * •

$\mathop{\rm Tor}(\mathbb{Z}_{k},\mathop{\rm{\bf
Z}_{\mathcal{F}}})\cong\mathop{\rm tor_{k}}(\mathop{\rm{\bf
Z}_{\mathcal{F}}})=\mathop{\rm{\bf
Z}_{\mathcal{F}}}(\mathbb{Z}_{k})\cong\mathop{\rm
Hom}(\Gamma,\mathbb{Z}_{k})$; the first isomorphism and the equality are
clear, the second isomorphism follows from the fact that $X$ is connected and
$\mathbb{Z}_{k}$ is totally disconnected,

  * •

$\mathop{\rm Tor}(\mathbb{Z}_{k},\mathop{\rm{\bf
H}_{\mathcal{F}}})\cong\mathop{\rm tor_{k}}(\mathop{\rm{\bf
H}_{\mathcal{F}}})$; this is clear,

  * •

$\mathbb{Z}_{k}\otimes\mathop{\rm{\bf
B}_{\mathcal{F}}}\cong\pi^{1}(X)/k\pi^{1}(X)$; this follows from the
isomorphisms $\mathop{\rm{\bf B}_{\mathcal{F}}}\cong
C_{z}(X,\mathbb{T}^{1})\cong C_{z}(X,\mathbb{R})\oplus\pi^{1}(X)$ and from the
divisibility of $C_{z}(X,\mathbb{R})$:

$\begin{split}\mathbb{Z}_{k}\otimes\mathop{\rm{\bf
B}_{\mathcal{F}}}&\cong\mathbb{Z}_{k}\otimes
C_{z}(X,\mathbb{T}^{1})\cong\mathbb{Z}_{k}\otimes\big{(}C_{z}(X,\mathbb{R})\oplus\pi^{1}(X)\big{)}\\\
&\cong\big{(}\mathbb{Z}_{k}\otimes
C_{z}(X,\mathbb{R})\big{)}\oplus\big{(}\mathbb{Z}_{k}\otimes\pi^{1}(X)\big{)}\\\
&\cong\mathbb{Z}_{k}\otimes\pi^{1}(X)\cong\pi^{1}(X)/k\pi^{1}(X),\end{split}$

  * •

$\mathbb{Z}_{k}\otimes\mathop{\rm{\bf Z}_{\mathcal{F}}}=0$; this follows from
the fact that the group $\mathop{\rm{\bf Z}_{\mathcal{F}}}$ is divisible by
the assumptions of the theorem.

Now consider the exact sequence for tensor and torsion products derived from
(2.27) and corresponding to the group $\mathbb{Q}/\mathbb{Z}$:

$\begin{split}0&\longrightarrow\mathop{\rm
Tor}(\mathbb{Q}/\mathbb{Z},\mathop{\rm{\bf
B}_{\mathcal{F}}})\longrightarrow\mathop{\rm
Tor}(\mathbb{Q}/\mathbb{Z},\mathop{\rm{\bf
Z}_{\mathcal{F}}})\longrightarrow\mathop{\rm
Tor}(\mathbb{Q}/\mathbb{Z},\mathop{\rm{\bf H}_{\mathcal{F}}})\\\
&\longrightarrow(\mathbb{Q}/\mathbb{Z})\otimes\mathop{\rm{\bf
B}_{\mathcal{F}}}\longrightarrow(\mathbb{Q}/\mathbb{Z})\otimes\mathop{\rm{\bf
Z}_{\mathcal{F}}}\longrightarrow(\mathbb{Q}/\mathbb{Z})\otimes\mathop{\rm{\bf
H}_{\mathcal{F}}}\longrightarrow 0.\end{split}$ (4.22)

Then, similarly to the preceding step of the proof, (4.22) yields (4.17) with
the help of the following observations:

  * •

$\mathop{\rm Tor}(\mathbb{Q}/\mathbb{Z},\mathop{\rm{\bf B}_{\mathcal{F}}})=0$;
this follows from the fact that the group $\mathop{\rm{\bf B}_{\mathcal{F}}}$
is torsion-free by connectedness of $X$,

  * •

$\mathop{\rm Tor}(\mathbb{Q}/\mathbb{Z},\mathop{\rm{\bf
Z}_{\mathcal{F}}})\cong\bigcup_{l\in\mathbb{N}}\mathop{\rm
Hom}(\Gamma,\mathbb{Z}_{l})$; this follows from the following isomorphisms by
connectedness of $X$:

$\begin{split}\mathop{\rm Tor}(\mathbb{Q}/\mathbb{Z},\mathop{\rm{\bf
Z}_{\mathcal{F}}})&\cong\mathop{\rm tor}(\mathop{\rm{\bf
Z}_{\mathcal{F}}})=\bigcup_{l\in\mathbb{N}}\text{tor}_{l}(\mathop{\rm{\bf
Z}_{\mathcal{F}}})=\bigcup_{l\in\mathbb{N}}\mathop{\rm{\bf
Z}_{\mathcal{F}}}(\mathbb{Z}_{l})\\\ &\cong\bigcup_{l\in\mathbb{N}}\mathop{\rm
Hom}(\Gamma,\mathbb{Z}_{l}),\end{split}$

  * •

$\mathop{\rm Tor}(\mathbb{Q}/\mathbb{Z},\mathop{\rm{\bf
H}_{\mathcal{F}}})\cong\mathop{\rm tor}(\mathop{\rm{\bf H}_{\mathcal{F}}})$;
this is clear,

  * •

$(\mathbb{Q}/\mathbb{Z})\otimes\mathop{\rm{\bf
B}_{\mathcal{F}}}\cong(\mathbb{Q}/\mathbb{Z})\otimes\pi^{1}(X)$; this follows
from the divisibility of $C_{z}(X,\mathbb{R})$:

$\begin{split}(\mathbb{Q}/\mathbb{Z})\otimes\mathop{\rm{\bf
B}_{\mathcal{F}}}&\cong(\mathbb{Q}/\mathbb{Z})\otimes
C_{z}(X,\mathbb{T}^{1})\\\
&\cong(\mathbb{Q}/\mathbb{Z})\otimes\big{(}C_{z}(X,\mathbb{R})\oplus\pi^{1}(X)\big{)}\\\
&\cong\big{(}(\mathbb{Q}/\mathbb{Z})\otimes
C_{z}(X,\mathbb{R})\big{)}\oplus\big{(}(\mathbb{Q}/\mathbb{Z})\otimes\pi^{1}(X)\big{)}\\\
&\cong(\mathbb{Q}/\mathbb{Z})\otimes\pi^{1}(X),\end{split}$

  * •

$(\mathbb{Q}/\mathbb{Z})\otimes\mathop{\rm{\bf Z}_{\mathcal{F}}}=0$; this
follows from the fact that $\mathop{\rm{\bf Z}_{\mathcal{F}}}$ is a divisible
group by the assumptions of the theorem and $\mathbb{Q}/\mathbb{Z}$ is a
torsion group.

Now assume that $\Gamma$ possesses no non-trivial finite abelian quotient
groups. Then $\mathop{\rm Hom}(\Gamma,\mathbb{Z}_{l})=0$ for every
$l\in\mathbb{N}$ and so the isomorphisms (4.18) and (4.19) follow from the
exactness of the sequences (4.16) and (4.17), respectively. To finish the
proof, we need to verify (4.20). The second isomorphism from (4.20) is a
consequence of (4.19). The first isomorphism from (4.20) follows from the
following facts.

  * •

The torsion subgroup $\mathop{\rm tor}(\mathop{\rm{\bf H}_{\mathcal{F}}})$ of
$\mathop{\rm{\bf H}_{\mathcal{F}}}$ is divisible. Indeed, by the assumptions
of the theorem, $\mathop{\rm{\bf Z}_{\mathcal{F}}}$ is a divisible group and
hence so is its quotient group $\mathop{\rm{\bf H}_{\mathcal{F}}}$. Thus,
$\mathop{\rm tor}(\mathop{\rm{\bf H}_{\mathcal{F}}})$ is divisible, being the
torsion subgroup of a divisible group.

  * •

The group $\mathop{\rm tor}(\mathop{\rm{\bf H}_{\mathcal{F}}})$ is contained
as a direct summand in $\mathop{\rm{\bf H}_{\mathcal{F}}}$ by divisibility.

  * •

Every complementary summand $\mathfrak{D}$ of $\mathop{\rm
tor}(\mathop{\rm{\bf H}_{\mathcal{F}}})$ in $\mathop{\rm{\bf
H}_{\mathcal{F}}}$ is divisible. This follows from the divisibility of both
$\mathop{\rm{\bf H}_{\mathcal{F}}}$ and $\mathop{\rm tor}(\mathop{\rm{\bf
H}_{\mathcal{F}}})$.

  * •

Given $\mathfrak{D}$ as above, it is always a subgroup of the groupoid
$\mathop{\rm{\bf MinH}_{\mathcal{F}}}$. Indeed, we have $\mathfrak{D}\setminus
1\subseteq\mathop{\rm{\bf H}_{\mathcal{F}}}\setminus\mathop{\rm
tor}(\mathop{\rm{\bf H}_{\mathcal{F}}})=\mathop{\rm{\bf
MinH}_{\mathcal{F}}}\setminus 1$.

This verifies (4.20). ∎

######  Corollary 4.43.

Let $\mathop{\rm\mathcal{F}\colon\Gamma\curvearrowright X}$ be a minimal flow
on a compact connected space $X$. Assume that $\Gamma$ has no non-trivial
finite abelian quotient groups and that the group $\mathop{\rm{\bf
Z}_{\mathcal{F}}}$ is divisible. Then the identity component
$F(\mathop{\rm\daleth_{\mathcal{F}}})_{0}$ of
$F(\mathop{\rm\daleth_{\mathcal{F}}})$ is a topological direct summand in
$F(\mathop{\rm\daleth_{\mathcal{F}}})$. In fact, under the notation from
Theorem 4.41, there are topological isomorphisms

$\begin{array}[]{ccccc}F(\mathop{\rm\daleth_{\mathcal{F}}})&\cong&F(\mathop{\rm\daleth_{\mathcal{F}}})_{0}&\times&F(\mathop{\rm\daleth_{\mathcal{F}}})/F(\mathop{\rm\daleth_{\mathcal{F}}})_{0}\\\
{}\hfil&\cong&\mathfrak{D}^{*}&\times&\mathop{\rm tor}\left(\mathop{\rm{\bf
H}_{\mathcal{F}}}\right)^{*}\\\
{}\hfil&\cong&\mathfrak{D}^{*}&\times&\left((\mathbb{Q}/\mathbb{Z})\otimes\pi^{1}(X)\right)^{*},\end{array}$

where all $\mathfrak{D}$, $\mathop{\rm tor}(\mathop{\rm{\bf
H}_{\mathcal{F}}})$ and $(\mathbb{Q}/\mathbb{Z})\otimes\pi^{1}(X)$ are assumed
to carry the discrete topology.

###### Proof.

By virtue of Remark 4.8 and by virtue of (4.20) from Theorem 4.41, the
following topological isomorphisms hold for an appropriate divisible subgroup
$\mathfrak{D}$ of the groupoid $\mathop{\rm{\bf MinH}_{\mathcal{F}}}$:

$\begin{split}F(\mathop{\rm\daleth_{\mathcal{F}}})=\left(\mathop{\rm{\bf
B}_{\mathcal{F}}}\right)^{\perp}\cong\left(\mathop{\rm{\bf
H}_{\mathcal{F}}}\right)^{*}&\cong(\mathfrak{D}\oplus\mathop{\rm
tor}(\mathop{\rm{\bf H}_{\mathcal{F}}}))^{*}\\\
&\cong\mathfrak{D}^{*}\times\mathop{\rm tor}\left(\mathop{\rm{\bf
H}_{\mathcal{F}}}\right)^{*}\\\
&\cong\mathfrak{D}^{*}\times\left((\mathbb{Q}/\mathbb{Z})\otimes\pi^{1}(X)\right)^{*}.\end{split}$
(4.23)

Since $\mathop{\rm tor}(\mathop{\rm{\bf H}_{\mathcal{F}}})$ is a torsion
group, its dual group $\mathop{\rm tor}(\mathop{\rm{\bf
H}_{\mathcal{F}}})^{*}$ is totally disconnected. Further, since $\mathfrak{D}$
is a subgroup of $\mathop{\rm{\bf MinH}_{\mathcal{F}}}$, it is torsion-free
and its dual group $\mathfrak{D}^{*}$ is therefore connected. Consequently,
under the isomorphisms from (4.23), the group $\mathfrak{D}^{*}$ corresponds
to $F(\mathop{\rm\daleth_{\mathcal{F}}})_{0}$ and the group $\mathop{\rm
tor}(\mathop{\rm{\bf H}_{\mathcal{F}}})^{*}$ corresponds to
$F(\mathop{\rm\daleth_{\mathcal{F}}})/F(\mathop{\rm\daleth_{\mathcal{F}}})_{0}$.
This proves the corollary. ∎

######  Theorem 4.44.

Let $\mathop{\rm\mathcal{F}\colon\Gamma\curvearrowright X}$ be a minimal flow
with $\Gamma\in\mathsf{LieGp}$ connected and with $X$ a compact manifold. Set
$n=\mathop{\rm rank}(H_{1}^{w}(\mathcal{F}))$, $n+m=\mathop{\rm
rank}(H_{1}^{w}(X))$ and denote by $d_{1},\dots,d_{n}$ the elementary divisors
of $H_{1}^{w}(\mathcal{F})$ in $H_{1}^{w}(X)$. If $k\geq 2$ is an integer and
$\delta_{i}=\gcd(d_{i},k)$ ($i=1,\dots,n$) then

$\text{$\mathop{\rm tor}$}_{k}\left(\mathop{\rm{\bf
H}_{\mathcal{F}}}\right)\cong\text{$\mathop{\rm
tor}$}_{k}\left(H_{1}^{w}(X)/H_{1}^{w}(\mathcal{F})\right)\oplus\left(\mathbb{Z}_{k}\right)^{m}\cong\mathbb{Z}_{\delta_{1}}\oplus\dots\oplus\mathbb{Z}_{\delta_{n}}\oplus\left(\mathbb{Z}_{k}\right)^{m}$
(4.24)

and

$\mathop{\rm tor}\left(\mathop{\rm{\bf
H}_{\mathcal{F}}}\right)\cong\mathop{\rm
tor}\left(H_{1}^{w}(X)/H_{1}^{w}(\mathcal{F})\right)\oplus\left(\mathbb{Q}/\mathbb{Z}\right)^{m}\cong\mathbb{Z}_{d_{1}}\oplus\dots\oplus\mathbb{Z}_{d_{n}}\oplus\left(\mathbb{Q}/\mathbb{Z}\right)^{m}.$
(4.25)

######  Remark 4.45.

We wish to add the following remarks.

  * •

The isomorphisms (4.24) and (4.25) are consistent with the isomorphisms (4.18)
and (4.19) from Theorem 4.41, respectively. For if $\Gamma$ is simply
connected then $\mathop{\rm{\bf Z}_{\mathcal{F}}}$ is a divisible group and we
have $n=\mathop{\rm rank}(H_{1}^{w}(\mathcal{F}))=0$, $m=\mathop{\rm
rank}(H_{1}^{w}(X))$ and $\pi^{1}(X)\cong H_{1}^{w}(X)$, which leads to
isomorphisms

$\begin{split}\mathop{\rm tor_{k}}(\mathop{\rm{\bf
H}_{\mathcal{F}}})&\cong\pi^{1}(X)/k\pi^{1}(X)\cong
H_{1}^{w}(X)/kH_{1}^{w}(X)\\\
&\cong\mathbb{Z}^{m}/k\mathbb{Z}^{m}\cong\left(\mathbb{Z}_{k}\right)^{m}\end{split}$

and

$\begin{split}\mathop{\rm tor}(\mathop{\rm{\bf
H}_{\mathcal{F}}})&\cong(\mathbb{Q}/\mathbb{Z})\otimes\pi^{1}(X)\cong(\mathbb{Q}/\mathbb{Z})\otimes
H_{1}^{w}(X)\\\
&\cong(\mathbb{Q}/\mathbb{Z})\otimes\mathbb{Z}^{m}\cong(\mathbb{Q}/\mathbb{Z})^{m}.\end{split}$

  * •

The isomorphism (4.24) can help us understand the structure of the set
$\mathop{\rm per}_{k}(\mathop{\rm{\bf H}_{\mathcal{F}}})$ of all cohomology
classes of extensions $\mathcal{C}\in\mathop{\rm{\bf Z}_{\mathcal{F}}}$ with
$F(\mathcal{C})=\mathbb{Z}_{k}$. Indeed, an element
$(z_{1},\dots,z_{n},\zeta_{1},\dots,\zeta_{m})$ of
$\mathbb{Z}_{\delta_{1}}\oplus\dots\oplus\mathbb{Z}_{\delta_{n}}\oplus(\mathbb{Z}_{k})^{m}$
corresponds to an element of $\mathop{\rm per}_{k}(\mathop{\rm{\bf
H}_{\mathcal{F}}})$ under the isomorphism (4.24) if and only if one of the
following conditions is satisfied:

    1. (i)

there is $1\leq i\leq n$ such that $k$ divides $d_{i}$ (that is,
$\delta_{i}=k$) and $z_{i}$ generates
$\mathbb{Z}_{\delta_{i}}=\mathbb{Z}_{k}$,

    2. (ii)

there is $1\leq j\leq m$ such that $\zeta_{j}$ generates $\mathbb{Z}_{k}$.

###### Proof of Theorem 4.44.

Throughout the whole proof, both $X$ and $\Gamma$ will be assumed pointed. The
base point of $X$ will be denoted by $z$ and the base point of $\Gamma$ will
be its identity $1$. We also recall that for abelian groups $A,C$ and their
subgroups $B\mathrel{\lhd}A$, $D\mathrel{\lhd}C$, $\mathop{\rm Hom}(A,B;C,D)$
stands for the group of all morphisms $h\colon A\to C$ with $h(B)\subseteq D$.
Now we turn to the proof, proceeding in three steps.

_1st step._ Fix a map $\xi\in C_{z}(X,\mathbb{T}^{1})$ and an integer $k\geq
2$. We show that the following conditions are equivalent:

  1. (a)

$\mathop{\rm co}(\xi)$ lifts across
$\kappa_{k}\colon\mathbb{T}^{1}\to\mathbb{T}^{1}$ to an extension
$\mathcal{C}\in\mathop{\rm{\bf Z}_{\mathcal{F}}}$,

  2. (b)

$\xi^{\sharp}H_{1}^{w}(\mathcal{F})\subseteq k\mathbb{Z}$, where
$\xi^{\sharp}$ is the induced morphism $H_{1}^{w}(X)\to
H_{1}^{w}(\mathbb{T}^{1})$ and $H_{1}^{w}(\mathbb{T}^{1})$ is identified with
$\mathbb{Z}$.

We claim that the following conditions are equivalent:

  * •

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

  * •

$\mathop{\rm co}(\xi)$ lifts across $\kappa_{k}$ to a continuous base point
preserving map $\Gamma\times X\to\mathbb{T}^{1}$,

  * •

$\mathop{\rm co}(\xi)^{\sharp}\pi_{1}(\Gamma\times
X)\subseteq\kappa_{k}^{\sharp}\pi_{1}(\mathbb{T}^{1})=k\mathbb{Z}$,

  * •

$\xi^{\sharp}\mathcal{F}_{z}^{\sharp}\pi_{1}(\Gamma)\subseteq k\mathbb{Z}$,

  * •

$\xi^{\sharp}H_{1}^{w}(\mathcal{F})\subseteq k\mathbb{Z}$.

Indeed, the equivalence of the first four of these conditions follows from
Lemmas 3.36 and 2.17. The equivalence of the last two conditions follows from
the following simple computation based on the commutativity of the diagram in
Lemma 2.17

$p_{\mathbb{T}^{1}}\left(\xi^{\sharp}\mathcal{F}_{z}^{\sharp}\pi_{1}(\Gamma)\right)=\xi^{\sharp}\mathcal{F}_{z}^{\sharp}p_{\Gamma}(\pi_{1}(\Gamma))=\xi^{\sharp}\mathcal{F}_{z}^{\sharp}H_{1}^{w}(\Gamma)=\xi^{\sharp}H_{1}^{w}(\mathcal{F})$

and from the equality $p_{\mathbb{T}^{1}}=\mathop{\rm id}_{\mathbb{Z}}$, which
holds under the standard identifications $\pi_{1}(\mathbb{T}^{1})\cong
H_{1}^{w}(\mathbb{T}^{1})\cong\mathbb{Z}$.

_2nd step._ We verify (4.24).

Given a propositional form $\mathcal{P}$ defined on the set
$C_{z}(X,\mathbb{T}^{1})$, we use the symbol $[\mathcal{P}(\xi)]$ to denote
the set of all $\xi\in C_{z}(X,\mathbb{T}^{1})$ such that the proposition
$\mathcal{P}(\xi)$ holds true. We shall arrive at (4.24) through a series of
isomorphisms. Each of these isomorphisms will be dealt with in a separate
item.

  1. (i)

By virtue of the equivalence (a)$\Leftrightarrow$(b) from the first step of
the proof,

$\begin{split}\mathop{\rm tor_{k}}(\mathop{\rm{\bf
H}_{\mathcal{F}}})&=\left\\{\mathcal{C}\in\mathop{\rm{\bf
Z}_{\mathcal{F}}}:\mathcal{C}^{k}\in\mathop{\rm{\bf
B}_{\mathcal{F}}}\right\\}/\mathop{\rm{\bf B}_{\mathcal{F}}}\\\
&=\left\\{\kappa_{k}^{-1}(\mathop{\rm
co}(\xi)):\xi^{\sharp}H_{1}^{w}(\mathcal{F})\subseteq
k\mathbb{Z}\right\\}/\mathop{\rm{\bf B}_{\mathcal{F}}},\end{split}$ (4.26)

where $\kappa_{k}$ stands for the $k$-endomorphism of $\mathop{\rm{\bf
Z}_{\mathcal{F}}}$. Observe that $\kappa_{k}$ is a monomorphism by
connectedness of $\Gamma$ and $X$. Consequently, by virtue of (4.26),

$\mathop{\rm tor_{k}}(\mathop{\rm{\bf
H}_{\mathcal{F}}})\cong\left\\{\mathop{\rm
co}(\xi):\xi^{\sharp}H_{1}^{w}(\mathcal{F})\subseteq
k\mathbb{Z}\right\\}/\kappa_{k}(\mathop{\rm{\bf B}_{\mathcal{F}}}).$ (4.27)

  2. (ii)

Under the usual isomorphism $\sigma\colon
C_{z}(X,\mathbb{T}^{1})\ni\xi\mapsto\mathop{\rm co}(\xi)\in\mathop{\rm{\bf
B}_{\mathcal{F}}}$, the subgroup $\kappa_{k}(\mathop{\rm{\bf
B}_{\mathcal{F}}})$ of $\mathop{\rm{\bf B}_{\mathcal{F}}}$ corresponds to the
subgroup $\kappa_{k}(C_{z}(X,\mathbb{T}^{1}))$ of $C_{z}(X,\mathbb{T}^{1})$.
The latter subgroup consists of all those maps $\xi\in
C_{z}(X,\mathbb{T}^{1})$, which lift across
$\kappa_{k}\colon\mathbb{T}^{1}\to\mathbb{T}^{1}$ to an element of
$C_{z}(X,\mathbb{T}^{1})$; this occurs if and only if
$\xi^{\sharp}H_{1}^{w}(X)\subseteq k\mathbb{Z}$. Thus, the isomorphism (4.27)
yields

$\mathop{\rm tor_{k}}(\mathop{\rm{\bf
H}_{\mathcal{F}}})\cong\frac{[\xi^{\sharp}H_{1}^{w}(\mathcal{F})\subseteq
k\mathbb{Z}]}{[\xi^{\sharp}H_{1}^{w}(X)\subseteq
k\mathbb{Z}]}\cong\frac{[\xi^{\sharp}H_{1}^{w}(\mathcal{F})\subseteq
k\mathbb{Z}]/[\xi^{\sharp}H_{1}^{w}(X)=0]}{[\xi^{\sharp}H_{1}^{w}(X)\subseteq
k\mathbb{Z}]/[\xi^{\sharp}H_{1}^{w}(X)=0]}.$ (4.28)

  3. (iii)

Consider the map $\varphi\colon
C_{z}(X,\mathbb{T}^{1})\ni\xi\mapsto\xi^{\sharp}\in\mathop{\rm
Hom}(H_{1}^{w}(X),\mathbb{Z})$. By our assumptions on $X$, $\varphi$ is an
epimorphism and $\ker(\varphi)=[\xi^{\sharp}H_{1}^{w}(X)=0]$. Consequently, it
follows from (4.28) by elementary group-theoretic arguments that

$\begin{split}\mathop{\rm tor_{k}}(\mathop{\rm{\bf
H}_{\mathcal{F}}})&\cong\frac{\mathop{\rm
Hom}(H_{1}^{w}(X),H_{1}^{w}(\mathcal{F});\mathbb{Z},k\mathbb{Z})}{\mathop{\rm
Hom}(H_{1}^{w}(X),H_{1}^{w}(X);\mathbb{Z},k\mathbb{Z})}\\\
&\cong\frac{\mathop{\rm
Hom}(\mathbb{Z}^{n}\oplus\mathbb{Z}^{m},d_{1}\mathbb{Z}\oplus\dots\oplus
d_{n}\mathbb{Z};\mathbb{Z},k\mathbb{Z})}{\mathop{\rm
Hom}(\mathbb{Z}^{n}\oplus\mathbb{Z}^{m},\mathbb{Z}^{n}\oplus\mathbb{Z}^{m};\mathbb{Z},k\mathbb{Z})},\end{split}$
(4.29)

where $d_{1}\mathbb{Z}\oplus\dots\oplus
d_{n}\mathbb{Z}\subseteq\mathbb{Z}^{n}\cong\mathbb{Z}^{n}\oplus
0\subseteq\mathbb{Z}^{n}\oplus\mathbb{Z}^{m}$.

  4. (iv)

Consider the isomorphism

$\psi\colon\mathop{\rm Hom}(\mathbb{Z}^{n}\oplus\mathbb{Z}^{m},\mathbb{Z})\ni
h\mapsto(h(e_{1}),\dots,h(e_{n+m}))\in\mathbb{Z}^{n}\oplus\mathbb{Z}^{m},$

where $e_{1},\dots,e_{n+m}$ is the standard basis for
$\mathbb{Z}^{n}\oplus\mathbb{Z}^{m}$. Then

$\psi(\mathop{\rm
Hom}(\mathbb{Z}^{n}\oplus\mathbb{Z}^{m},d_{1}\mathbb{Z}\oplus\dots\oplus
d_{n}\mathbb{Z};\mathbb{Z},k\mathbb{Z}))=\frac{k}{\delta_{1}}\mathbb{Z}\oplus\dots\oplus\frac{k}{\delta_{n}}\mathbb{Z}\oplus\mathbb{Z}^{m}$

and

$\psi(\mathop{\rm
Hom}(\mathbb{Z}^{n}\oplus\mathbb{Z}^{m},\mathbb{Z}^{n}\oplus\mathbb{Z}^{m};\mathbb{Z},k\mathbb{Z}))=k\mathbb{Z}^{n}\oplus
k\mathbb{Z}^{m}.$

Consequently, by virtue of (4.29),

$\begin{split}\mathop{\rm tor_{k}}(\mathop{\rm{\bf
H}_{\mathcal{F}}})&\cong\left(\frac{k}{\delta_{1}}\mathbb{Z}\oplus\dots\oplus\frac{k}{\delta_{n}}\mathbb{Z}\oplus\mathbb{Z}^{m}\right)/\left(k\mathbb{Z}^{n}\oplus
k\mathbb{Z}^{m}\right)\\\
&\cong\mathbb{Z}_{\delta_{1}}\oplus\dots\oplus\mathbb{Z}_{\delta_{n}}\oplus\left(\mathbb{Z}_{k}\right)^{m}\\\
&\cong\mathop{\rm
tor_{k}}(\mathbb{Z}_{d_{1}}\oplus\dots\oplus\mathbb{Z}_{d_{n}})\oplus\left(\mathbb{Z}_{k}\right)^{m}\\\
&\cong\mathop{\rm
tor_{k}}(H_{1}^{w}(X)/H_{1}^{w}(\mathcal{F}))\oplus\left(\mathbb{Z}_{k}\right)^{m}.\end{split}$
(4.30)

This verifies the two isomorphisms from (4.24) and finishes the second step of
the proof.

$\begin{CD}\kappa_{k}^{-1}(\mathop{\rm{\bf
B}_{\mathcal{F}}})@>{\sigma^{-1}\kappa_{k}}>{}>[\xi^{\sharp}H_{1}^{w}(\mathcal{F})\subseteq
k\mathbb{Z}]@>{\psi\varphi}>{}>\frac{k}{\delta_{1}}\mathbb{Z}\oplus\dots\oplus\frac{k}{\delta_{n}}\mathbb{Z}\oplus\mathbb{Z}^{m}\\\
{\rotatebox{90.0}{$\subseteq$}}{\rotatebox{90.0}{$\subseteq$}}{\rotatebox{90.0}{$\subseteq$}}\\\
\mathop{\rm{\bf
B}_{\mathcal{F}}}@>{}>{\sigma^{-1}\kappa_{k}}>[\xi^{\sharp}H_{1}^{w}(X)\subseteq
k\mathbb{Z}]@>{}>{\psi\varphi}>k\mathbb{Z}^{n}\oplus k\mathbb{Z}^{m}\end{CD}$
Figure 4.14: Verifying (4.24)

Since we shall have to refer to the procedure just finished and also modify it
in a certain way, we summarize its steps in the form of a diagram in Figure
4.14. (We emphasize that all the arrows in the diagram are epimorphisms.)

_3rd step._ We verify (4.25).

Let $k,l\geq 2$ be integers, $l$ divisible by $k$ and $k$ divisible by
$d=d_{n}$. Then $\gcd(d_{i},k)=\gcd(d_{i},l)=d_{i}$ for $i=1,\dots,n$ and
hence, by the second step of the proof, there are isomorphisms

$\text{tor}_{k}(\mathop{\rm{\bf
H}_{\mathcal{F}}})\cong\mathbb{Z}_{d_{1}}\oplus\dots\oplus\mathbb{Z}_{d_{n}}\oplus\left(\mathbb{Z}_{k}\right)^{m}$
(4.31)

and

$\text{tor}_{l}(\mathop{\rm{\bf
H}_{\mathcal{F}}})\cong\mathbb{Z}_{d_{1}}\oplus\dots\oplus\mathbb{Z}_{d_{n}}\oplus\left(\mathbb{Z}_{l}\right)^{m}.$
(4.32)

We need to determine what form the inclusion $\mathop{\rm
tor}_{k}(\mathop{\rm{\bf H}_{\mathcal{F}}})\subseteq\mathop{\rm
tor}_{l}(\mathop{\rm{\bf H}_{\mathcal{F}}})$ takes after passing to the right
hand sides of (4.31) and (4.32). To this end, we invoke the procedure from the
second step of the proof, which is summarized in the diagram in Figure 4.14,
but we apply the monomorphism $\kappa_{l}$ instead of $\kappa_{k}$. This leads
to the inclusions depicted in the diagram in Figure 4.15, as we now show.
(Similarly to Figure 4.14, all the arrows in Figure 4.15 are claimed to be
epimorphisms.)

$\begin{CD}\kappa_{k}^{-1}(\mathop{\rm{\bf
B}_{\mathcal{F}}})@>{\sigma^{-1}\kappa_{l}}>{}>\kappa_{l/k}[\xi^{\sharp}H_{1}^{w}(\mathcal{F})\subseteq
k\mathbb{Z}]@>{\psi\varphi}>{}>\frac{l}{d_{1}}\mathbb{Z}\oplus\dots\oplus\frac{l}{d_{n}}\mathbb{Z}\oplus\frac{l}{k}\mathbb{Z}^{m}\\\
{\rotatebox{90.0}{$\subseteq$}}{\rotatebox{90.0}{$\subseteq$}}{\rotatebox{90.0}{$\subseteq$}}\\\
\mathop{\rm{\bf
B}_{\mathcal{F}}}@>{\sigma^{-1}\kappa_{l}}>{}>[\xi^{\sharp}H_{1}^{w}(X)\subseteq
l\mathbb{Z}]@>{\psi\varphi}>{}>l\mathbb{Z}^{n}\oplus l\mathbb{Z}^{m}\\\
{\rotatebox{90.0}{$\supseteq$}}{\rotatebox{90.0}{$\supseteq$}}{\rotatebox{90.0}{$\supseteq$}}\\\
\kappa_{l}^{-1}(\mathop{\rm{\bf
B}_{\mathcal{F}}})@>{\sigma^{-1}\kappa_{l}}>{}>[\xi^{\sharp}H_{1}^{w}(\mathcal{F})\subseteq
l\mathbb{Z}]@>{\psi\varphi}>{}>\frac{l}{d_{1}}\mathbb{Z}\oplus\dots\oplus\frac{l}{d_{n}}\mathbb{Z}\oplus\mathbb{Z}^{m}\end{CD}$
Figure 4.15: Towards the inclusion $\mathop{\rm tor}_{k}(\mathop{\rm{\bf
H}_{\mathcal{F}}})\subseteq\mathop{\rm tor}_{l}(\mathop{\rm{\bf
H}_{\mathcal{F}}})$

The second and the third line of the diagram follow immediately from the
second step of the proof. We verify the first line, proceeding in two steps.

  * •

Recall from the first step of the proof that

$\kappa_{k}^{-1}(\mathop{\rm{\bf
B}_{\mathcal{F}}})=\left\\{\kappa_{k}^{-1}(\mathop{\rm
co}(\xi)):\xi^{\sharp}H_{1}^{w}(\mathcal{F})\subseteq k\mathbb{Z}\right\\}.$

By applying $\sigma^{-1}\kappa_{l}$ to this equality, we obtain

$\begin{split}\sigma^{-1}\kappa_{l}\left(\kappa_{k}^{-1}(\mathop{\rm{\bf
B}_{\mathcal{F}}})\right)&=\sigma^{-1}\left(\left\\{\mathop{\rm
co}\left(\xi^{l/k}\right):\xi^{\sharp}H_{1}^{w}(\mathcal{F})\subseteq
k\mathbb{Z}\right\\}\right)\\\
&=\left\\{\xi^{l/k}:\xi^{\sharp}H_{1}^{w}(\mathcal{F})\subseteq
k\mathbb{Z}\right\\}\\\
&=\kappa_{l/k}[\xi^{\sharp}H_{1}^{w}(\mathcal{F})\subseteq
k\mathbb{Z}].\end{split}$

  * •

By the second step of the proof,

$\begin{split}\psi\varphi[\xi^{\sharp}H_{1}^{w}(\mathcal{F})\subseteq
k\mathbb{Z}]&=\frac{k}{\delta_{1}}\mathbb{Z}\oplus\dots\oplus\frac{k}{\delta_{n}}\mathbb{Z}\oplus\mathbb{Z}^{m}\\\
&=\frac{k}{d_{1}}\mathbb{Z}\oplus\dots\oplus\frac{k}{d_{n}}\mathbb{Z}\oplus\mathbb{Z}^{m}.\end{split}$

Consequently,

$\begin{split}\psi\varphi\left(\kappa_{l/k}[\xi^{\sharp}H_{1}^{w}(\mathcal{F})\subseteq
k\mathbb{Z}]\right)&=\kappa_{l/k}\left(\psi\varphi[\xi^{\sharp}H_{1}^{w}(\mathcal{F})\subseteq
k\mathbb{Z}]\right)\\\
&=\kappa_{l/k}\left(\frac{k}{d_{1}}\mathbb{Z}\oplus\dots\oplus\frac{k}{d_{n}}\mathbb{Z}\oplus\mathbb{Z}^{m}\right)\\\
&=\frac{l}{d_{1}}\mathbb{Z}\oplus\dots\oplus\frac{l}{d_{n}}\mathbb{Z}\oplus\frac{l}{k}\mathbb{Z}^{m}.\end{split}$

These two observations verify the first line of the diagram in Figure 4.15.

Set
$\varrho=\psi\varphi\sigma^{-1}\kappa_{l}\colon\kappa_{l}^{-1}(\mathop{\rm{\bf
B}_{\mathcal{F}}})\to\mathbb{Z}^{n}\oplus\mathbb{Z}^{m}$. Following the
results summarized in Figure 4.15, we claim that $\varrho$ induces
isomorphisms

$\begin{split}\text{tor}_{k}(\mathop{\rm{\bf
H}_{\mathcal{F}}})&=\frac{\kappa_{k}^{-1}(\mathop{\rm{\bf
B}_{\mathcal{F}}})}{\mathop{\rm{\bf
B}_{\mathcal{F}}}}\cong\frac{\varrho(\kappa_{k}^{-1}(\mathop{\rm{\bf
B}_{\mathcal{F}}}))}{\varrho(\mathop{\rm{\bf B}_{\mathcal{F}}})}\\\
&=\left(\frac{l}{d_{1}}\mathbb{Z}\oplus\dots\oplus\frac{l}{d_{n}}\mathbb{Z}\oplus\frac{l}{k}\mathbb{Z}^{m}\right)/(l\mathbb{Z}^{n}\oplus
l\mathbb{Z}^{m})\end{split}$ (4.33)

and

$\begin{split}\text{tor}_{l}(\mathop{\rm{\bf
H}_{\mathcal{F}}})&=\frac{\kappa_{l}^{-1}(\mathop{\rm{\bf
B}_{\mathcal{F}}})}{\mathop{\rm{\bf
B}_{\mathcal{F}}}}\cong\frac{\varrho(\kappa_{l}^{-1}(\mathop{\rm{\bf
B}_{\mathcal{F}}}))}{\varrho(\mathop{\rm{\bf B}_{\mathcal{F}}})}\\\
&=\left(\frac{l}{d_{1}}\mathbb{Z}\oplus\dots\oplus\frac{l}{d_{n}}\mathbb{Z}\oplus\mathbb{Z}^{m}\right)/(l\mathbb{Z}^{n}\oplus
l\mathbb{Z}^{m}).\end{split}$ (4.34)

In order to prove this claim, it suffices to show that
$\ker(\varrho)\subseteq\mathop{\rm{\bf B}_{\mathcal{F}}}$. So let
$\mathcal{C}\in\kappa_{l}^{-1}(\mathop{\rm{\bf B}_{\mathcal{F}}})$, $\xi\in
C_{z}(X,\mathbb{T}^{1})$ be the transfer function of $\kappa_{l}(\mathcal{C})$
and assume that $\varrho(\mathcal{C})=0$. Then $\xi^{\sharp}=0$ and hence
$\xi\in\mathop{\rm Div}(C_{z}(X,\mathbb{T}^{1}))$. Since $\sigma\colon
C_{z}(X,\mathbb{T}^{1})\to\mathop{\rm{\bf B}_{\mathcal{F}}}$ is an
isomorphism, it follows that $\mathop{\rm co}(\xi)=\sigma(\xi)\in\mathop{\rm
Div}(\mathop{\rm{\bf B}_{\mathcal{F}}})$. Consequently,
$\mathcal{C}=\kappa_{l}^{-1}(\mathop{\rm co}(\xi))\in\mathop{\rm{\bf
B}_{\mathcal{F}}}$, as was to be shown.

Now, under the natural isomorphisms $\mathbb{Z}/l\mathbb{Z}\cong\mathop{\rm
tor}_{l}(\mathbb{T}^{1})$ and
$((l/k)\mathbb{Z})/l\mathbb{Z}\cong\mathbb{Z}/k\mathbb{Z}\cong\mathop{\rm
tor}_{k}(\mathbb{T}^{1})$, the inclusion
$((l/k)\mathbb{Z})/l\mathbb{Z}\subseteq\mathbb{Z}/l\mathbb{Z}$ takes the form
of the inclusion $\mathop{\rm tor}_{k}(\mathbb{T}^{1})\subseteq\mathop{\rm
tor}_{l}(\mathbb{T}^{1})$. Thus, by virtue of (4.33) and (4.34), the inclusion
$\mathop{\rm tor}_{k}(\mathop{\rm{\bf H}_{\mathcal{F}}})\subseteq\mathop{\rm
tor}_{l}(\mathop{\rm{\bf H}_{\mathcal{F}}})$ takes the form of the right hand
side of Figure 4.16.

$\begin{CD}\mathop{\rm tor}_{k}(\mathop{\rm{\bf
H}_{\mathcal{F}}}){}={}\kappa_{k}^{-1}(\mathop{\rm{\bf
B}_{\mathcal{F}}})/\mathop{\rm{\bf
B}_{\mathcal{F}}}{}\cong{}\mathbb{Z}_{d_{1}}\oplus\dots\oplus\mathbb{Z}_{d_{n}}\oplus\mathop{\rm
tor}_{k}(\mathbb{T}^{1})^{m}\\\
{\rotatebox{90.0}{$\supseteq$}}{\rotatebox{90.0}{$\supseteq$}}{\rotatebox{90.0}{$\supseteq$}}\\\
\mathop{\rm tor}_{l}(\mathop{\rm{\bf
H}_{\mathcal{F}}}){}={}\kappa_{l}^{-1}(\mathop{\rm{\bf
B}_{\mathcal{F}}})/\mathop{\rm{\bf
B}_{\mathcal{F}}}{}\cong{}\mathbb{Z}_{d_{1}}\oplus\dots\oplus\mathbb{Z}_{d_{n}}\oplus\mathop{\rm
tor}_{l}(\mathbb{T}^{1})^{m}\end{CD}$ Figure 4.16: Another form of the
inclusion $\mathop{\rm tor}_{k}(\mathop{\rm{\bf
H}_{\mathcal{F}}})\subseteq\mathop{\rm tor}_{l}(\mathop{\rm{\bf
H}_{\mathcal{F}}})$

Let $(k_{j})_{j\in\mathbb{N}}$ be a sequence of positive integers with the
following properties:

  * •

$d$ divides $k_{1}$ and $k_{j}$ divides $k_{j+1}$ for every $j\in\mathbb{N}$,

  * •

for every $k\in\mathbb{N}$ there is $j\in\mathbb{N}$ such that $k$ divides
$k_{j}$.

Then the results of this step of the proof yield isomorphisms

$\begin{split}\mathop{\rm tor}(\mathop{\rm{\bf
H}_{\mathcal{F}}})&=\bigcup_{k\in\mathbb{N}}\text{$\mathop{\rm
tor}$}_{k}(\mathop{\rm{\bf
H}_{\mathcal{F}}})=\bigcup_{j\in\mathbb{N}}\text{$\mathop{\rm
tor}$}_{k_{j}}(\mathop{\rm{\bf
H}_{\mathcal{F}}})\cong\bigcup_{j\in\mathbb{N}}\left(\mathbb{Z}_{d_{1}}\oplus\dots\oplus\mathbb{Z}_{d_{n}}\oplus\text{tor}_{k_{j}}(\mathbb{T}^{1})^{m}\right)\\\
&=\mathbb{Z}_{d_{1}}\oplus\dots\oplus\mathbb{Z}_{d_{n}}\oplus\mathop{\rm
tor}(\mathbb{T}^{1})^{m}=\mathbb{Z}_{d_{1}}\oplus\dots\oplus\mathbb{Z}_{d_{n}}\oplus\left(\mathbb{Q}/\mathbb{Z}\right)^{m}\\\
&\cong\mathop{\rm
tor}\left(H_{1}^{w}(X)/H_{1}^{w}(\mathcal{F})\right)\oplus(\mathbb{Q}/\mathbb{Z})^{m}.\end{split}$

This verifies (4.25). ∎

###  4.7 Torsions over topologically free flows

In this section we continue our study of the cohomology group $\mathop{\rm{\bf
H}_{\mathcal{F}}}$, focusing now on the situation when
$\mathop{\rm\mathcal{F}\colon\Gamma\curvearrowright X}$ is a topologically
free minimal flow with $\Gamma\in\mathsf{LieGp}$ connected and with $X$ a
compact (connected) manifold. We start by determining the complementary
summand of $\mathop{\rm Div}(\mathop{\rm{\bf Z}_{\mathcal{F}}})$ in
$\mathop{\rm{\bf Z}_{\mathcal{F}}}$ and that of $\mathop{\rm
Div}(\mathop{\rm{\bf H}_{\mathcal{F}}})$ in $\mathop{\rm{\bf
H}_{\mathcal{F}}}$. Then we express $\mathop{\rm{\bf H}_{\mathcal{F}}}$ as a
direct sum of its torsion subgroup $\mathop{\rm tor}(\mathop{\rm{\bf
H}_{\mathcal{F}}})$ and a divisible subgroup $\mathfrak{D}$ of the groupoid
$\mathop{\rm{\bf MinH}_{\mathcal{F}}}$.

######  Theorem 4.46.

Let $\mathop{\rm\mathcal{F}\colon\Gamma\curvearrowright X}$ be a minimal
topologically free flow with $\Gamma\in\mathsf{LieGp}$ connected and with $X$
a compact manifold. Set $n=\mathop{\rm rank}(H_{1}^{w}(\mathcal{F}))$,
$n+m=\mathop{\rm rank}(H_{1}^{w}(X))$ and denote by $d_{1},\dots,d_{n}$ the
elementary divisors of $H_{1}^{w}(\mathcal{F})$ in $H_{1}^{w}(X)$. Then there
are direct sums

$\mathop{\rm{\bf Z}_{\mathcal{F}}}=\mathop{\rm Div}(\mathop{\rm{\bf
Z}_{\mathcal{F}}})\oplus H_{1}^{w}(\mathcal{F})\cong\mathop{\rm
Div}(\mathop{\rm{\bf Z}_{\mathcal{F}}})\oplus\mathbb{Z}^{n}$ (4.35)

and

$\mathop{\rm{\bf H}_{\mathcal{F}}}=\mathop{\rm Div}(\mathop{\rm{\bf
H}_{\mathcal{F}}})\oplus\mathop{\rm
tor}\Big{(}H_{1}^{w}(X)/H_{1}^{w}(\mathcal{F})\Big{)}\cong\mathop{\rm
Div}(\mathop{\rm{\bf
H}_{\mathcal{F}}})\oplus\mathbb{Z}_{d_{1}}\oplus\dots\oplus\mathbb{Z}_{d_{n}}.$
(4.36)

Moreover, for every $G\in\mathsf{CAGp}$, the exact sequence
$(\ref{Eq:exact.sequence})$ from Theorem 4.36 takes the form

$\begin{split}0&\longrightarrow\mathop{\rm{\bf
B}_{\mathcal{F}}}(G)\stackrel{{\scriptstyle\mu_{G}}}{{\longrightarrow}}\mathop{\rm{\bf
Z}_{\mathcal{F}}}(G)\stackrel{{\scriptstyle\kappa_{G}}}{{\longrightarrow}}\mathop{\rm
Hom}(G^{*},\mathop{\rm{\bf H}_{\mathcal{F}}})\longrightarrow\\\
&\longrightarrow\mathop{\rm
Ext}(G^{*},\mathbb{Z})^{n+m}\longrightarrow\mathop{\rm
Ext}(G^{*},\mathbb{Z})^{n}\longrightarrow\bigoplus_{i=1}^{n}\mathop{\rm
Ext}(G^{*},\mathbb{Z}_{d_{i}})\longrightarrow 0.\end{split}$ (4.37)

###### Proof.

Before turning to the proof we fix some notation. By definition of $n$, $m$
and $d_{1},\dots,d_{n}$, the inclusion $H_{1}^{w}(\mathcal{F})\subseteq
H_{1}^{w}(X)$ can be expressed in the form $d_{1}\mathbb{Z}\oplus\dots\oplus
d_{n}\mathbb{Z}\subseteq\mathbb{Z}^{n}\cong\mathbb{Z}^{n}\oplus
0\subseteq\mathbb{Z}^{n}\oplus\mathbb{Z}^{m}$. The standard basis of
$\mathbb{Z}^{n}\oplus\mathbb{Z}^{m}\cong\mathbb{Z}^{n+m}$ will be denoted by
$e_{1},\dots,e_{n+m}$ and we shall write $\eta=d_{1}\dots d_{n}$. We assume
throughout the whole proof that the space $X$ is pointed and we denote its
base point by $z$. Recall the notation
$\mathcal{F}_{z}=\mathcal{F}(-,z)\colon\Gamma\to X$ and
$\mathcal{C}_{z}=\mathcal{C}(-,z)\colon\Gamma\to\mathbb{T}^{1}$ for every
$\mathcal{C}\in\mathop{\rm{\bf Z}_{\mathcal{F}}}$. We shall use freely the
usual identifications $\pi_{1}(\mathbb{T}^{1})\cong
H_{1}^{w}(\mathbb{T}^{1})\cong\mathbb{Z}$. Finally, observe that by our
assumptions on $X$, for every $h\in\mathop{\rm Hom}(H_{1}^{w}(X),\mathbb{Z})$
there is $\xi\in C_{z}(X,\mathbb{T}^{1})$ such that $\xi^{\sharp}=h$.

_1st step._ We collect some useful facts about lifts of extensions from
$\mathop{\rm{\bf Z}_{\mathcal{F}}}$ and about the divisible subgroup
$\mathop{\rm Div}(\mathop{\rm{\bf Z}_{\mathcal{F}}})$ of $\mathop{\rm{\bf
Z}_{\mathcal{F}}}$.

Let $\mathcal{C}\in\mathop{\rm{\bf Z}_{\mathcal{F}}}$, $\xi\in
C_{z}(X,\mathbb{T}^{1})$ and $l\in\mathbb{N}$. Then the following statements
hold:

  1. (a)

we have

$\mathcal{C}^{\sharp}\pi_{1}(\Gamma\times
X)=\mathcal{C}_{z}^{\sharp}H_{1}^{w}(\Gamma)\subseteq\mathbb{Z};$

this follows from Lemma 2.17,

  2. (b)

we have

$\mathop{\rm co}(\xi)^{\sharp}\pi_{1}(\Gamma\times
X)=\xi^{\sharp}\mathcal{F}_{z}^{\sharp}H_{1}^{w}(\Gamma)=\xi^{\sharp}H_{1}^{w}(\mathcal{F})\subseteq\mathbb{Z};$

this follows from Lemma 2.17,

  3. (c)

$\mathcal{C}$ lifts across $\kappa_{l}\colon\mathbb{T}^{1}\to\mathbb{T}^{1}$
to an element of $\mathop{\rm{\bf Z}_{\mathcal{F}}}$ if and only if
$\mathcal{C}_{z}^{\sharp}H_{1}^{w}(\Gamma)\subseteq l\mathbb{Z}$ (in
particular, $\mathcal{C}\in\mathop{\rm Div}(\mathop{\rm{\bf
Z}_{\mathcal{F}}})$ if and only if
$\mathcal{C}_{z}^{\sharp}H_{1}^{w}(\Gamma)=0$); this follows from Lemma 3.36
and from (a),

  4. (d)

$\mathop{\rm co}(\xi)$ lifts across
$\kappa_{l}\colon\mathbb{T}^{1}\to\mathbb{T}^{1}$ to an element of
$\mathop{\rm{\bf Z}_{\mathcal{F}}}$ if and only if
$\xi^{\sharp}H_{1}^{w}(\mathcal{F})\subseteq l\mathbb{Z}$ (in particular,
$\mathop{\rm co}(\xi)\in\mathop{\rm Div}(\mathop{\rm{\bf Z}_{\mathcal{F}}})$
if and only if $\xi^{\sharp}H_{1}^{w}(\mathcal{F})=0$); this follows from
Lemma 3.36 and from (b).

_2nd step._ For $i=1,\dots,n$ fix $\xi_{i}\in C_{z}(X,\mathbb{T}^{1})$ so that
$(\xi_{i})^{\sharp}(e_{j})=\delta_{ij}\eta/d_{j}$ for $j=1,\dots,n$ and
$(\xi_{i})^{\sharp}(e_{j})=0$ for $j=n+1,\dots,n+m$, and let
$\mathcal{C}_{i}\in\mathop{\rm{\bf Z}_{\mathcal{F}}}$ be the lift of
$\mathop{\rm co}(\xi_{i})$ across
$\kappa_{\eta}\colon\mathbb{T}^{1}\to\mathbb{T}^{1}$. We claim that
$\mathcal{C}_{1},\dots,\mathcal{C}_{n}$ form a basis for a free abelian
subgroup $\mathcal{S}$ of $\mathop{\rm{\bf Z}_{\mathcal{F}}}$ and that there
is a direct sum $\mathop{\rm{\bf Z}_{\mathcal{F}}}=\mathop{\rm
Div}(\mathop{\rm{\bf Z}_{\mathcal{F}}})\oplus\mathcal{S}$. This will verify
(4.35).

First notice that by virtue of (d) from the first step of the proof, all
$\mathcal{C}_{i}$ are well defined. Now, in order to prove the two claims, it
suffices to verify the following statements.

  1. ($\alpha$)

For all $k_{1},\dots,k_{n}\in\mathbb{Z}$,
$\sum_{i=1}^{n}k_{i}\mathcal{C}_{i}\in\mathop{\rm Div}(\mathop{\rm{\bf
Z}_{\mathcal{F}}})$ implies $k_{1}=\dots=k_{n}=0$.

  2. ($\beta$)

The set $\mathop{\rm Div}(\mathop{\rm{\bf Z}_{\mathcal{F}}})\cup\mathcal{S}$
generates the group $\mathop{\rm{\bf Z}_{\mathcal{F}}}$.

We begin be verifying ($\alpha$). So fix $k_{1},\dots,k_{n}\in\mathbb{Z}$ with
$\sum_{i=1}^{n}k_{i}\mathcal{C}_{i}\in\mathop{\rm Div}(\mathop{\rm{\bf
Z}_{\mathcal{F}}})$. Then

$\mathop{\rm
co}\left(\sum_{i=1}^{n}k_{i}\xi_{i}\right)=\sum_{i=1}^{n}k_{i}\mathop{\rm
co}(\xi_{i})=\eta\left(\sum_{i=1}^{n}k_{i}\mathcal{C}_{i}\right)\in\mathop{\rm
Div}(\mathop{\rm{\bf Z}_{\mathcal{F}}})$

and hence, by virtue of (d),
$\sum_{i=1}^{n}k_{i}(\xi_{i})^{\sharp}=(\sum_{i=1}^{n}k_{i}\xi_{i})^{\sharp}$
vanishes on $H_{1}^{w}(\mathcal{F})$. By our choice of the maps $\xi_{i}$,
this leads to $k_{1}=\dots=k_{n}=0$.

To verify ($\beta$), fix $\mathcal{C}\in\mathop{\rm{\bf Z}_{\mathcal{F}}}$.
Since the flow $\mathcal{F}$ is topologically free by the assumptions of the
theorem, the morphism $\mathcal{F}_{z}^{\sharp}\colon H_{1}^{w}(\Gamma)\to
H_{1}^{w}(\mathcal{F})$ is an isomorphism. Thus, since
$\eta\mathcal{C}_{z}^{\sharp}=(\eta\mathcal{C})_{z}^{\sharp}\colon
H_{1}^{w}(\Gamma)\to\mathbb{Z}$ takes its values in $\eta\mathbb{Z}$, it
follows that so does
$(\eta\mathcal{C})_{z}^{\sharp}(\mathcal{F}_{z}^{\sharp})^{-1}\colon
H_{1}^{w}(\mathcal{F})\to\mathbb{Z}$. Consequently,
$(\eta\mathcal{C})_{z}^{\sharp}(\mathcal{F}_{z}^{\sharp})^{-1}$ extends to a
morphism $\varrho\colon H_{1}^{w}(X)\to\mathbb{Z}$ with
$\varrho(e_{j})\in(\eta/d_{j})\mathbb{Z}$ for $j=1,\dots,n$ and
$\varrho(e_{j})=0$ for $j=n+1,\dots,n+m$. By our choice of the maps $\xi_{i}$,
there are integers $k_{1},\dots,k_{n}$ with
$\varrho=\sum_{i=1}^{n}k_{i}(\xi_{i})^{\sharp}=(\sum_{i=1}^{n}k_{i}\xi_{i})^{\sharp}$.
Thus, with the help of Lemma 2.17, we obtain

$\eta\mathcal{C}_{z}^{\sharp}=(\eta\mathcal{C})_{z}^{\sharp}=\varrho\mathcal{F}_{z}^{\sharp}=\Big{(}\sum_{i=1}^{n}k_{i}\xi_{i}\Big{)}^{\sharp}\mathcal{F}_{z}^{\sharp}=\mathop{\rm
co}\Big{(}\sum_{i=1}^{n}k_{i}\xi_{i}\Big{)}_{z}^{\sharp}=\eta\Big{(}\sum_{i=1}^{n}k_{i}\mathcal{C}_{i}\Big{)}_{z}^{\sharp},$

whence it follows that

$\Big{(}\mathcal{C}-\sum_{i=1}^{n}k_{i}\mathcal{C}_{i}\Big{)}_{z}^{\sharp}=\mathcal{C}_{z}^{\sharp}-\Big{(}\sum_{i=1}^{n}k_{i}\mathcal{C}_{i}\Big{)}_{z}^{\sharp}=0.$

In view of (c) from the first step of the proof, this means that
$\mathcal{C}-\sum_{i=1}^{n}k_{i}\mathcal{C}_{i}\in\mathop{\rm
Div}(\mathop{\rm{\bf Z}_{\mathcal{F}}})$, which verifies ($\beta$).

_3rd step._ We show that $\mathcal{S}\cap\mathop{\rm{\bf
B}_{\mathcal{F}}}=\langle
d_{1}\mathcal{C}_{1},\dots,d_{n}\mathcal{C}_{n}\rangle$.

First we verify inclusion “$\subseteq$”. So fix
$k_{1},\dots,k_{n}\in\mathbb{Z}$ and assume that
$\sum_{i=1}^{n}k_{i}\mathcal{C}_{i}\in\mathop{\rm{\bf B}_{\mathcal{F}}}$; we
show that $d_{i}$ divides $k_{i}$ for every $i=1,\dots,n$. Then

$\eta\left(\sum_{i=1}^{n}k_{i}\mathcal{C}_{i}\right)=\sum_{i=1}^{n}k_{i}\mathop{\rm
co}(\xi_{i})=\mathop{\rm co}\left(\sum_{i=1}^{n}k_{i}\xi_{i}\right)$

and hence $\mathop{\rm co}(\sum_{i=1}^{n}k_{i}\xi_{i})$ lifts across
$\kappa_{\eta}\colon\mathbb{T}^{1}\to\mathbb{T}^{1}$ to an element of
$\mathop{\rm{\bf B}_{\mathcal{F}}}$. Consequently,
$\sum_{i=1}^{n}k_{i}\xi_{i}$ lifts across $\kappa_{\eta}$ to an element of
$C_{z}(X,\mathbb{T}^{1})$, whence it follows that
$(\sum_{i=1}^{n}k_{i}\xi_{i})^{\sharp}=\sum_{i=1}^{n}k_{i}(\xi_{i})^{\sharp}$
takes its values in $\eta\mathbb{Z}$. By our choice of the maps $\xi_{i}$ it
follows that $d_{i}$ divides $k_{i}$ for every $i=1,\dots,n$.

To verify inclusion “$\supseteq$”, it suffices to show that
$d_{i}\mathcal{C}_{i}\in\mathop{\rm{\bf B}_{\mathcal{F}}}$ for every
$i=1,\dots,n$. By our choice of the maps $\xi_{i}$,
$(d_{i}\xi_{i})^{\sharp}(H_{1}^{w}(X))=d_{i}(\xi_{i})^{\sharp}(H_{1}^{w}(X))\subseteq\eta\mathbb{Z}$
for every $i=1,\dots,n$. Consequently, $d_{i}\xi_{i}$ lifts across
$\kappa_{\eta}$ to an element of $C_{z}(X,\mathbb{T}^{1})$ and hence
$\mathop{\rm co}(d_{i}\xi_{i})$ lifts across $\kappa_{\eta}$ to an element of
$\mathop{\rm{\bf B}_{\mathcal{F}}}$. Since
$\kappa_{\eta}(d_{i}\mathcal{C}_{i})=d_{i}(\kappa_{\eta}\mathcal{C}_{i})=d_{i}\mathop{\rm
co}(\xi_{i})=\mathop{\rm co}(d_{i}\xi_{i})$, it follows by uniqueness of a
lift that, indeed, $d_{i}\mathcal{C}_{i}\in\mathop{\rm{\bf B}_{\mathcal{F}}}$.

_4th step._ Let $\mathcal{M}$ consist of the maps $\xi\in
C_{z}(X,\mathbb{T}^{1})$ with $\xi^{\sharp}H_{1}^{w}(\mathcal{F})=0$. Clearly,
$\mathcal{M}$ is a subgroup of $C_{z}(X,\mathbb{T}^{1})$ and hence
$\sigma(\mathcal{M})$ is a subgroup of $\mathop{\rm{\bf B}_{\mathcal{F}}}$,
where $\sigma$ is the usual isomorphism
$C_{z}(X,\mathbb{T}^{1})\to\mathop{\rm{\bf B}_{\mathcal{F}}}$. We show that
$\mathop{\rm{\bf
B}_{\mathcal{F}}}=\sigma(\mathcal{M})\oplus(\mathcal{S}\cap\mathop{\rm{\bf
B}_{\mathcal{F}}})=\sigma(\mathcal{M})\oplus\langle
d_{1}\mathcal{C}_{1},\dots,d_{n}\mathcal{C}_{n}\rangle$.

To verify the desired direct sum, we need to check the following statements:

  1. (i)

$\sigma(\mathcal{M})\cap(\mathcal{S}\cap\mathop{\rm{\bf B}_{\mathcal{F}}})=0$,

  2. (ii)

$\mathop{\rm{\bf
B}_{\mathcal{F}}}=\sigma(\mathcal{M})+(\mathcal{S}\cap\mathop{\rm{\bf
B}_{\mathcal{F}}})$.

We begin with statement (i). So fix $\xi\in\mathcal{M}$ and
$k_{1},\dots,k_{n}\in\mathbb{Z}$ with $\mathop{\rm
co}(\xi)=\sum_{i=1}^{n}k_{i}(d_{i}\mathcal{C}_{i})$; we show that
$k_{1}=\dots=k_{n}=0$ and $\xi=0$. From the last equality we obtain

$\mathop{\rm co}(\eta\xi)=\eta\mathop{\rm
co}(\xi)=\sum_{i=1}^{n}k_{i}d_{i}(\eta\mathcal{C}_{i})=\sum_{i=1}^{n}k_{i}d_{i}\mathop{\rm
co}(\xi_{i})=\mathop{\rm co}\left(\sum_{i=1}^{n}k_{i}d_{i}\xi_{i}\right),$

whence $\eta\xi=\sum_{i=1}^{n}k_{i}d_{i}\xi_{i}$ and so
$\eta\xi^{\sharp}=\sum_{i=1}^{n}k_{i}d_{i}(\xi_{i})^{\sharp}$. By applying the
last equality to the elements $e_{1},\dots,e_{n}$ and by using the definition
of the maps $\xi_{1},\dots\xi_{n}$, we get $k_{1}=\dots=k_{n}=0$. This yields
$\mathop{\rm co}(\xi)=0$ and hence also $\xi=0$.

To verify statement (ii), fix a map $\zeta\in C_{z}(X,\mathbb{T}^{1})$; we
show that $\mathop{\rm
co}(\zeta)\in\sigma(\mathcal{M})+(\mathcal{S}\cap\mathop{\rm{\bf
B}_{\mathcal{F}}})$. Set $k_{i}=\zeta^{\sharp}(e_{i})$ for $i=1,\dots,n$. By
our choice of the maps $\xi_{i}$, the morphism
$(\sum_{i=1}^{n}k_{i}d_{i}\xi_{i})^{\sharp}=\sum_{i=1}^{n}k_{i}d_{i}(\xi_{i})^{\sharp}$
takes its values in $\eta\mathbb{Z}$ and so the map
$\sum_{i=1}^{n}k_{i}d_{i}\xi_{i}$ lifts across
$\kappa_{\eta}\colon\mathbb{T}^{1}\to\mathbb{T}^{1}$ to an element $\vartheta$
of $C_{z}(X,\mathbb{T}^{1})$.

Set $\xi=\zeta-\vartheta$; we show that $\xi\in\mathcal{M}$. For every
$j=1,\dots,n$,

$\eta\xi^{\sharp}(e_{j})=\eta\zeta^{\sharp}(e_{j})-(\eta\vartheta)^{\sharp}(e_{j})=\eta\zeta^{\sharp}(e_{j})-\sum_{i=1}^{n}k_{i}d_{i}(\xi_{i})^{\sharp}(e_{j})=\eta
k_{j}-k_{j}d_{j}\frac{\eta}{d_{j}}=0$

and hence $\xi^{\sharp}(e_{j})=0$. Consequently,
$\xi^{\sharp}H_{1}^{w}(\mathcal{F})=0$ and, indeed, $\xi\in\mathcal{M}$.

Now, since $\mathop{\rm co}(\zeta)=\mathop{\rm co}(\xi)+\mathop{\rm
co}(\vartheta)$ and $\mathop{\rm co}(\xi)\in\sigma(\mathcal{M})$ by our
discussion above, it remains to show that $\mathop{\rm
co}(\vartheta)\in\mathcal{S}\cap\mathop{\rm{\bf B}_{\mathcal{F}}}$. By
definition of $\vartheta$,

$\eta\mathop{\rm co}(\vartheta)=\mathop{\rm co}(\eta\vartheta)=\mathop{\rm
co}\left(\sum_{i=1}^{n}k_{i}d_{i}\xi_{i}\right)=\sum_{i=1}^{n}k_{i}d_{i}\mathop{\rm
co}(\xi_{i})=\eta\left(\sum_{i=1}^{n}k_{i}(d_{i}\mathcal{C}_{i})\right),$

which yields $\mathop{\rm
co}(\vartheta)=\sum_{i=1}^{n}k_{i}(d_{i}\mathcal{C}_{i})\in\langle
d_{1}\mathcal{C}_{1},\dots,d_{n}\mathcal{C}_{n}\rangle=\mathcal{S}\cap\mathop{\rm{\bf
B}_{\mathcal{F}}}$.

_5th step._ We verify the isomorphism (4.36).

By virtue of (d) from the first step of the proof, $\sigma(\mathcal{M})$ is a
subgroup of $\mathop{\rm Div}(\mathop{\rm{\bf Z}_{\mathcal{F}}})$. Further, by
the second and the third step of the proof, there is an isomorphism
$\mathcal{S}/(\mathcal{S}\cap\mathop{\rm{\bf
B}_{\mathcal{F}}})\cong\mathbb{Z}_{d_{1}}\oplus\dots\oplus\mathbb{Z}_{d_{n}}$.
Consequently, there are isomorphisms of groups

$\begin{split}\mathop{\rm{\bf H}_{\mathcal{F}}}&=\mathop{\rm{\bf
Z}_{\mathcal{F}}}/\mathop{\rm{\bf B}_{\mathcal{F}}}=\frac{\mathop{\rm
Div}(\mathop{\rm{\bf
Z}_{\mathcal{F}}})\oplus\mathcal{S}}{\sigma(\mathcal{M})\oplus(\mathcal{S}\cap\mathop{\rm{\bf
B}_{\mathcal{F}}})}\cong\frac{\mathop{\rm Div}(\mathop{\rm{\bf
Z}_{\mathcal{F}}})}{\sigma(\mathcal{M})}\oplus\frac{\mathcal{S}}{\mathcal{S}\cap\mathop{\rm{\bf
B}_{\mathcal{F}}}}\\\ &\cong\frac{\mathop{\rm Div}(\mathop{\rm{\bf
Z}_{\mathcal{F}}})}{\sigma(\mathcal{M})}\oplus\left(\mathbb{Z}_{d_{1}}\oplus\dots\oplus\mathbb{Z}_{d_{n}}\right).\end{split}$

By divisibility of $\mathop{\rm Div}(\mathop{\rm{\bf Z}_{\mathcal{F}}})$, the
group $\mathop{\rm Div}(\mathop{\rm{\bf
Z}_{\mathcal{F}}})/\sigma(\mathcal{M})$ corresponds to the divisible subgroup
$\mathop{\rm Div}(\mathop{\rm{\bf H}_{\mathcal{F}}})$ of $\mathop{\rm{\bf
H}_{\mathcal{F}}}$ under the isomorphism above. This verifies (4.36).

_6th step._ We finish the proof of the theorem by verifying (4.37).

The existence of an exact sequence (4.37) follows from the exactness of (4.8)
with the help of the following three observations. First, since
$\pi^{1}(X)\cong H_{1}^{w}(X)\cong\mathbb{Z}^{n+m}$,

$\mathop{\rm Ext}(G^{*},\pi^{1}(X))\cong\mathop{\rm
Ext}(G^{*},H_{1}^{w}(X))\cong\mathop{\rm
Ext}(G^{*},\mathbb{Z}^{n+m})\cong\mathop{\rm Ext}(G^{*},\mathbb{Z})^{n+m}.$

Further, since $\mathop{\rm{\bf Z}_{\mathcal{F}}}\cong\mathop{\rm
Div}(\mathop{\rm{\bf Z}_{\mathcal{F}}})\oplus\mathbb{Z}^{n}$ by the second
step of the proof,

$\mathop{\rm Ext}(G^{*},\mathop{\rm{\bf Z}_{\mathcal{F}}})\cong\mathop{\rm
Ext}(G^{*},\mathop{\rm Div}(\mathop{\rm{\bf
Z}_{\mathcal{F}}}))\oplus\mathop{\rm
Ext}(G^{*},\mathbb{Z}^{n})\cong\mathop{\rm Ext}(G^{*},\mathbb{Z})^{n}.$

Finally, since $\mathop{\rm{\bf H}_{\mathcal{F}}}\cong\mathop{\rm
Div}(\mathop{\rm{\bf
H}_{\mathcal{F}}})\oplus(\mathbb{Z}_{d_{1}}\oplus\dots\oplus\mathbb{Z}_{d_{n}})$
by the fifth step of the proof,

$\mathop{\rm Ext}(G^{*},\mathop{\rm{\bf H}_{\mathcal{F}}})\cong\mathop{\rm
Ext}(G^{*},\mathop{\rm Div}(\mathop{\rm{\bf
H}_{\mathcal{F}}}))\oplus\mathop{\rm
Ext}\left(G^{*},\bigoplus_{i=1}^{n}\mathbb{Z}_{d_{i}}\right)\cong\bigoplus_{i=1}^{n}\mathop{\rm
Ext}(G^{*},\mathbb{Z}_{d_{i}}).$

This verifies the exactness of (4.37). ∎

######  Theorem 4.47.

Under the assumptions and notation from Theorem 4.46, there exists a divisible
subgroup $\mathfrak{D}$ of the groupoid $\mathop{\rm{\bf MinH}_{\mathcal{F}}}$
such that $\mathop{\rm{\bf H}_{\mathcal{F}}}=\mathfrak{D}\oplus\mathop{\rm
tor}(\mathop{\rm{\bf H}_{\mathcal{F}}})$. Moreover, there are isomorphisms of
groups

$\begin{split}\mathop{\rm tor}(\mathop{\rm{\bf
H}_{\mathcal{F}}})&\cong(\mathbb{Q}/\mathbb{Z})^{m}\oplus\mathop{\rm
tor}\left(H_{1}^{w}(X)/H_{1}^{w}(\mathcal{F})\right)\\\
&\cong(\mathbb{Q}/\mathbb{Z})^{m}\oplus\Big{(}\mathbb{Z}_{d_{1}}\oplus\dots\oplus\mathbb{Z}_{d_{n}}\Big{)}.\end{split}$
(4.38)

Finally, there are topological isomorphisms

$\begin{split}F(\mathop{\rm\daleth_{\mathcal{F}}})&\cong\mathfrak{D}^{*}\times\mathop{\rm
tor}\left(\mathop{\rm{\bf
H}_{\mathcal{F}}}\right)^{*}\cong\mathfrak{D}^{*}\times\left((\mathbb{Q}/\mathbb{Z})^{*}\right)^{m}\times\mathbb{Z}_{d_{1}}\times\dots\times\mathbb{Z}_{d_{n}},\end{split}$
(4.39)

where all $\mathfrak{D}$, $\mathop{\rm tor}(\mathop{\rm{\bf
H}_{\mathcal{F}}})$, $\mathbb{Q}/\mathbb{Z}$ and $\mathbb{Z}_{d_{i}}$
($i=1,\dots,n$) are assumed to carry the discrete topology.

######  Remark 4.48.

We wish to add the following observations.

  * •

By the conclusion of the theorem, $\mathfrak{D}$ is a divisible group.
Moreover, being a complementary summand to $\mathop{\rm tor}(\mathop{\rm{\bf
H}_{\mathcal{F}}})$ in $\mathop{\rm{\bf H}_{\mathcal{F}}}$, $\mathfrak{D}$ is
also torsion-free. Thus, algebraically, $\mathfrak{D}$ is a rational linear
space.

  * •

Since the group $\mathfrak{D}$ is divisible and torsion-free, its dual group
$\mathfrak{D}^{*}$ is torsion-free and connected. Further, since $\mathop{\rm
tor}(\mathop{\rm{\bf H}_{\mathcal{F}}})$ is a torsion group, its dual group
$\mathop{\rm tor}(\mathop{\rm{\bf H}_{\mathcal{F}}})^{*}$ is totally
disconnected. Consequently, under the first of the two isomorphisms in (4.39),
$\mathfrak{D}^{*}$ corresponds to $F(\mathop{\rm\daleth_{\mathcal{F}}})_{0}$.
In particular, $F(\mathop{\rm\daleth_{\mathcal{F}}})_{0}$ is a topological
direct summand in $F(\mathop{\rm\daleth_{\mathcal{F}}})$ and
$F(\mathop{\rm\daleth_{\mathcal{F}}})/F(\mathop{\rm\daleth_{\mathcal{F}}})_{0}$
is topologically isomorphic to $\mathop{\rm tor}(\mathop{\rm{\bf
H}_{\mathcal{F}}})^{*}$.

###### Proof of Theorem 4.47.

Since $\mathop{\rm Div}(\mathop{\rm{\bf H}_{\mathcal{F}}})$ is divisible by
definition, its torsion subgroup $\mathop{\rm tor}(\mathop{\rm
Div}(\mathop{\rm{\bf H}_{\mathcal{F}}}))$ is a direct summand in $\mathop{\rm
Div}(\mathop{\rm{\bf H}_{\mathcal{F}}})$ with a divisible torsion-free
complementary summand $\mathfrak{D}$. Moreover,

$\begin{split}\mathfrak{D}\setminus 1&\subseteq\mathop{\rm
Div}(\mathop{\rm{\bf H}_{\mathcal{F}}})\setminus\mathop{\rm tor}(\mathop{\rm
Div}(\mathop{\rm{\bf H}_{\mathcal{F}}}))=\mathop{\rm Div}(\mathop{\rm{\bf
H}_{\mathcal{F}}})\setminus\left(\mathop{\rm tor}(\mathop{\rm{\bf
H}_{\mathcal{F}}})\cap\mathop{\rm Div}(\mathop{\rm{\bf
H}_{\mathcal{F}}})\right)\\\ &=\mathop{\rm Div}(\mathop{\rm{\bf
H}_{\mathcal{F}}})\setminus\mathop{\rm tor}(\mathop{\rm{\bf
H}_{\mathcal{F}}})\subseteq\mathop{\rm{\bf
H}_{\mathcal{F}}}\setminus\mathop{\rm tor}(\mathop{\rm{\bf
H}_{\mathcal{F}}})=\mathop{\rm{\bf MinH}_{\mathcal{F}}}\setminus 1\end{split}$

and hence $\mathfrak{D}$ is a subgroup of the groupoid $\mathop{\rm{\bf
MinH}_{\mathcal{F}}}$.

Now, by virtue of (4.37) from Theorem 4.46, $\mathop{\rm{\bf
H}_{\mathcal{F}}}$ contains a subgroup $\mathfrak{A}$ isomorphic to
$\mathop{\rm
tor}(H_{1}^{w}(X)/H_{1}^{w}(\mathcal{F}))\cong\mathbb{Z}_{d_{1}}\oplus\dots\oplus\mathbb{Z}_{d_{n}}$
such that $\mathop{\rm{\bf H}_{\mathcal{F}}}=\mathop{\rm Div}(\mathop{\rm{\bf
H}_{\mathcal{F}}})\oplus\mathfrak{A}$. Consequently,

$\mathop{\rm{\bf H}_{\mathcal{F}}}=\mathop{\rm Div}(\mathop{\rm{\bf
H}_{\mathcal{F}}})\oplus\mathfrak{A}=\mathfrak{D}\oplus\mathop{\rm
tor}(\mathop{\rm Div}(\mathop{\rm{\bf H}_{\mathcal{F}}}))\oplus\mathfrak{A}.$

Thus, $\mathop{\rm tor}(\mathop{\rm{\bf H}_{\mathcal{F}}})=\mathop{\rm
tor}(\mathop{\rm Div}(\mathop{\rm{\bf H}_{\mathcal{F}}}))\oplus\mathfrak{A}$
is a direct summand in $\mathop{\rm{\bf H}_{\mathcal{F}}}$ and $\mathfrak{D}$
is its complementary direct summand. Finally, by virtue of (4.25) from Theorem
4.44,

$\mathop{\rm tor}(\mathop{\rm Div}(\mathop{\rm{\bf
H}_{\mathcal{F}}}))=\mathop{\rm Div}(\mathop{\rm tor}(\mathop{\rm{\bf
H}_{\mathcal{F}}}))\cong\mathop{\rm
Div}\left((\mathbb{Q}/\mathbb{Z})^{m}\oplus\mathbb{Z}_{d_{1}}\oplus\dots\oplus\mathbb{Z}_{d_{n}}\right)\cong(\mathbb{Q}/\mathbb{Z})^{m},$

which finishes the proof of (4.38)

The two isomorphisms from (4.39) follow immediately from the topological
isomorphism $F(\mathop{\rm\daleth_{\mathcal{F}}})\cong(\mathop{\rm{\bf
H}_{\mathcal{F}}})_{d}^{*}$ obtained in Remark 4.8 and from the first part of
Theorem 4.47. ∎

##  Chapter 5 Algebraic-Topological Aspects

###  5.1 Lifts of transfer functions

Let $\mathop{\rm\mathcal{F}\colon\Gamma\curvearrowright X}$ be a minimal flow.
For an extension $\mathcal{C}\in\mathop{\rm\mathsf{CAGpZ}_{\mathcal{F}}}$ of
$\mathcal{F}$ with a totally disconnected section $F(\mathcal{C})$, we
investigate the relation between $F(\mathcal{C})$ on one side and the
existence of lifts of transfer functions for certain quotients of
$\mathcal{C}$ on the other side. This relation is described in Proposition
5.3.

In order to formulate the next result we fix some notation. Given
$G\in\mathsf{CAGp}$ and $E\mathrel{\lhd}K\mathrel{\lhd}G$, we denote by
$p_{K}$ and $p_{E}$ the canonical quotient morphisms $G\to G/K$ and $G\to
G/E$, respectively. Also, $q_{E}$ stands for the (unique) morphism $G/E\to
G/K$ with $q_{E}p_{E}=p_{K}$. Notice that $q_{E}$ is a quotient morphism with
kernel $\ker(q_{E})=p_{E}(K)\cong K/E$ and it is thus equivalent to the
canonical quotient morphism $G/E\to(G/E)/(K/E)$.

######  Lemma 5.1.

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 conditions are
equivalent for every $E\mathrel{\lhd}K$:

  1. (a)

$p_{E}\mathcal{C}\in\mathop{\rm{\bf B}_{\mathcal{F}}}(G/E)$ (that is,
$F(\mathcal{C})\subseteq E$),

  2. (b)

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

The situation is depicted in Figure 5.17.

######  Remark 5.2.

Notice the following facts.

  * •

The assumption on $\Gamma$ from Lemma 5.1 is satisfied for instance if
$\Gamma$ is connected, which is the situation of our main interest. Of course,
there are also disconnected groups without non-trivial finite abelian quotient
groups, say, the discrete group of rationals $\mathbb{Q}_{d}$. More generally,
a locally compact abelian group $\Gamma$ has no non-trivial finite (abelian)
quotient groups if and only if its dual group $\Gamma^{*}$ is torsion-free. In
case when $\Gamma$ is discrete abelian the latter condition is equivalent with
the divisibility of $\Gamma$.

  * •

In connection with condition (b) observe that $\xi$ lifts across $q_{E}$ to a
continuous map $X\to G/E$ if and only if it lifts to a continuous base point
preserving map $X\to G/E$. For if $\eta\colon X\to G/E$ is continuous and
satisfies $q_{E}\eta=\xi$ then $\eta^{\prime}:=\eta\eta(z)^{-1}\colon X\to
G/E$ is continuous, preserves the base points and, since $\eta(z)\in
q_{E}^{-1}(\xi(z))=\ker(q_{E})$, $\eta^{\prime}$ is also a lift of $\xi$
across $q_{E}$.

$\ctdiagram{\ctinnermid\ctv 0,0:{G/E}\ctv-50,-50:{G/K}\ctv-50,50:{G}\ctv
92,0:{\Gamma\times
X}\ctv-58,0:{p_{K}}\ctv-30,20:{p_{E}}\ctv-30,-17:{q_{E}}\ctv
33,-8:{p_{E}\mathcal{C}}\ctv 32,-35:{p_{K}\mathcal{C}=\mathop{\rm
co}(\xi)}\ctv
22,33:{\mathcal{C}}\put(75.0,0.0){\vector(-1,0){63.0}}\put(75.0,3.0){\vector(-1,0){119.0}}\put(75.0,-3.0){\vector(-1,0){112.0}}\put(-50.0,44.0){\vector(0,-1){87.0}}\put(-45.0,47.0){\vector(1,-1){35.0}}\put(-10.0,-5.0){\vector(-1,-1){32.0}}\ctnohead}$
Figure 5.17: Lifts of transfer functions of coboundaries.

###### Proof of Lemma 5.1.

Fix a closed subgroup $E$ of $K$. As has been observed at the beginning of