arXiv:1804.01029v2 [math.GN] 4 Apr 2018

Continuous Cocycles Endowed with
Point-Open Topology∗
Kayvan Nejabati Zenouz†
5th Apr, 2018
Abstract
Given a topological group G and a Hausdorff topological group A
on which G acts continuously and compatibly with the group operation
of A, we study the set of continuous cocycles of G with value in A. This
set is a function space and can be endowed with several topologies. By
imposing a suitable function space topology on the set of cocycles of
G with value in A, we propose a topological study of this set, and we
prove, as our first main result, that if A is a compact group having a
presentation as an inverse limit of compact and Hausdorff topological
groups Ar , for r in a directed poset R, on which G acts continuously
and compatibly with the group operation of Ar and equivariantly with
respect to the transition maps, then one has a natural identification
between the first nonabelian cohomology set of G with coefficients in
the inverse limit A and the inverse limit of first nonabelian cohomology
sets of G with coefficients in Ar . Furthermore, we prove, as our second
main result, that if G is compact and Hausdorff, and A is abelian therefore one can define cohomology groups for all n ≥ 1 - then under
a certain condition on Ar one has a natural identification between
cohomology group with coefficients in the inverse limit A and the
inverse limit of cohomology groups with coefficients in Ar , for all n ≥
1.
∗

The author obtained the results in this paper while studying for a PhD degree at the
University of Exeter funded by an EPSRC Doctoral Training Grant. This research was
partially supported by the ERC Advanced grant 320974.
†
School of Mathematics, University of Edinburgh, 5605, JCMB, Edinburgh,
EH9 3FD, Email: knejabat@ed.ac.uk, Website: http://www.maths.ed.ac.uk/school-ofmathematics/people?person=644

CONTENTS

2

Contents
1 Introduction

2

2 Continuous Cocycles and Torsors
2.1 Function Spaces . . . . . . . . . . . . . . . . . . .
2.2 G-Groups . . . . . . . . . . . . . . . . . . . . . .
2.3 Continuous Cocycles and Torsors . . . . . . . . .
2.4 Cohomology Map . . . . . . . . . . . . . . . . . .
2.5 First Nonabelian Continuous Cohomology Pointed

. . . . .
. . . . .
. . . . .
. . . . .
Spaces .

.
.
.
.
.

.
.
.
.
.

4
4
10
12
17
22

3 The First Main Theorem
25
3.1 Applications of The First Main Theorem . . . . . . . . . . . . 29
4 G-Modules and The Second Main Theorem
31
4.1 One Application of The Second Main Theorem . . . . . . . . . 42

1

Introduction

It is a theorem cf. [NSW08, p. 141, (2.7.5) Theorem] that if G is a profinite
group, and A is a compact G-module having a presentation A = limn∈N An
←−
as a countable inverse limit of finite discrete G-modules An , then there exist
a natural exact sequence
1

lim1n∈N H i−1 (G, An )
←−

i
(G, A)
Hcts

limn∈N H i (G, An )
←−

1;

however, in the investigations presented in [NSW08, p. 136, §7] it has not
been taken into consideration the fact that there are several natural ways
i
to make Hcts
(G, A) into a topological space, and to study some of the topoi
logical properties of Hcts
(G, A); hence the fundamental aim of this paper is
to investigate generalisations of the above theorem by introducing a suitable
i
topology on the groups Hcts
(G, A), and studying their topological properties.
However, we first begin by studying cohomology with coefficients in nonabelian groups, and then move onto cohomology with coefficients in abelian
groups, and eventually, as a corollary of our results, we give a relationship
between Gi−1 and An for all n under which one has lim1n∈N H i−1 (G, An ) = 1
←−
on the exact sequence above.

Kayvan Nejabati Zenouz
Knejabat@ed.ac.uk

Compiled on 5th Apr, 2018 at 00:27

1 Introduction

3

We start with a continuous action of a topological group G on a Hausdorff
topological group A compatible with the group operation of A (in such case
we shall refer to A as a G-group), and we study the properties of the topo1
(G, A), the set of continuous 1-cocycles of G in A, endowed
logical space Zcts
with the point-open topology (aka the topology of pointwise convergence);
1
(G, A), of cohomology classes of continwe also study the quotient space Hcts
uous cocycles of G in A, endowed with the quotient topology. In particular,
1
investigating the properties of the quotient space Hcts
(G, A) we prove, as our
first main result the following.
Theorem 1. Let A be a compact G-group. Assume A has a presentation
A = limr∈R Ar . Denote by ϕr : A −→ Ar the natural projections. Then there
←−
exists a natural continuous bijection
1
1
Θ : Hcts
(G, A)po −→ lim Hcts
(G, Ar )po given by [a] 7−→ ([ϕr∗ (a)]).
←−
r∈R

In particular, if G is a compact (even locally compact) and Hausdorff space,
and A is evenly continuous with respect to G, then Θ is a homeomorphism.
That is if A = limr∈R Ar is given as an inverse limit of compact topo←−
logical G-groups Ar , for r in a directed poset R, then there exists a natural
continuous bijection of pointed sets
1
1
Θ = Θ1 : Hcts
(G, A)po −→ lim Hcts
(G, Ar )po ,
←−
r∈R

which is a homeomorphism, when G a is compact (even locally compact) and
Hausdorff group and the set of continuous maps from G −→ Ar is an evenly
continuous set for each r ∈ R.
Next we apply our methodology to obtain some results in abelian cohomology. In the case where A is a compact abelian G-group, i.e., A is a
compact G-module, for a fixed n ≥ 1, we study the group of n-cocycles of G
in A endowed with the point-open topology, and we prove if A = limr∈R Ar is
←−
given as above, G is compact and Hausdorff, and when the set of continuous
function from Gn−1 −→ Ar is evenly continuous for all r ∈ R, then that there
exists a continuous group isomorphism
n
n
Θn : Hcts
(G, A)po −→ lim Hcts
(G, Ar )po ,
←−
r∈R

Kayvan Nejabati Zenouz
Knejabat@ed.ac.uk

Compiled on 5th Apr, 2018 at 00:27

2 Continuous Cocycles and Torsors

4

which is a homeomorphism, when the set of continuous function from Gn −→
Ar is evenly continuous for all r ∈ R. More precesely we prove as our second
main theorem the following.
Theorem 2. Let A a compact G-module and fix n ≥ 1. Assume A has a
presentation A = limr∈R Ar , where Ar is evenly continuous with respect to
←−
Gn−1 for all r ∈ R. Denote by ϕr : A −→ Ar the natural projections. Then
there exists a natural continuous bijection
n
n
(G, Ar )po given by [a] 7−→ ([ϕr∗ (a)]).
(G, A)po −→ lim Hcts
Θn : Hcts
←−
r∈R

In particular, if A is evenly continuous with respect to Gn , then Θn is a
homeomoprhism.
This document is organised as follows. In Section 2 we provide background information relating to function spaces, and we develop a general theory for studying the topological space of continuous 1-cocycles and the first
nonabelian continuous cohomology space endowed with point-open topology.
We prove several topological properties of these spaces; we also briefly discuss
the, well-known, one-to-one correspondence between the set of continuous 1cocycles and the set of continuous torsors. In Section 3 we focus on torsors
under a compact and Hausdorff topological group endowed with the pointopen topology and provide the proof for our first main theorem, we also we
discuss two immediate applications of our first main theorem. Finally, in
Section 4 we investigate the abelian situation, we prove our second main
theorem, and provide an application of this theorem.

2
2.1

Continuous Cocycles and Torsors
Function Spaces

Let A and B be topological spaces. Denote by M (B,
Q A) the set of all maps
from B −→ A. As a set we have M (B, A) = b∈B Ab i.e., M (B, A) is
the product of copies of A one for each b ∈ B. It is known that the set
M (B, A) can be endowed with several topologies. For example M (B, A) can
be endowed with the compact-open topology, the point-open topology, and
if A is a uniform space; e.g., a metric space or a topological group, then the

Kayvan Nejabati Zenouz
Knejabat@ed.ac.uk

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2.1 Function Spaces

5

set M (B, A) can also be endowed the topology of uniform convergence, or
the topology of compact convergence.
A subbase for compact-open topology on M (B, A) is given by subsets of
def
the form β(K, V ) = {f ∈ M (B, A) : f (K) ⊆ V }, where K ⊆ B is a compact
subspace and V ⊆ A is an open subset. Therefore, an element of the base
for the compact-open topology on M (B, A) is given by a finite intersection
of elements of the subbase cf. [Kel75, p. 221]. We shall write M (B, A)co for
the set M (B, A) endowed with compact-open topology.
A coarser/smaller topology than compact-open on M (B, A) is the pointopen topology. The point-open topology on M (B, A) is obtained by taking
as subbase subsets of the form β({b}, V ), for a point b ∈ B and an open set
V ⊆ A. We shall denote the set M (B, A) endowed with point-open topology
by M (B, A)po . In particular, if we let
def

εb : M (B, A)po −→ A be the map given by f 7−→ fb = f (b),
which is the natural
projections,
Q then we have a natural homeomorphism
Q
∼
M (B, A)po = b∈B Ab , where b∈B Ab is endowed with product (Tychonoff)
topology cf. [Kel75, p. 217, Chapter 7]. As such the map εb is continuous
for all b ∈ B since for any open subset V ⊂ A we have ε−1
b (V ) = β({b}, V ),
1
which is an open subset of M (B, A)po .
Since the point-open topology on M (B, A) is coarser than the compactopen topology, the map id : M (B, A)co −→ M (B, A)po is continuous, and for
all b ∈ B, we have a commutative diagram of continuous maps
M (B, A)co

id

M (B, A)po
εb

εb id

A.

We denote by Mcts (B, A) the subset of M (B, A) containing all the continuous maps from B −→ A. For any subset F = F (B, A) ⊆ M (B, A) we
let
def
Fcts = Fcts (B, A) = F ∩ Mcts (B, A).
We further denote by Mcts (B, A)po the set Mcts (B, A) together with the subspace topology inherited from M (B, A)po ; similarly, we denote by Mcts (B, A)co
the set Mcts (B, A) together with the subspace topology inherited from M (B, A)co .
1

When A is a uniform space see [Kel75, p. 226] for the definition of the topology of
uniform convergence on M (B, A), and see [Kel75, p. 229] for the definition of the topology
of compact convergence on M (B, A).
Kayvan Nejabati Zenouz
Knejabat@ed.ac.uk

Compiled on 5th Apr, 2018 at 00:27

2.1 Function Spaces

6

For the rest of the document in places we shall use the word ”space” by
which we shall always mean a ”topological space” when the topology under
consideration is understood.
We note that the point-open topology on M (B, A) is completely determined by the topology of A, so by working with this topology we are often
ignoring the information offered by the topology of B i.e., we are assuming B
is a set; therefore if one would like to take into consideration the information
offered by the topology of B, then the compact-open topology provides this
opportunity. It also happens that these two topologies coincide for many
cases; for example, if A is a Hausdorff space, then one finds that M (B, A)po
is a Hausdorff space; now in a given situation when M (B, A)co is a compact
space, then it follows that the map id is a homeomorphism, and therefore
one has M (B, A)co ∼
= M (B, A)po .
We mainly focus on working with the space M (B, A)po ; however, most
of our investigations can also be carried out when replacing the point-open
topology with the compact-open topology. We first recall a few basic facts
relating to point-open topology.
Lemma 2.1. Let B be a topological space. Assume ϕ : A −→ A0 is a
continuous map between topological spaces. Then the map
def

ϕ∗ : M (B, A)po −→ M (B, A0 )po given by f 7−→ ϕf = ϕ ◦ f is continuous.
In particular, the map ϕ∗ restricts to a continuous map (also denoted by ϕ∗ )
ϕ∗ : Mcts (B, A)po −→ Mcts (B, A0 )po .
Proof. To show the continuity of ϕ∗ , it suffices to check that the inverse
image of an element of the subbase for the point-open topology on M (B, A0 )
is an open subset of M (B, A)po . Fix a point b ∈ B and an open subset
V ⊆ A0 . Note since ϕ is continuous ϕ−1 (V ) ⊆ A is an open subset. Let
β({b}, V ) ⊆ M (B, A0 )po be an element of the subbase for the point-open
topology on M (B, A0 ). Then we have
ϕ−1
∗ (β({b}, V )) = {f ∈ M (B, A) : ϕ(f (b)) ∈ V };
it follows that
−1
ϕ−1
∗ (β({b}, V )) = β({b}, ϕ (V ))

Kayvan Nejabati Zenouz
Knejabat@ed.ac.uk

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2.1 Function Spaces

7

−1
since if f ∈ ϕ−1
∗ (β({b}, V )), then ϕ(f (b)) ∈ V so f (b) ∈ ϕ (V ) i.e., f ∈
−1
−1
β({b}, ϕ (V )). Conversely, if g ∈ β({b}, ϕ (V )), then g(b) ∈ ϕ−1 (V ), and
so
ϕ(g(b)) = ϕg(b) ∈ ϕ(ϕ−1 (V )) ⊆ V

i.e., ϕg = ϕ∗ (g) ∈ β({b}, V ) so g ∈ ϕ−1
∗ (β({b}, V )).
Now since β({b}, ϕ−1 (V )) ⊆ M (B, A)po is an element of the subbase for
the topology, which follows from continuity of ϕ, we conclude that ϕ∗ is a
continuous map. In particular, since the composition of two continuous maps
is a continuous map; if f ∈ Mcts (B, A)po , then ϕf ∈ Mcts (B, A0 )po ; now the
second statement follows since restriction of a continuous map to a subspace
is a continuous map cf. [Kel75, p. 85], and this proves the lemma.
Lemma 2.2. Let A be a topological space. Assume ψ : B −→ B 0 is a
continuous map between topological spaces. Then the map
def

ψ ∗ : M (B 0 , A)po −→ M (B, A)po given by f 7−→ f ψ = f ◦ ψ is continuous.
In particular, the map ψ ∗ restricts to a continuous map (also denoted by ψ ∗ )
ψ ∗ : Mcts (B 0 , A)po −→ Mcts (B, A)po .
Proof. Similarly to Lemma 2.1, we check the continuity of ψ ∗ on the elements
of the subbase for the topology of M (B, A)po . Fix a point b ∈ B and an open
subset V ⊆ A. Let β({b}, V ) ⊆ M (B, A)po be an element of the subbase for
the point-open topology on M (B, A). Then we have
ψ∗

−1

(β({b}, V )) = {f ∈ M (B 0 , A) : f (ψ(b)) ∈ V };

it follows that
ψ∗

−1

(β({b}, V )) = β({ψ(b)}, V )

since if f ∈ ψ ∗ −1 (β({b}, V )), then f ψ(b) ∈ V so f ∈ β({ψ(b)}, V ). Conversely, if g ∈ β({ψ(b)}, V ), then gψ(b) ∈ V , and so gψ = ψ ∗ (g) ∈ β({b}, V );
hence, g ∈ ψ ∗ −1 (β({b}, V )).
Now since β({ψ(b)}, V ) ⊆ M (B 0 , A)po is an element of the subbase for the
topology, we conclude that ψ ∗ is a continuous map. In particular, since the
composition two continuous maps is a continuous map, if f ∈ Mcts (B 0 , A)po ,
then f ψ ∈ Mcts (B, A)po . Now the second statement follows since restriction
of a continuous map to a subspace is a continuous map cf. [Kel75, p. 85],
and this proves the lemma.
Kayvan Nejabati Zenouz
Knejabat@ed.ac.uk

Compiled on 5th Apr, 2018 at 00:27

2.1 Function Spaces

8

Lemma 2.3. Suppose A and B are topological spaces.
i) If A is a Hausdorff space, then the spaces M (B, A)po and Mcts (B, A)po
are Hausdorff.
ii) If A is a compact space, then M (B, A)po is a compact topological space.
Q
Proof. Note we have a natural identification M (B, A)po ∼
= b∈B Ab , and
so M (B, A)po is a product of Hausdorff spaces; therefore it is a Hausdorff
space cf. [Wil04, p. 87, 13.8 Theorem. b)]. Now since Mcts (B, A)po ⊆
M (B, A)po is a subspace of a Hausdorff space, we conclude that Mcts (B, A)po
is a Hausdorff space cf. [Wil04, p. 87, 13.8 Theorem.Qa)], and this proves i).
If A is a compact space, then since M (B, A)po ∼
= b∈B Ab is a product of
compact spaces, we conclude that M (B, A)po is a compact space cf. [Wil04,
p. 120, 17.8 Theorem (Tychonoff)], and this proves ii).
We shall fix a directed poset R and assume {Ar , ϕrs , R} is an inverse
system of topological spaces see [RZ10, p. 1, for definition] for the rest of the
article
Lemma 2.4. Let A and B be topological spaces. Assume A has a presentation A = limr∈R Ar . Denote by ϕr : A −→ Ar the natural projection. Then
←−
there exists a continuous bijection
θ : Mcts (B, A)po −→ lim Mcts (B, Ar )po given by f 7−→ (ϕr∗ (f )).
←−
r∈R

In particular, if A is a compact space and Ar is Hausdorff for each r ∈ R,
then θ is a homeomorphism.
Proof. By Lemma 2.1 the continuous transition maps ϕrt : Ar −→ At for
r, t ∈ R and r ≥ t induce a commutative diagram of continuous maps
Mcts (B, Ar )po
M (B, Ar )po

ϕrt∗

ϕrt∗

Mcts (B, At )po
M (B, At )po ;

therefore we see that the sets {Mcts (B, Ar ), ϕrt∗ , R} and {M (B, Ar ), ϕrt∗ , R}
are inverse system of topological spaces; hence the inverse limits exists and is
unique up to isomorphism cf. [RZ10, p. 2, Proposition 1.1.1]. In particular,
Kayvan Nejabati Zenouz
Knejabat@ed.ac.uk

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2.1 Function Spaces

9

by the universal property of the inverse limit, we obtain a unique continuous
commutative diagram
Mcts (B, A)po

θ

limr∈R Mcts (B, Ar )po
←−

M (B, A)po

θ

limr∈R M (B, Ar )po ,
←−

it is clear that θ on the bottom horizontal arrow is a continuous bijection;
also one has θ(Mcts (B, A)po ) = limr∈R Mcts (B, Ar )po since given an element
←−
(fr ) ∈ limr∈R Mcts (B, Ar )po , then by the universal property of the inverse
←−
limit limr∈R Ar there exists a unique continuous map f : B −→ A such that
←−
ϕr∗ (f ) = fr for all r ∈ R cf. [RZ10, p. 2].
In particular, if A is a compact space, then by Lemma 2.3, ii), M (B, A)po
is a compact space; further if Ar is Hausdorff for each r ∈ R, then by Lemma
2.3, i), the space Mcts (B, Ar ) is Hausdorff for each r ∈ R; therefore
Y
M (B, Ar )po
lim M (B, Ar )po ⊆
←−
r∈R

r∈R

is a Hausdorff topological space cf. [Wil04, p. 87, 13.8 Theorem. a) &
b)]; thus in such case θ is a continuous bijection from a compact space to a
Hausdorff space, which implies that θ is a homeomorphism cf. [Wil04, p. 123,
17.14 Theorem], this proves the lemma.
Definition 2.5 (Evenly Continuous Sets). We say a subset F ⊆ M (B, A) is
evenly continuous if for each b ∈ B, each a ∈ A, and each neighbourhood U
of a there exists a neighbourhood V of b and a neighbourhood W of a such
that for all f ∈ F we have f (V ) ⊆ U whenever f (b) ∈ W . cf. [Kel75, p. 235].
We say A is evenly continuous with respect to a topological space B if the set
Mcts (B, A) is evenly continuous.
Two obvious examples of evenly continuous sets are as follows. For two
topological spaces A, and B, denote by M (B 0 , A) ⊆ M (B, A) the of all
constant maps from B −→ A. Then M (B 0 , A) is an evenly continuous
set since let b ∈ B, a ∈ A, and U be a neighbourhood of a. Then B
is a neighbourhood of b and U is a neighbourhood of a such that for all
f ∈ M 0 (B, A) we have f (B) = af ∈ U whenever f (b) = af ∈ U , and
this show that M (B 0 , A) is evenly continuous; also if A and B are discrete
Kayvan Nejabati Zenouz
Knejabat@ed.ac.uk

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2.2 G-Groups

10

spaces, then A is evenly continuous with respect to B since let b ∈ B, a ∈ A
and U a neighbourhood of a, then {b} is a neighbourhood of b and U is
a neighbourhood of a such that f ({b}) ⊆ U whenever f (b) ∈ U for all
f ∈ Mcts (B, A).
Lemma 2.6. Suppose A and B are compact and Hausdorff spaces. If A is
an evenly continuous space with respect to B, then Mcts (B, A)po is a compact
space. In particular, if A has a presentation A = limr∈R Ar , where Ar is a
←−
compact, Hausdorff, which is an evenly continuous space with respect to B
for each r ∈ R, then Mcts (B, A)po is a compact space.
Proof. If A and B are compact and Hausdorff spaces, then A and B are
T4 -spaces cf. [Wil04, p. 121, 17.1 Theorem]; therefore A and B are regular
spaces; in particular A and B are also locally compact cf. [Wil04, p. 130, 18.2
Theorem]. Now if A is evenly continuous with respect to B, then Mcts (B, A)co
is a compact space cf. [Kel75, p. 236, 21 ASCOLI THEOREM]. It also follows
from Lemma 2.3, i), that Mcts (B, A)po is a Hausdorff space. Now the map
Id : Mcts (B, A)co −→ Mcts (B, A)po is a continuous bijection from a compact
space to a Hausdorff space, and therefore it is a homeomorphism cf. [Wil04,
p. 123, 17.14 Theorem], which implies that Mcts (B, A)po is a compact space.
In particular, if A has a presentation A = limr∈R Ar , where Ar is a com←−
pact, Hausdorff and evenly continuous space with respect to B for each
r ∈ R, then by above Mcts (B, Ar )po is a compact space and by Lemma
2.3, i), Mcts (B, Ar )po is a Hausdorff space; further by Lemma 2.4 we have a
homeomorphism
θ : Mcts (B, A)po −→ lim Mcts (B, Ar )po given by f 7−→ (ϕr,∗ (f )).
←−
r∈R

Q
Now since limr∈R Mcts (B, Ar )po is a closed subset of r∈R Mcts (B, Ar )po cf.
←−
Q
[RZ10, p. 3, Lemma 1.1.2], and r∈R Mcts (B, Ar )po is a compact space cf.
[Wil04, p. 120, 17.8 Theorem (Tychonoff)], we have that limr∈R Mcts (B, Ar )po
←−
is a compact space cf. [Wil04, p. 120, 17.5 Theorem. a)]; therefore we
conclude that Mcts (B, A)po is a compact space, which proves the lemma.

2.2

G-Groups

We shall be concerned with a fixed topological group G for the rest of the
document. We denote the continuous inversion and multiplication of a topological group H by ιH : H −→ H and mH : H × H −→ H respectively.
Kayvan Nejabati Zenouz
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2.2 G-Groups

11

Definition 2.7 ((Continuous) G-Group, G-Space, and G-Map). 2 A (topological) G-group is a Hausdorff topological group A on which G acts (on the
left) continuously and compatibly with the group operation of A. In other
words, A is a G-group if A is a Hausdorff topological group together with a
continuous map ac : G × A −→ A (where G × A is endowed with product
def
topology), and we shall write s x = ac(s, x), such that the following holds:
i) ac(1, x) = x and ac(mG (s, t), x) = ac(s, ac(t, x)) i.e., 1 x = x and st x =
s t
( x), for all s, t ∈ G and x ∈ A.
ii) ac(s, mA (x, y)) = mA (ac(s, x), ac(s, y)) i.e., s (xy) =
and x, y ∈ A.

s s

x y, for all s ∈ G

A Hausdorff topological space A0 together with a continuous map ac :
G × A0 −→ A0 such that condition i) above holds is known as a (topological)
G-space.
A G-group homomorphism between two G-groups A and A0 is a continuous group homomorphism ϕ : A −→ A0 such that ϕ commutes with the
action of G, i.e., ϕ(ac(s, x)) = ac(s, ϕ(x)), for all s ∈ G and x ∈ A. Similarly, a map of G-spaces is a continuous map which commutes with the action
of G.
Lemma 2.8. Let A be a G-group. Then the map
def

ρ : G −→ M (A, A)po given by s 7−→ ρ(s) = ac|({s}×A)
is continuous. In particular, one has Im ρ ⊆ Autcts (A)po as a subgroup.
Proof. To show ρ is continuous, fix a point x ∈ A and an open subset V ⊆ A.
Let β({x}, V ) ⊆ M (A, A)po be an element of the subbase for the pointopen topology on M (A, A). Let s ∈ ρ−1 (β({x}, V )). Then ac|({s}×A) (x) =
ac(s, x) ∈ V , so (s, x) ∈ ac−1 (V ). By continuity of ac the set ac−1 (V ) is
an open subset of G × A; therefore there exist open subsets U1 ⊆ G and
V1 ⊂ A with (s, x) ⊆ U1 × V1 ⊆ ac−1 (V ). Now s ∈ U1 , and if t ∈ U1 ,
then ρ(t)(x) = ac(t, x) ∈ V , so ρ(t) ∈ β({x}, V ) i.e., t ∈ ρ−1 (β({x}, V )),
therefore U1 ⊆ ρ−1 (β({x}, V )). This shows ρ−1 (β({x}, V )) is an open subset
of G; hence ρ is a continuous map. In particular, since the action of G
respects the multiplication of A i.e., since ac|({s}×A) is a continuous group
2

Compare with [NSW08, p. 7, (1.7.1) Definition] & [Ser02, p. 45, 5.1].

Kayvan Nejabati Zenouz
Knejabat@ed.ac.uk

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2.3 Continuous Cocycles and Torsors

12

automorphism of A (upon composing with the homeomorphism {s} × A ∼
=
A) for every s ∈ G, and ac|({st}×A) = ac|({s}×A) ac|({t}×A) , we must also have
Im ρ ⊆ Autcts (A)po . This shows that Im ρ ⊆ Autcts (A)po as a subgroup,
which proves the lemma.

2.3

Continuous Cocycles and Torsors

We define the set of continuous cocycles endowed with point-open topology
and investigate its topological properties. We also define the set of continuous torsors and briefly discuss their well-known relationship to continuous
cocycles.
Definition 2.9 (Cocycles and Continuous Cocycle Pointed Spaces). 3 Let A
be a G-group. Define the set of (1-)cocycles of G with values in A by
def

Z 1 (G, A)po = {a ∈ M (G, A)po : ast = as s at for all s, t ∈ G}.
def

1
(G, A)po = Z 1 (G, A)po ∩ Mcts (G, A)po .
We Let Zcts
1
The sets Z 1 (G, A)po , and Zcts
(G, A)po are topological pointed sets, whose
topologies are the subspace topology inherited from M (G, A)po , and their
distinguished point is given by the trivial cocycle a1 , which is the constant
map a1,s = 1 for all s ∈ G.

Remark 2.10. Given a G-group homomorphism ϕ : A −→ A0 between Ggroups A and A0 as defined in Definition 2.9, it follows from Lemma 2.1, and
the fact that ϕ commutes with the action of G i.e., ϕ( s x) = s ϕ(x) for all
s ∈ G and x ∈ A, that ϕ induces a continuous maps
ϕ∗ : Z 1 (G, A)po −→ Z 1 (G, A0 )po ,
which restricts to a continuous map (also denoted by ϕ∗ )
1
1
ϕ∗ : Zcts
(G, A)po −→ Zcts
(G, A0 )po .

In addition, if ψ : H −→ G is a continuous group homomorphism and A
is a G-group, then A can be regraded as a H-group, and using Lemma 2.2,
ψ induces a continuous map
ψ ∗ : Z 1 (G, A)po −→ Z 1 (H, A)po ,
3

Compare with [Ser02, p. 45, 5.1]

Kayvan Nejabati Zenouz
Knejabat@ed.ac.uk

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2.3 Continuous Cocycles and Torsors

13

which restricts to a continuous map (also denoted by ψ ∗ )
1
1
ψ ∗ : Zcts
(G, A)po −→ Zcts
(H, A)po .

Let A be a G-group. We briefly recall the well-known relationship between
continuous cocycles of G in A and continuous G-torsors under A.
Definition 2.11 (Continuous G-Torsor). 4 Let A be a G-group. Then a
continuous G-torsor under A is a topological space P , which is a left Gspace with action
def s

acG : G × P −→ P given by (s, p) 7−→ acG (s, p) =

p

and a right A-space with action
def

acA : P × A −→ P given by (p, x) 7−→ acA (p, x) = p.x
such that the map
φA : P × A −→ P × P given by (p, x) 7−→ (p, p.x)
is a homeomorphism of G-spaces.
We denote by TorsG
cts (A) the set of continuous G-torsors under A. A map
of continuous G-torsors under A is a continuous map which commutes with
the actions of G and A.
Note if P is a continuous G-torsor under A as in Definition 2.11, then the
map φA is a G-space map if and only if acA is a map of G-spaces; since we
have
φA (s p, s x) = (s p, s p.s x) = (s p, s (p.x))
for all s ∈ G, p ∈ P and x ∈ A if and only if s p.s x =
p ∈ P and x ∈ A.

s

(p.x) for all s ∈ G,

Remark 2.12 (Correspondence between continuous cocycles and torsors). Let
1
a ∈ Zcts
(G, A)po . Then one can construct a continuous G-torsor under A
denoted by a P as follows. The underlying set of a P is A on which G acts on
the left by
def
def
(s, p) 7−→ s0 p = as s p = mA (as , s p),
4

Compare with [Ser02, p. 46, 5.2].

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Knejabat@ed.ac.uk

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2.3 Continuous Cocycles and Torsors

14

and A acts on the right by
def

(p, x) 7−→ p.x = mA (p, x).
Therefore, we have s0 (p.x) = as s (px) = as s p s x = s0 p. s x, i.e., the action
of A is a G-space map. One verifies immediately that a P is a continuous
1
G-torsor. In particular, for a, b ∈ Zcts
(G, A)po we have a P = b P if and only
if a = b. This gives a well-defined injective map of sets
1
T : Zcts
(G, A)po −→ TorsG
cts (A).

Now let P be a G-torsor under A as defined in Definition 2.11. Choose a
point p0 ∈ P . Then we obtain a continuous map a : G −→ A, which makes
the following diagram commutative
G×P

(s,p)7→(p, s p)

G∼
= G × {p0 }

P ×P
a

φ−1
A

P ×A
A,

where the left vertical arrow is the natural inclusion and the right vertical
arrow is the projection to the second factor. Therefore, for any s ∈ G,
as ∈ A is the unique element satisfying s p0 = p0 .as (this is possible since
φA is a bijection). In particular, since φA is a map of G-spaces and for each
s, t ∈ G, we have that ast ∈ A is the unique element satisfying st p0 = p0 .ast ,
we find
st

p0 = s (t p0 ) = s (p0 .at ) = s p0 .s at = p0 .as . s at = p0 .(as s at );

since φ is injective, we have ast = as s at for all s, t ∈ G; from this it follows
that a is a continuous cocycle.
Furthermore, by restriction of φA to the subset {p0 } × A, we obtain a
canonical homeomorphism of G,A-spaces
A∼
= {p0 } × A −→ {p0 } × P ∼
= P given by x 7−→ p0 .x,
where A acts on itself by right multiplication, and G acts on A by s0 x = as s x
i.e., P ∼
= a P as G,A-spaces. This shows that the map T is a bijection of sets.
Therefore, we may introduce a topology on TorsG
cts (A) by declaring a subset
1
to be open if it is the image of an open subset of Zcts
(G, A)po under T . In
such case T will be a homeomorphism, and we shall denote by TorsG
cts (A)po
G
the set Torscts (A) with the topology imposed by the bijection T .
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Knejabat@ed.ac.uk

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2.3 Continuous Cocycles and Torsors

15

Lemma 2.13. Let A be a G-group. Then Z 1 (G, A)po ⊆ M (G, A)po is a
1
closed subset; so Zcts
(G, A)po ⊆ Mcts (G, A)po is a closed subset. In particular,
if G is a compact and Hausdorff space, and A is a compact G-group evenly
1
(G, A)po is a compact and Hausdorff
continuous with respect to G, then Zcts
space.
Proof. To show Z 1 (G, A)po ⊆ M (G, A)po is a closed subset, let s, t ∈ G and
define a map Fs,t : M (G, A)po −→ A by
def

def

Fs,t (f ) = fs s ft fst−1 = mA (fs , mA (ac(s, ft ), ιA (fst ))).
We claim that the map Fs,t is continuous. We shall check the continuity of
Fs,t directly.
−1
Let V ⊆ A be an open subset and f ∈ Fs,t
(V ). Then
mA (fs , mA (ac(s, ft ), ιA (fst ))) ∈ V,
so (fs , mA (ac(s, ft ), ιA (fst ))) ∈ m−1
A (V ), and by continuity of mA the subset
−1
mA (V ) is open in A × A; therefore there exist open subsets U1 , V1 ⊆ A such
that
(fs , mA (ac(s, ft ), ιA (fst ))) ∈ U1 × V1 ⊆ m−1
A (V ),
so f ∈ β({s}, U1 ), and mA (ac(s, ft ), ιA (fst )) ∈ V1 , hence (ac(s, ft ), ιA (fst )) ∈
m−1
A (V1 ); since mA is continuous, and V1 ⊆ A is an open subset, we can find
open subsets U2 , V2 ⊆ A such that
(ac(s, ft ), ιA (fst )) ∈ U2 × V2 ⊆ m−1
A (V1 ),
i.e., (s, ft ) ∈ ac−1 (U2 ) and fst ∈ ιA (V2 ); therefore (s, ft ) ∈ ac−1 (U2 ) and
f ∈ β({st}, ιA (V2 )).
Now since ac is a continuous map, ac−1 (U2 ) is an open subset of G × A,
so there exist open subsets U3 ⊆ G and V3 ⊆ A such that
(s, ft ) ∈ U3 × V3 ⊆ ac−1 (U2 ) ,
hence s ∈ U3 and f ∈ β({t}, V3 ). Now
f ∈ β({s}, U1 ) ∩ β({st}, ιA (V2 )) ∩ β({t}, V3 ),
which is an open subset of M (G, A)po , and if
g ∈ β({s}, U1 ) ∩ β({st}, ιA (V2 )) ∩ β({t}, V3 ),
Kayvan Nejabati Zenouz
Knejabat@ed.ac.uk

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2.3 Continuous Cocycles and Torsors

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−1
−1
then gs ∈ U1 , gst
∈ V2 , and gt ∈ V3 , so s gt ∈ U2 and so s gt gst
∈ V1 and
−1
−1
s
gs gt gst ∈ V , this implies that Fs,t (g) ∈ V , hence g ∈ Fs,t (V ). From
−1
this it follows that Fs,t
(V ) is an open subset of M (G, A)po , and so Fs,t is a
continuous map; since s, t where arbitrary, Fs,t is a continuous map for all
s, t ∈ G.
Now since A is a Hausdorff space, the subset {1} ⊆ A is closed, so
−1
Fs,t ({1}) ⊆ M (G, A)po is a closed subset by the continuity of Fs,t . In particular, we have
\
−1
Z 1 (G, A)po =
Fs,t
({1}),
s,t∈G

which is an intersection of closed subsets of M (G, A)po , which implies that
def
1
(G, A)po = Z 1 (G, A)po ∩
Z 1 (G, A)po is a closed subset of M (G, A)po ; so Zcts
Mcts (G, A)po is a closed subset of Mcts (G, A)po . Furthermore, recall A is
a Hausdorff space by Definition 2.7, so by Lemma 2.3 i), Mcts (G, A)po is
1
a Hausdorff space, which implies that Zcts
(G, A)po is a Hausdorff space cf.
[Wil04, p. 87, 13.8 Theorem. a)].
Now if G is compact and Hausdorff space, and A is a compact G-group,
which is evenly continuous with respect to G, then Mcts (G, A)po is a compact
1
space by Lemma 2.6, and since Zcts
(G, A)po is closed subset of a compact
1
space, we conclude that Zcts (G, A)po is also a compact space cf. [Wil04,
p. 119, 17.14 Theorem. a)], which proves the lemma.
Lemma 2.14. Let A be a G-group and B any topological space. Then
Mcts (B, A)po is a G-group.
Proof. Note A is a Hausdorff space, so Mcts (B, A)po is a Hausdorff space by
Lemma 2.3, i). Since mA , ιA and ac are continuous maps, the maps
mpo : Mcts (B, A)po × Mcts (B, A)po −→ Mcts (B, A)po
given by (f, g) 7−→ mA (f, g),
ιpo : Mcts (B, A)po −→ Mcts (B, A)po given by f 7−→ ιA (f ), and
acpo : G×Mcts (B, A)po −→ Mcts (B, A)po given by (s, f ) 7−→ ac(s, f ) = ac|{s}×A f
are well-defined. We shall directly check each of the maps above is continuous.
Then checking the maps mpo , ιpo , and acpo make Mcts (B, A)po a group on
which G acts compatibly with the group operation follows from properties of

Kayvan Nejabati Zenouz
Knejabat@ed.ac.uk

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2.4 Cohomology Map

17

mA , ιA , and ac. Note the continuity of ιpo follows immediately from Lemma
2.1.
Now fix a point b ∈ B and an open set V ⊆ A. Let βcts ({b}, V ) ⊆
Mcts (B, A)po be an element of the subbase for the point-open topology on
Mcts (B, A)po . In order to show mpo is continuous, we shall show that every element of m−1
po (βcts ({b}, V )) has an open neighbourhood contained in
−1
mpo (βcts ({b}, V )). Let (f, g) ∈ m−1
po (βcts ({b}, V )). Then mA (f (b), g(b)) ∈ V
−1
i.e., (f (b), g(b)) ∈ mA (V ). Since mA is continuous, m−1
A (V ) ⊆ A × A is an
open subset; therefore there exist open subsets U1 , V1 ⊆ A with
(f (b), g(b)) ∈ U1 × V1 ⊆ m−1
A (V );
hence f ∈ βcts ({b}, U1 ) and g ∈ βcts ({b}, V1 ). Now (f, g) ∈ βcts ({b}, U1 ) ×
βcts ({b}, V1 ), which is an open subset of Mcts (B, A)po × Mcts (B, A)po , and if
(f1 , g1 ) ∈ βcts ({b}, U1 ) × βcts ({b}, V1 ),
then (f1 (b), g1 (b)) ∈ U1 × V1 ⊆ m−1
A (V ) so mpo (f1 , g1 ) ∈ βcts ({b}, V ) hence,
−1
(f1 , g1 ) ∈ mpo (βcts ({b}, V )). Therefore,
m−1
po (βcts ({b}, V )) ⊆ Mcts (B, A)po × Mcts (B, A)po
is an open subset, and hence mpo is a continuous map.
Finally, to show the continuity of acpo , let (s, f ) ∈ ac−1
po (βcts ({b}, V )).
−1
−1
Then ac(s, f (b)) ∈ V , i.e., (s, f (b)) ∈ ac (V ), and ac (V ) is an open
subset of G × A since ac is continuous. Therefore, there exist open subsets
U2 ⊆ G and V2 ⊆ A with (s, f (b)) ∈ U2 × V2 ⊆ ac−1 (V ). Now s ∈ U2 and
f ∈ βcts ({b}, V2 ), hence (s, f ) ∈ U2 × βcts ({b}, V2 ), which is an open subset
of G × Mcts (B, A)po , and if (t, g) ∈ U2 × βcts ({b}, V2 ), then
(t, g(b)) ∈ U2 × V2 ⊆ ac−1
po (βcts ({b}, V )),
−1
so acpo (t, g) ∈ βcts ({b}, V ) i.e., (t, g) ∈ ac−1
po (βcts ({b}, V )). This shows acpo (βcts ({b}, V ))
is an open subset of G × Mcts (B, A)po ; therefore acpo is a continuous map,
and so the lemma follows.

2.4

Cohomology Map

In this subsection we define, for a G-group A, a continuous action of A
1
on the right of Zcts
(G, A)po , and discuss some of the consequences that the
continuity of this action entails.
Kayvan Nejabati Zenouz
Knejabat@ed.ac.uk

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2.4 Cohomology Map

18

Definition 2.15 (Cohomology Map). Let A be a G-group. Define the cohomology map of A by
def

1
cb = cbA : Zcts
(G, A)po ×A −→ M 1 (G, A)po , given by (a, x) 7−→ cb(a, x) = a.x
def

such that a.x is given by (a.x)s = x−1 as s x, for all s ∈ G. Two elements
1
(G, A)po are said to be (A)-cohomologous if there exists x ∈ A with
a, b ∈ Zcts
a = b.x.
Proposition 2.16. Let A be a G-group, and let cb be the map of Definition
2.15. Then we have
1
1
cb : Zcts
(G, A)po × A −→ Zcts
(G, A)po ,
1
(G, A)po × A)
and cb defines a continuous (for the product topology on Zcts
1
right action of A on Zcts (G, A)po .

Proof. We first show the map cb is a well-defined action of A on the right of
1
Zcts
(G, A)po . Recall we have
def

def

(a.x)s = x−1 as s x = mA (ιA (x), mA (as , ac(s, x)));
therefore a.x can be given as the following composition of continuous maps
(ιA (x),a,ac|

G

G×{x}

)

A×A×A

mA ×idA

A×A

mA

A.

This shows a.x is a continuous map from G to A. Now a.x is a cocycle since
(a.x)st = x−1 ast st x = x−1 as s at s ( t x) = x−1 as s x s x−1 s at s ( t x)
= x−1 as s x s (x−1 at t x) = (a.x)s s (a.x)t , for all s, t ∈ G.
Above arguments show that the map cb is well-defined. To see cb defines a
right action, note (a.1)s = 1−1 as s 1 = as for all s ∈ G, so a.1 = a for all
a ∈ Z 1 (G, A)co , also
(a.xy)s = (xy)−1 as s (xy) = y −1 x−1 as s x s y = y −1 (a.x)s s y = ((a.x).y)s
for all s ∈ G so a.xy = (a.x).y, for all x, y ∈ A and all a ∈ Z 1 (G, A)co .
Therefore, it remains to show that the map cb is continuous. To check
the continuity of cb it suffices to show that the inverse image of any element
Kayvan Nejabati Zenouz
Knejabat@ed.ac.uk

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2.4 Cohomology Map

19

1
of the subbase for the topology of Zcts
(G, A)po , via cb, is an open subset of
1
Zcts (G, A)po × A. Fix a point s ∈ G and an open subset V ⊆ A, and let
def
1
βcts ({s}, V ) = {a ∈ Zcts
(G, A)po : as ∈ V } be an element of subbase for
1
the point-open topology on Zcts
(G, A)po . We shall show every element of
−1
cb (βcts ({s}, V )) has a neighbourhood contained in cb−1 (βcts ({s}, V )). Let
(a, x) ∈ cb−1 (βcts ({s}, V )). Then mA (ιA (x), mA (as , ac(s, x))) ∈ V , and so

(ιA (x), mA (as , ac(s, x))) ∈ m−1
A (V ).
Since mA is a continuous map, m−1
A (V ) is an open subset of A × A;
therefore there exist open subsets U1 , V1 ⊆ A such that
(ιA (x), mA (as , ac(s, x))) ∈ U1 × V1 ⊆ m−1
A (V ).
Thus, x ∈ ιA (U1 ) and (as , ac(s, x)) ∈ m−1
A (V1 ); once again, since mA is
−1
continuous, mA (V1 ) is an open subset of A × A, so we can find open subsets
U2 , V2 ⊆ A such that
(as , ac(s, x)) ∈ U2 × V2 ⊆ m−1
A (V1 ).
Hence, a ∈ βcts ({s}, U2 ) and (s, x) ∈ ac−1 (V2 ), and since ac is a continuous
map, there exist open subsets V3 ⊆ G and U3 ⊆ A with (s, x) ∈ V3 × U3 ⊆
ac−1 (V2 ).
Now x ∈ U3 ∩ ιA (U1 ) and a ∈ βcts ({s}, U2 ), so
(a, x) ∈ βcts ({s}, U2 ) × U3 ∩ ιA (U1 ),
1
which is an open subset for the product topology on Zcts
(G, A)po × A, and if

(b, y) ∈ βcts ({s}, U2 ) × U3 ∩ ιA (U1 ),
then b(s) ∈ U2 , y ∈ U3 , and y −1 ∈ U1 , so ac(s, y) ∈ V2 , and therefore
mA (b(s), ac(s, y)) ∈ V1 , hence mA (y −1 , mA (b(s), ac(s, y))) = y −1 b(s) s y ∈ V
which implies that (b.y)(s) ∈ V i.e., cb(b, y) = b.y ∈ βcts ({s}, V ), and so
(b, y) ∈ cb−1 (βcts ({s}, V )). This shows that cb−1 (βcts ({s}, V )) is an open
1
subset of Zcts
(G, A)po × A, which proves that cb is a continuous map and
establishes the proposition.
1
1
Corollary 2.17. Let A be a G-group, and cb : Zcts
(G, A)po ×A −→ Zcts
(G, A)po
1
be the continuous action of A on Zcts
(G, A)po as in Proposition 2.16. Fix an
1
element a ∈ Zcts
(G, A)po , and denote by StabA (a) ⊆ A the subset of element
of A fixing a. Then the subset StabA (a) ⊆ A is a closed subgroup. In particular, if A is a compact G-group, then StabA (a) ⊆ A is a nonempty, compact,
and Hausdorff space.

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2.4 Cohomology Map

20

Proof. The subset StabA (a) ⊆ A is a subgroup. We shall show that StabA (a) ⊆
1
A is a closed subset. Recall we have the continuous map cb : Zcts
(G, A)po ×
1
A −→ Zcts
(G, A)po by Proposition 2.16, so we obtain, by restriction of cb to
1
the subset {a}×A, a continuous map cb|{a}×A : {a}×A −→ Zcts
(G, A)po . Note
1
Zcts (G, A)po is a Hausdorff space, which follows from Lemma 2.3 i); this im1
(G, A)po is a closed subset. Hence cb−1
plies that {a} ⊆ Zcts
|{a}×A ({a}) ⊆ {a}×A
∼
is a closed subset, but cb−1
|{a}×A ({a}) = StabA (a) under the homeomorphism
{a} × A ∼
= A, so StabA (a) ⊆ A is a closed subset.
Alternatively, one can directly prove every element y ∈ A \ StabA (a) has
a neighbourhood contained in A \ StabA (a) as follows. If y ∈ A \ StabA (a),
then a.y 6= a, so there exists s ∈ G with y −1 as s y 6= as , which implies
(y −1 as ,s y) ∈
/ m−1
A ({as }). The space A is Hausdorff, so {as } ⊂ A is a closed
subset; therefore m−1
A ({as }) ⊆ A × A is a closed subset of A × A; thus,
−1
A×A\mA ({as }) is an open subset of A×A. In particular, A×A\m−1
A ({as })
contains the element (y −1 as ,s y), so there exist open subsets U, V ⊂ A with
y −1 as ∈ U and s y ∈ V , and U × V ⊆ A × A \ m−1
A ({as }). Therefore,
−1
s−1
y ∈ as U ∩
V . Now since A is a topological group as U −1 is an open
−1
subset of A, and G acting continuously on A implies s V is an open subset
−1
of A, hence as U −1 ∩ s V is an open neighbourhood of y.
Note we have U × V ∩ m−1
/ mA (U × V ) = U V . Now
A ({as }) = ∅, so as ∈
−1
−1
s−1
−1
given z ∈ as U ∩
V , we have z = as u = s v for some u ∈ U and
s s−1
v) = uv 6= as i.e., (a.z)s 6= as , so
v ∈ V , so (a.z)s = z −1 as s z = ua−1
s as (
−1
z∈
/ StabA (a); thus as U −1 ∩ s V ⊂ A \ StabA (a), which shows A \ StabA (a)
is an open subset of A; therefore StabA (a) is a closed subset of A.
Now a closed subset of a compact and Hausdorff space is a compact and
Hausdorff space cf. [Wil04, p. 119, 17.14 Theorem. a)] & cf. [Wil04, p. 87,
13.8 Theorem. a)]; so if A is a compact G-group, then StabA (a) ⊆ A is a
nonempty, compact, and Hausdorff space.
Corollary 2.18. Let A be a G-group. Then the subset AG ⊆ A, the fixed
subgroup of A under the action of G, is a closed.
1
Proof. By Corollary 2.17, the stabiliser of each element of Zcts
(G, A)co is a
G
closed subgroup of A, so to prove A ⊆ A is a closed subset, it suffices to
show AG = StabA (a1 ). But this is immediate, since let x ∈ AG . Then s x = x
for all s ∈ G i.e., x−1 s x = 1 for all s ∈ G so x−1 a1,s s x = a1,s for all
s ∈ G i.e., a1 .x = a1 so x ∈ StabA (a1 ). Conversely, if x ∈ StabA (a1 ), then
x−1 a1,s s x = a1,s for all s ∈ G so s x = x for all s ∈ G, hence x ∈ AG .

Kayvan Nejabati Zenouz
Knejabat@ed.ac.uk

Compiled on 5th Apr, 2018 at 00:27

2.4 Cohomology Map

21

1
Corollary 2.19. Let A be a compact G-group. Fix a ∈ Zcts
(G, A)po , and let
OrbA (a) be the orbit of the element a under the action of A. Then the subset
1
OrbA (a) ⊆ Zcts
(G, A)po , is a nonempty, compact, and Hausdorff subspace.
1
1
Proof. By Proposition 2.16, the map cb : Zcts
(G, A)po × A −→ Zcts
(G, A)po
is a continuous action. Therefore, we obtain a continuous map
1
cb|{a}×A : {a} × A −→ Zcts
(G, A)po ,

by restriction of cb to subspace {a} × A ⊆ Z 1 (G, A)po × A. In particular, the
image of cb|{a}×A is OrbA (a). Therefore, we have a surjective continuous map
cb|{a}×A : {a} × A −→ OrbA (a),
and cb|{a}×A (a, x) = cb|{a}×A (a, y) if and only if xy −1 ∈ StabA (a). Hence we
have a commutative diagram
{a} × A

cb|

{a} × A/StabA (a)

{a}×A

cb|

{a}×A

OrbA (a)
OrbA (a).

Note A is a compact and Hausdorff space, so by Corollary 2.17, StabA (a)
is a compact and Hausdorff topological group. Now StabA (a) acts continuously on the right of A, then A/StabA (a) is a compact space being
continuous image of the compact space A under the continuous open map
A  A/StabA (a) cf. [Wil04, p. 119, Theorem 17.1], and A/StabA (a) is Haus1
(G, A)po is a Hausdorff
dorff cf. [Bre72, p. 38, Theorem 3.1, (1)]. Since Zcts
1
space, OrbA (a) ⊆ Zcts (G, A)po is a Hausdorff space cf. [Wil04, p. 87, 13.8
Theorem. a)]. Now the map
cb|{a}×A : {a} × A/StabA (a) −→ OrbG (a)
is a continuous bijection from a compact space to a Hausdorff space, so
cb|{a}×A is a homeomorphism cf. [Wil04, p. 123, 17.14 Theorem.]; therefore
A/StabA (a) ∼
= {a} × A/StabA (a) ∼
= OrbA (a).
The above shows OrbA (a) is a compact and Hausdorff space, it contains
a, so it is a nonempty, compact, and Hausdorff space.
Kayvan Nejabati Zenouz
Knejabat@ed.ac.uk

Compiled on 5th Apr, 2018 at 00:27

2.5 First Nonabelian Continuous Cohomology Pointed Spaces

22

Remark 2.20. Recall a G-group homomorphism ϕ : A −→ A0 between G1
1
groups A and A0 induces a continuous map ϕ∗ : Zcts
(G, A)po −→ Zcts
(G, A0 )po
by Remark 2.10. Now for (a, x) ∈ Z 1 (G, A)po × A we have ϕ∗ cbA (a, x) =
cbA0 (ϕ∗ (a), ϕ(x)) i.e., ϕ induces the following commutative diagram
1
Zcts
(G, A)po × A

ϕ∗ ×ϕ

cbA0

cbA
1
(G, A)po
Zcts

2.5

1
Zcts
(G, A0 )po × A0

ϕ∗

1
(G, A0 )po .
Zcts

First Nonabelian Continuous Cohomology Pointed
Spaces

In this subsection, for a G-group A, we define the first nonabelian continuous
cohomology pointed space of G with coefficients in A as the quotient space
1
(G, A)po by the continuous action of A given by cb, and we investigate the
Zcts
properties this space. But first we note that the zeroth cohomology group of
def
0
0
G with coefficients in a G-group A is defined as Hcts
(G, A)po = Hcts
(G, A) =
AG , which is a closed subgroup of A by Corollary 2.18.
Definition 2.21 (First Nonabelian Continuous Cohomology Pointed Spaces).
5
Let A be a G-group. Then we define the first nonabelian continuous cohomology pointed space of G with coefficients in A as the quotient space
1
Zcts
(G, A)po by the action of A i.e.,
def

1
1
Hcts
(G, A)po = Zcts
(G, A)po /A.
1
1
In other words, we have Hcts
(G, A)po = Zcts
(G, A)po / ∼, where for a, b ∈
1
we write a ∼ b if a and b are cohomologous. The set Hcts
(G, A)po
is endowed with the quotient topology; in particular, the natural surjective
map
1
1
πA : Zcts
(G, A)po −→ Hcts
(G, A)po ,
1
Zcts
(G, A)po

1
which sends an element a ∈ Zcts
(G, A)po to its orbit under the action of A
is a continuous open map cf. [Bre72, p. 37]. If we denote the image of an
1
element a ∈ Zcts
(G, A)po under πA by [a], then the class of trivial cocycle [a1 ]
1
gives a distinguished point of Hcts
(G, A)po .
5

Compare with [Ser02, p. 45, 5.1].

Kayvan Nejabati Zenouz
Knejabat@ed.ac.uk

Compiled on 5th Apr, 2018 at 00:27

2.5 First Nonabelian Continuous Cohomology Pointed Spaces

23

Remark 2.22. Recall that by Remark 2.20 a G-group homomorphism ϕ :
A −→ A0 between G-groups A and A0 induces a commutative
1
(G, A)po × A
Zcts

ϕ∗ ×ϕ

1
(G, A0 )po × A0
Zcts
cbA0

cbA
1
Zcts
(G, A)po

ϕ∗

1
Zcts
(G, A0 )po .

Therefore, ϕ induces a commutative diagram
1
Zcts
(G, A)po

ϕ∗

1
Zcts
(G, A0 )po
π A0

πA
1
(G, A)po
Hcts

ϕ∗

1
(G, A0 )po ,
Hcts
def

1
(G, A)po we set ϕ∗ ([a]) = [ϕ∗ (a)]. The map ϕ∗ is conwhere for [a] ∈ Hcts
1
(G, A0 )po be an open subset. Then by the
tinuous as follows. Let V ⊆ Hcts
commutativity of the diagram above we have ϕ∗ πA = πA0 ϕ∗ , so
−1
−1
−1
(ϕ∗ πA )−1 (V ) = πA−1 (ϕ−1
∗ (V )) = (πA0 ϕ∗ ) (V ) = ϕ∗ (πA0 (V )),

since πA is surjective, applying πA to the both side of equation above we find
−1
−1
−1
πA (πA−1 (ϕ−1
∗ (V ))) = ϕ∗ (V ) = πA (ϕ∗ (πA0 (V ))).

Now since πA0 and ϕ∗ are continuous maps and πA is an open map we conclude
1
that ϕ−1
∗ (V ) is an open subset of Hcts (G, A)po .
Similarly, if ψ : H −→ G is a continuous group homomorphism, by
Remark 2.10 and calculations similar to the one above, one obtains a commutative diagram of continuous maps
1
Zcts
(G, A)po

ψ∗

1
Zcts
(H, A)po

πA
1
Hcts
(G, A)po

πA
ψ

∗

∗

1
Hcts
(H, A)po ,
def

1
where for [a] ∈ Hcts
(G, A)po we set ψ ([a]) = [ψ ∗ (a)].

Next remark discusses the well-known correspondence between cohomology classes of continuous cocycles and isomorphism classes of torsors.
Kayvan Nejabati Zenouz
Knejabat@ed.ac.uk

Compiled on 5th Apr, 2018 at 00:27

2.5 First Nonabelian Continuous Cohomology Pointed Spaces

24

Remark 2.23. Let TorsG
cts (A) be the set of continuous G-torsors under A as
in Definition 2.11. Then by Remark 2.12 we have a homeomorphism
1
T : Zcts
(G, A)po −→ TorsG
cts (A)po .
1
Now suppose for a, b ∈ Zcts
(G, A)po , we have a = b.x for some x ∈ A.
Then we obtain a homeomorphism of G,A-spaces
bP

−→

aP

def

given by p 7−→ x−1 p = mA (x−1 , p).

Conversely, given two G-torsors P1 , P2 ∈ TorsG
cts (A)po and a homeomorphism of G,A -spaces α : P1 −→ P2 . Then by Remark 2.12 we can find
canonical homeomorphisms of G,A -spaces P1 ∼
= a P and P2 ∼
= b P for some
1
a, b ∈ Zcts (G, A)po . Therefore, α induces a homeomorphism of G,A-spaces
(also denoted by α), α : a P −→ b P . In particular, since α(p.x) = α(p).x,
we see that α(p) = α(1p) = α(1)p i.e., α is just multiplication on the left on
A by α(1). Further, we must have α( s0 p) = s0 α(p), so
α( s0 p) = α(as s p) = α(1)as s p =

s0

α(p) = bs s α(p) = bs s α(1) s p,

Therefore, as = α(1)−1 bs s α(1) for all s ∈ G i.e., a = b.α(1). Let TorsG
cts (A)po /Isom
be the set of G,A-isomorphism classes of G-torsors under A endowed with
quotient topology, and denote by [P ] the G,A-isomorphism classes of a Gtorsors P . Then the above arguments shows that T descends to a homeomorphism
1
Hcts
(G, A)po −→ TorsG
cts (A)po /Isom given by [a] 7−→ [ a P ].

Lemma 2.24. Suppose A is a compact G-group. Then
1
i) The topological space Hcts
(G, A)po is Hausdorff.
1
ii) The topological space Zcts
(G, A)po is compact if and only if the topological
1
space Hcts (G, A)po is compact; in particular if G is compact and Hasu1
dorff, and A is evenly continuous with respect to G, then Hcts
(G, A)po is
a compact space.
1
Proof. By Proposition 2.16, the cohomology map cb : Zcts
(G, A)po × A −→
1
1
Zcts (G, A)po is continuous; by Lemma 2.3, Zcts (G, A)po is a Hausdorff space.
Now A is a compact and Hausdorff topological group which acts continuously

Kayvan Nejabati Zenouz
Knejabat@ed.ac.uk

Compiled on 5th Apr, 2018 at 00:27

3 The First Main Theorem

25

1
1
on a Hausdorff space Zcts
(G, A)po . Therefore Zcts
(G, A)po is an A-space with
A a compact space, which matches the definition of [Bre72, p. 32, 1]. Now i)
follows from [Bre72, p. 38, Theorem 3.1, 1)], and ii) follows from [Bre72, p. 38,
Theorem 3.1, 4)]; in particular, if G is compact and Hausdorff space, and A
1
(G, A)po
is evenly continuous space with respect to G, by Lemma 2.13, Zcts
1
is a compact space, which implies that the space Hcts (G, A)po is a compact
space cf. [Wil04, p. 119, Theorem 17.1].

3

The First Main Theorem

In this section we shall work with compact G-groups. Recall R is a fixed directed poset and we assume {Ar , ϕrs , R} is an inverse system of compact topological G-groups, where the transition maps are G-group homomorphism.
Then the first main result of this article is the following.
Theorem 3.1. Let A be a compact G-group. Assume A has a presentation
A = limr∈R Ar . Denote by ϕr : A −→ Ar the natural projections. Then there
←−
exists a continuous bijection
1
1
Θ : Hcts
(G, A)po −→ lim Hcts
(G, Ar )po given by [a] 7−→ ([ϕr∗ (a)]).
←−
r∈R

In particular, if G is a compact and Hausdorff space, and A is evenly continuous with respect to G, then Θ is a homeomorphism.
We prove Theorem 3.1, using 3 lemmas, by showing there exists a welldefined natural continuous map Θ, induced by ϕr for r ∈ R, such that Θ is
a continuous bijection.
Lemma 3.2. Let A be a compact G-group satisfying the assumption of Theorem 3.1. Then there exists a well-defined continuous map
1
1
Θ : Hcts
(G, A)po −→ lim Hcts
(G, Ar )po given by [a] 7−→ ([ϕr∗ (a)])
←−
r∈R

and we have a commutative diagram
1
Zcts
(G, A)po

θ

πA
1
Hcts
(G, A)po

Kayvan Nejabati Zenouz
Knejabat@ed.ac.uk

1
limr∈R Zcts
(G, Ar )po
←−
lim

Θ

π

←
−r∈R Ar
1
limr∈R Hcts
(G, Ar )po .
←−
Compiled on 5th Apr, 2018 at 00:27

3 The First Main Theorem

26

Proof. Since ϕrt : Ar −→ At for r ≥ t are continuous G-group homomorphisms, by Remark 2.22, they induce a commutative diagram of continuous
maps
ϕrt∗
1
1
Zcts
(G, Ar )po
Zcts
(G, At )po
π At

π Ar
1
Hcts
(G, Ar )po

ϕrt∗

1
Hcts
(G, At )po .

1
Note ϕrt are transition maps, and so {Hcts
(G, Ar )po , ϕrt∗ , R} is inverse system
of topological pointed sets.
Now the continuous maps ϕr : A −→ Ar for r ∈ R are G-group homomorphisms, so by Remark 2.22 they induces a commutative diagram of
continuous maps
1
Zcts
(G, A)po

ϕr∗

1
Zcts
(G, Ar )po
π Ar

πA
1
Hcts
(G, A)po

ϕr∗

1
Hcts
(G, Ar )po ,

1
1
(G, Ar )po , ϕrt∗ , R}.
(G, Ar )po , ϕrt∗ , R} and {Hcts
compatible with transition maps of {Zcts
Taking the inverse limit of the above diagram we obtain a unique commutative diagram
1
1
Zcts
(G, A)po θ limr∈R Zcts
(G, Ar )po
←−
πA
1
Hcts
(G, A)po

lim

Θ

π

←
−r∈R Ar
1
(G, Ar )po .
limr∈R Hcts
←−

This shows that the map Θ is well-defined and is given by [a] 7−→ ([ϕr∗ (a)])
1
for [a] ∈ Hcts
(G, A)po . Now it follows from Lemma 2.4 that the map θ is
continuous (homeomorphism in fact since A is compact), and limr∈R πAr is
←−
a continuous map (being inverse limit of continuous maps), also since πA is
an open map, by preforming a similar calculation to what is done is Remark
2.22, we find that Θ is a continuous map.
Lemma 3.3. Let A be a compact G-group satisfying the assumption of Theorem 3.1. Then the continuous map
1
1
(G, A)po −→ lim Hcts
(G, Ar )po ,
Θ : Hcts
←−
r∈R

defined in Lemma 3.2, is injective.
Kayvan Nejabati Zenouz
Knejabat@ed.ac.uk

Compiled on 5th Apr, 2018 at 00:27

3 The First Main Theorem

27

1
Proof. Let [a], [b] ∈ Hcts
(G, A)po , and suppose

Θ([a]) = ([ϕr a]) = Θ([b]) = ([ϕr b]).
Then for all r ∈ R there exists xr ∈ Ar such that ϕr a = ϕr b.xr . Therefore,
the set Sr = {e
x ∈ Ar : ϕr a = ϕr b.e
x} is nonempty for all r ∈ R. Now
Sr is equal to the coset StabAr (ϕr b)xr , where StabAr (ϕr b) is the stabiliser
1
(G, Ar )po . The set StabAr (ϕr b) ⊆ Ar is
of ϕr b for the action of Ar on Zcts
a closed, compact, and Hausdorff subgroup by Corollary 2.17. Therefore,
Sr = StabAr (ϕr b)xr is nonempty, compact, and Hausdorff space for each
r ∈ R. In particular, if r ≥ t and x
e ∈ Sr , then
ϕt b.ϕrt (e
x) = ϕrt ϕr b.ϕrt (e
x) = ϕrt (ϕr b.e
x) = ϕrt ϕr a = ϕt a,
so ϕrt (e
x) ∈ St . Hence, the set {Sr , ϕrt , R} is an inverse system of nonempty,
compact, and Hausdorff spaces. Therefore, limr∈R Sr is nonempty, compact,
←−
and Hausdorff cf. [RZ10, p. 4, Proposition 1.1.4]. Now for x ∈ limr∈R Sr ,→
←−
limr∈R Ar = A, one has
←−
ϕr b.x = ϕr b.ϕr (x) = ϕr a
for all r ∈ R, so θ(b.x) = θ(a), and since by Lemma 2.4 the map θ is injective,
we conclude that b.x = a; therefore [a] = [b], and this proves that the map
Θ is injective.
Lemma 3.4. Let A be a compact G-group satisfying the assumption of Theorem 3.1. Then the continuous map
1
1
Θ : Hcts
(G, A)po −→ lim Hcts
(G, Ar )po ,
←−
r∈R

defined in Lemma 3.2, is surjective.
Proof. Recall we have surjective continuous maps
1
1
πAr : Zcts
(G, Ar )po −→ Hcts
(G, Ar )po
1
for each r ∈ R, which are compatible with the transition maps of {Zcts
(G, Ar )po , ϕrt∗ , R}
1
1
and {Hcts (G, Ar )po , ϕrt∗ , R}. Let ([ar ]) ∈ limr∈R Hcts (G, Ar )po . Then since Ar
←−
is compact and Hausdorff for each r ∈ R, the set πA−1r ([ar ]) = OrbAr (ar ) is a
nonempty, compact, and Hausdorff space for each r ∈ R by Corollary 2.19. In

Kayvan Nejabati Zenouz
Knejabat@ed.ac.uk

Compiled on 5th Apr, 2018 at 00:27

3 The First Main Theorem

28

particular the set {OrbAr (ar ), ϕrt∗ , R} is an inverse system of nonempty, compact, and Hausdorff spaces. Therefore, the set limr∈R OrbAr (ar ) is nonempty,
←−
compact, and Hausdorff cf. [RZ10, p. 4, Proposition 1.1.4]. Now for
1
a ∈ lim OrbAr (ar ) ,→ lim Zcts
(G, Ar )po
←−
←−
r∈R

r∈R

1
(G, A)po and ΘπA θ−1 (a) = ([ar ]); this proves that
we have πA θ−1 (a) ∈ Hcts
the map Θ is surjective.

Now we provide the proof of the Theorem 3.1. We recall the statement
of Theorem 3.1 again.
Theorem 3.5. Let A be a compact G-group. Assume A has a presentation
A = limr∈R Ar . Denote by ϕr : A −→ Ar the natural projections. Then there
←−
exists a continuous bijection
1
1
Θ : Hcts
(G, A)po −→ lim Hcts
(G, Ar )po given by [a] 7−→ ([ϕr∗ (a)]).
←−
r∈R

In particular, if G is a compact and Hausdorff space, and A is evenly continuous with respect to G, then Θ is a homeomorphism.
Proof. By Lemma 3.2, there exists a commutative diagram of continuous
maps
1
1
(G, A)co θ limr∈R Zcts
(G, Ar )co
Zcts
←−
πA
1
(G, A)
Hcts

lim

Θ

π

←
−r∈R Ar
1
(G, Ar ),
limr∈R Hcts
←−

by Lemma 3.3 the map Θ is injective, and by Lemma 3.4 the map Θ is
surjective. Therefore, Θ is a continuous bijection.
1
In particular, by Lemma 2.24, i), the space Hcts
(G, Ar ) is Hausdorff for
1
all r ∈ R; hence limr∈R Hcts (G, Ar ) is a Hausdorff space cf. [Wil04, p. 87,
←−
13.8 Theorem. a) & b)]; now if G is a compact and Hausdorff space, and A
is evenly continuous with respect to G, then By Lemma 2.24, ii), the space
1
Hcts
(G, A) is a compact space. Therefore, in such case Θ is a continuous
bijection from a compact space to a Hausdorff space, which implies that Θ
is a homeomorphism cf. [Wil04, p. 123, 17.14 Theorem].

Kayvan Nejabati Zenouz
Knejabat@ed.ac.uk

Compiled on 5th Apr, 2018 at 00:27

3.1 Applications of The First Main Theorem

3.1

29

Applications of The First Main Theorem

As an application of Theorem 3.1, one can relate the continuous cohomology
sets with coefficients in a finitely generated profinite group, to the inverse
limit of cohomology sets with coefficients in finite discrete groups. Another
application of Theorem 3.1 is concerned with the cohomology set with coefficients in a prosolvable group. We first discuss a general lemma which applies
to both of these applications of Theorem 3.1. Recall a profinite groups is a
compact, Hausdorff and totally disconnected topological group, which is also
an evenly continuous space by Corollary 2.19.
Proposition 3.6. Let A be a profinite G-group, and assume a collection
{Nr }r∈R of characteristic closed subgroups of A is given such that ∩r∈R Nr = 1
and Nr ⊆ Nt whenever r > t. Let Ar = A/Nr . Then A has a presentation
A by profinite G-groups Ar , and limr∈R Ar satisfies the assumpA∼
= lim
←−
←−r∈R r
tion of Theorem 3.1; in particular there exists a continuous bijection
1
1
Θ : Hcts
(G, A)po −→ lim Hcts
(G, Ar )po given by [a] 7−→ ([ϕr,∗ (a)]);
←−
r∈R

further if G is a compact and Hausdorff space, and A is evenly continuous
with respect to G, then Θ is a homeomorphism.
Proof. Since Nr is a closed normal subgroup of A for all r ∈ R, the quotient
group Ar is a profinite group for all r ∈ R cf. [RZ10, p. 28, Proposition 2.2.1,
(a)]. By Lemma 2.8, the group G acts by automorphism on A; in particular,
since Nr is a characteristic subgroup of A, the exact sequence
1

Nr

A

Ar

1,

is an exact sequence of G-groups. Therefore, the set {Ar }r∈R is an inverse
system, where the obvious transition Ar  At , whenever r ≥ t, are G-group
homomorphisms. Now since limr∈R is an exact functor on the category of
←−
profinite groups cf. [RZ10, p. 31, Proposition 2.2.4], taking the inverse limit
we obtain an exact sequence
1

limr∈R Nr
←−

limr∈R A
←−

limr∈R Ar
←−

1,

where limr∈R Nr ∼
limr∈R A ∼
= ∩r∈R Nr = 1, and ←
= A, so we have a natural
←−
−
∼
homeomorphism A = limr∈R Ar .
←−
Kayvan Nejabati Zenouz
Knejabat@ed.ac.uk

Compiled on 5th Apr, 2018 at 00:27

3.1 Applications of The First Main Theorem

30

Now limr∈R Ar satisfies the assumption of Theorem 3.1; therefore, we have
←−
a continuous bijection
1
1
Θ : Hcts
(G, lim Ar )po −→ lim Hcts
(G, Ar )po .
←−
←−
r∈R

r∈R

We further have a homeomorphism
1
1
Hcts
(G, A)po ∼
(G, lim Ar )po
= Hcts
←−
r∈R

induced by the homeomorphism A ∼
A . Hence, the map
= lim
←−r∈R r
1
1
Hcts
(G, A)po −→ lim Hcts
(G, Ar )po ,
←−
r∈R

which is the composition
1
1
1
Hcts
(G, A)po ∼
(G, lim Ar )po −→ lim Hcts
(G, Ar )po
= Hcts
←−
←−
r∈R

r∈R

is a continuous bijection. In particular, according to Theorem 3.1, if G is a
compact and Hausdorff space, and A is evenly continuous with respect to G,
then Θ is a homeomorphism.
Corollary 3.7. Suppose A is a finitely generated profinite G-group. Then A
has a presentation A ∼
A by finite G-groups Ar for r ∈ N satisfying
= lim
←−r∈N r
the assumption of Theorem 3.1; so there exists a there exist a continuous
bijection
1
1
Θ : Hcts
(G, A)po −→ lim Hcts
(G, Ar )po ,
←−
r∈N

and if G is a compact and Hausdorff space, and A is evenly continuous with
respect to G, then Θ is a homeomorphism.
Proof. Since A is a finitely generated profinite group, 1 ∈ A has a fundamental system of neighbourhoods consisting of a countable chain of open
characteristic subgroups
A = N0 ⊇ N1 ⊇ N2 , ...
cf. [RZ10, p. 44, Proposition 2.5.1]. Therefore, we have a natural homeomorphism A ∼
A , and limr∈N Ar satisfies the assumption of Theorem
= lim
←−r∈N r
←−
3.1. Now the corollary follows from Proposition 3.6.
Kayvan Nejabati Zenouz
Knejabat@ed.ac.uk

Compiled on 5th Apr, 2018 at 00:27

4 G-Modules and The Second Main Theorem

31

Corollary 3.8. Suppose A is a prosolvable6 G-group. Then A has a presentation A ∼
A by profinite G-groups Ai , for i ∈ N, satisfying the
= lim
←−i≥1 i
assumption of Theorem 3.1; so there exists a continuous bijection
1
1
(G, A)po −→ lim Hcts
(G, Ai )po ,
Hcts
←−
i≥1

and if G is a compact and Hausdorff space, and A is evenly continuous with
respect to G, then Θ is a homeomorphism.
Proof. Suppose A is a prosolvable group. Then one can write A = limr∈R Ar
←−
with Ar finite solvable group for each r ∈ R, and continuous surjective homomorphisms ϕr : A −→ Ar such that ∩r∈R Ker ϕr = 1 cf. [RZ10, p. 22,
Theorem 2.1.3, (c)].
def
Define A(i + 1) = [A(i) : A(i)] to be the closure of the commutator subgroup of A(i), i.e., [A(i) : A(i)] is generated by commutator elements of A(i),
for i ∈ N, and let A(0) = A. By induction on i, one can show that A(i) is
a characteristic subgroup of A for every i ≥ 1; we set Ai = A/A(i). Now
ϕr (A(i)) ⊆ Ar (i) for all i ≥ 1, and since for all r ∈ R there exists dr ∈ N
with Ar (dr ) = 1 (since Ar is solvable for each r ∈ R) one has ϕr (A(dr )) = 1,
hence ∩i≥1 A(i) ⊂ A(dr ) ⊂ Ker ϕr for all r ∈ R i.e., ∩i≥1 A(i) ⊆ ∩r Ker ϕr = 1.
Therefore, one has a natural homeomorphism A ∼
A , and limi≥1 Ai
= lim
←−i≥1 i
←−
satisfies the assumption of Theorem 3.1. Now the corollary follows from
Proposition 3.6.

4

G-Modules and The Second Main Theorem

In this section we shall fix G to be a compact and Hausdorff topological group,
and work with abelian G-groups, which also known, and we shall refer to,
as G-modules, also R is a fixed directed poset and we assume {Ar , ϕrs , R}
is an inverse system of compact topological G-modules, where the transition
maps are G-group homomorphism. Given a G-module A, one can define the
n
continuous cohomology sets Hcts
(G, A) for all n ≥ 0 cf. [NSW08, p. 137].
n
Denote by G the product of n copies of G endowed with product topology.
Then the main result of this article is the following.
6

See [RZ10, p. 19] for definition.

Kayvan Nejabati Zenouz
Knejabat@ed.ac.uk

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4 G-Modules and The Second Main Theorem

32

Theorem 4.1. Let A a compact G-module and fix n ≥ 1. Assume A has a
presentation A = limr∈R Ar , where Ar is evenly continuous with respect to
←−
Gn−1 for all r ∈ R. Denote by ϕr : A −→ Ar the natural projections. Then
there exists a continuous bijection
n
n
Θn : Hcts
(G, A)po −→ lim Hcts
(G, Ar )po given by [a] 7−→ ([ϕr∗ (a)]).
←−
r∈R

In particular, if A is evenly continuous with respect to Gn , then Θn is a
homeomoprhism.
The proof of Theorem 4.1 is somewhat similar to the proof of Theorem
3.1. We fix n ≥ 1 and start by introducing the point-open topology on the
n
set of n-cocycles of G with values in A, Zcts
(G, A). Since Ar are compact
G-modules, which are evenly continuous with respect to Gn−1 and for all
r ∈ R, we have that Mcts (Gn−1 , A) is a compact and Hausdorff space. Then
n
n
(G, A) by a continuous
(G, A) as a quotient space of Zcts
we interpret Hcts
action of a compact and Hausdorff topological group Mcts (Gn−1 , A), and
from these we deduce our theorem.
We fix a G-module A, sometimes we shall use ”+” instead of the operation
of mA of A. As before, we denote by Mcts (Gn , A)po the set of all continuous
maps from Gn −→ A endowed with point-open topology. Recall we let
Mcts (G0 , A)po be the set of all constant maps from G −→ A, so we have
natural identification Mcts (G0 , A)po ∼
= A. It follows from Lemma 2.14 that
n
Mcts (G , A)po is a G-module for all n ≥ 1. Now there exists a well-known
differential homomorphism
dn : Mcts (Gn−1 , A)po −→ Mcts (Gn , A)po given by f 7−→ dn f
for n ≥ 1, defined by
def s1

dn fs1 ,...,sn =

fs2 ,...,sn +

n−1
X

(−1)i fs1 ,...,si si+1 ,...sn + (−1)n fs1 ,..,sn−1 ,

i=1

which has the property that dn+1 dn = 0.
Lemma 4.2. Let A be a G-module. Then for each n ≥ 1 the map
dn : Mcts (Gn−1 , A)po −→ Mcts (Gn , A)po given by f 7−→ dn f,
defined above, is a continuous group homomorphism.
Kayvan Nejabati Zenouz
Knejabat@ed.ac.uk

Compiled on 5th Apr, 2018 at 00:27

4 G-Modules and The Second Main Theorem

33

Proof. Fix n ≥ 1. Then the map dn is a group homomorphism, so we only
def
require to that show dn is a continuous map. Fix a point s = (s1 , ..., sn ) ∈ Gn
and an open set V ⊆ A. Let βcts ({s}, V ) ⊆ Mcts (Gn , A)po be an element of the subbase for the point-open topology on Mcts (Gn , A)po . Let
f ∈ d−1
n (βcts ({s}, V )). Then
!!
n−1
X
def
(−1)i fs1 ,...,si si+1 ,...sn , (−1)n fs1 ,..,sn−1
∈ V,
dn fs1 ,...,sn = mA s1 fs2 ,...,sn , mA
i=1

since mA is a continuous map and V is an open subset, there exists open
subsets U1 , V1 ⊆ A such that
!!
n−1
X
s1
fs2 ,...,sn , mA
(−1)i fs1 ,...,si si+1 ,...sn , (−1)n fs1 ,..,sn−1
∈ U1 ×V1 ⊆ m−1
A (V ).
i=1


Pn−1
i
n
Therefore, s1 fs2 ,...,sn ∈ U1 and mA
i=1 (−1) fs1 ,...,si si+1 ,...sn , (−1) fs1 ,..,sn−1 ∈
V1 . Now since the action map ac : G × A 7−→ A is continuous, there exists
open subsets U2 ⊆ G and V2 ⊆ A such that
(s1 , fs2 ,...,sn ) ∈ U2 × V2 ⊆ ac−1 (U1 ),
so s1 ∈ U2 and f ∈ βcts ({(s2 , ..., sn )}, V2 ). Now since
mA

n−1
X

!
(−1)i fs1 ,...,si si+1 ,...sn , (−1)n fs1 ,..,sn−1

∈ V1 ,

i=1

by continuity of mA , there exists open subsets U3 , V3 ⊆ A such that
!
n−1
X
(−1)i fs1 ,...,si si+1 ,...sn , (−1)n fs1 ,..,sn−1 ∈ V3 × U3 ⊆ m−1
A (V1 ),
i=1

so f ∈ βcts ({(s1 , ..., sn−1 )}, (−1)n U3 ) and
n−1
X
def
(−1)i fs1 ,...,si si+1 ,...sn = mA
i=1

n−1
X
−fs1 s2 ,...,sn ,
(−1)i fs1 ,...,si si+1 ,...sn

!
∈ V3 .

i=2

Again by continuity of mA , there exists open subsets U4 , V4 ⊆ A such that
!
n−1
X
−fs1 s2 ,...,sn ,
(−1)i fs1 ,...,si si+1 ,...sn ∈ U4 × V4 ⊆ m−1
A (V3 ),
i=2
Kayvan Nejabati Zenouz
Knejabat@ed.ac.uk

Compiled on 5th Apr, 2018 at 00:27

4 G-Modules and The Second Main Theorem

34

so f ∈ βcts ({(s1 s2 , ..., sn )}, −U4 ) and
n−1
X

def

(−1)i fs1 ,...,si si+1 ,...sn = mA

n−1
X
fs1 ,s2 s3 ,...,sn ,
(−1)i fs1 ,...,si si+1 ,...sn

i=2

!
∈ V4 .

i=3

And again by the continuity of mA , there exists open subsets U5 , V5 ⊆ A such
that
!
n−1
X
fs1 ,s2 s3 ,...,sn ,
(−1)i fs1 ,...,si si+1 ,...sn ∈ U5 × V5 ⊆ m−1
A (V4 ),
i=3

so f ∈ βcts ({(s1 , s2 s3 , ..., sn )}, U5 ) and
n−1
X

def

(−1)i fs1 ,...,si si+1 ,...sn = mA

fs1 ,s2 ,s3 s4 ,...,sn ,

i=3

n−1
X

!
(−1)i fs1 ,...,si si+1 ,...sn

∈ V5 .

i=4
def

Therefore, continuing in this form, and if we denote by sj = (s1 , ..., sj sj+1 , ..., sn ) ∈
Gn−1 , we find that open subset Uj+3 , Vj+3 ⊆ A such that
!
n−1
X
fs1 ,...,sj sj+1 ,...,sn ,
(−1)i fs1 ,...,si si+1 ,...sn ∈ Uj+3 × Vj+3 ⊆ m−1
A (Vj+2 ),
i=j

so fsj ∈ (−1)j U3+j and
n−1
X

def

(−1)i fs1 ,...,si si+1 ,...sn = mA

n−1
X
fs1 ,...,sj sj+1 ,...,sn ,
(−1)i fs1 ,...,si si+1 ,...sn

i=j

!
∈ Vj+3 ,

i=j

for j = 1, ..., n − 2; therefore, we find


j
n−1
f ∈ ∩n−2
Vn+1 )∩βcts ({(s2 , ..., sn )}, V2 )∩βcts ({(s1 , ..., sn−1 )}, (−1)n U3 ),
j=1 βcts ({sj }, (−1) U3+j ) ∩βcts ({sn−1 }, (−1)

which is an open subset of Mcts (Gn−1 , A)po , and if we take


j
n−1
g ∈ ∩n−2
Vn+1 )∩βcts ({(s2 , ..., sn )}, V2 )∩βcts ({(s1 , ..., sn−1 )}, (−1)n U3 ),
j=1 βcts ({sj }, (−1) U3+j ) ∩βcts ({sn−1 }, (−1)
−1
then one checks that g ∈ d−1
n (βcts ({s}, V )), which show that dn (βcts ({s}, V ))
is an open subset of Mcts (Gn−1 , A)po i.e., dn is a continuous map.

Kayvan Nejabati Zenouz
Knejabat@ed.ac.uk

Compiled on 5th Apr, 2018 at 00:27

4 G-Modules and The Second Main Theorem

35

Now since A is a Hausdorff space, by Lemma 2.14 Mcts (Gn , A)po is a
Hausdorff topological group for all n ≥ 0, so the subset containing the zero
map {0} ⊆ Mcts (Gn+1 , A)po is a closed subset. We define
def

def

n
(G, A)po = d−1
Zcts
n+1 ({0}) = Ker dn+1 ,

by the continuity of the homomorphism dn+1 , as proved in Lemma 4.11, the
n
subset Zcts
(G, A)po ⊆ Mcts (Gn , A)po is closed. In particular, if A is a compact
and evenly continuous with respect to Gn , then we deduce, using Lemma 2.6,
n
(G, A)po and Mcts (Gn , A)po are compact spaces for all n ≥ 0.
that both Zcts
Proposition 4.3. Let A be a G-module. Then for each n ≥ 1 we have a
map
n
n
cbn : Zcts
(G, A)po ×Mcts (Gn−1 , A)po −→ Zcts
(G, A)po given by (a, f ) 7−→ a+dn f,
n
which defines a continuous (for the product topology on Zcts
(G, A)po ×Mcts (Gn−1 , A)po )
n
right action of Mcts (Gn−1 , A)po on Zcts
(G, A)po .
def

n
Proof. Fix n ≥ 1. Note first that since dn+1 dn = 0 and Zcts
(G, A)po =
d−1
({0}),
the
map
cb
is
well-defined.
Now
It
follows
from
Lemma
2.6, i),
n
n+1
n−1
n
n
and the definition of Zcts (G, A)po , that both Zcts (G, A)po and Mcts (G , A)po
n
(G, A)po
are Hausdorff topological groups. Therefore, the group operation of Zcts
is a continuous map; since by Lemma 4.2 dn is a continuous map, we conclude that cbn is a continuous map; finally, it is clear that cbn is an action of
n
Mcts (Gn−1 , A)po on Zcts
(G, A)po .

Corollary 4.4. Let A be a G-module, fix n ≥ 1 and an element a ∈
n
(G, A)po , and denote by Stabn,A (a) the subset of elements of Mcts (Gn−1 , A)po
Zcts
fixing a. Then the subset Stabn,A (a) ⊆ Mcts (Gn−1 , A)po is a closed subgroup.
In particular, if A is a compact G-module, which is evenly continuous with
respect to Gn−1 , then Stabn,A (a) ⊆ Mcts (Gn−1 , A)po is a nonempty, compact,
and Huasdorff space.
def

n−1
Proof. Note we have Stabn,A (a) = Ker dn = Zcts
(G, A)po which is a closed
subgroup of Mcts (Gn−1 , A)po . Now if A is a compact G-module, which
is evenly continuous with respect to Gn−1 , then by Lemma 2.6 the space
n−1
Mecst (Gn−1 , A)po is compact, and since Zcts
(G, A)po is a closed subset of a
compact space, we conclude that it is also compact cf. [Wil04, p. 119, 17.5
Theorem. a)], now the proof of the corollary follows.

Kayvan Nejabati Zenouz
Knejabat@ed.ac.uk

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4 G-Modules and The Second Main Theorem

36

Corollary 4.5. Fix n ≥ 1. Let A be a compact G-module, which is evenly
n
continuous with respect to Gn−1 , and let a ∈ Zcts
(G, A)po . Let Orbn,A (a) be
the orbit of the element a under the action of Mcts (Gn−1 , A)po . If A , then
n
(G, A)po , is a nonempty, compact, and Hausdorff
the subset Orbn,A (a) ⊆ Zcts
subspace.
n
(G, A)po × Mcts (Gn−1 , A)po −→
Proof. By Proposition 4.3, the map cbn : Zcts
n
Zcts (G, A)po is continuous. Therefore, we obtain a continuous map

cb|{a}×M

n−1 ,A)po
cts (G

n
: {a} × Mcts (Gn−1 , A)po −→ Zcts
(G, A)po ,

by restriction of cbn to subspace {a} × Mcts (Gn−1 , A)po ⊆ Z n (G, A)po ×
Mcts (Gn−1 , A)po . In particular, the image of cb|{a}×A is Orbn,A (a). Therefore,
we have a surjective continuous map
cb|{a}×M

n−1 ,A)po
cts (G

: {a} × Mcts (Gn−1 , A)po −→ Orbn,A (a),

and cb|{a}×M (Gn−1 ,A)po (a, x) = cb|{a}×M (Gn−1 ,A)po (a, y) if and only if xy −1 ∈
cts
cts
Stabn,A (a). Hence we have a commutative diagram
cb|

{a} × Mcts (Gn−1 , A)po

{a}×Mcts (Gn−1 ,A)po

Orbn,A (a)

cb|

{a} × Mcts (Gn−1 , A)po /Stabn,A (a)

{a}×Mcts (Gn−1 ,A)po

Orbn,A (a).

By Corollary 4.4, Stabn,A (a) is a compact and Hausdorff topological
group. Now Stabn,A (a) acts continuously on the right of Mcts (Gn−1 , A)po ,
then Mcts (Gn−1 , A)po /Stabn,A (a) is a compact space being continuous image of the compact space Mcts (Gn−1 , A)po under the continuous open map
Mcts (Gn−1 , A)po  Mcts (Gn−1 , A)po /Stabn,A (a) cf. [Wil04, p. 119, Theorem
17.1], and Mcts (Gn−1 , A)po /Stabn,A (a) is Hausdorff cf. [Bre72, p. 38, Theorem
n
n
3.1, (1)]. Since Zcts
(G, A)po is a Hausdorff space, Orbn,A (a) ⊆ Zcts
(G, A)po
is a Hausdorff space cf. [Wil04, p. 87, 13.8 Theorem. a)]. Now
cb|{a}×M

n−1 ,A)po
cts (G

: {a} × Mcts (Gn−1 , A)po /Stabn,A (a) −→ Orbn,A (a)

is a continuous bijection from a compact space to a Hausdorff space, so
cb|{a}×M (Gn−1 ,A)po is a homeomorphism cf. [Wil04, p. 123, 17.14 Theorem];
cts
therefore
Mcts (Gn−1 , A)po /Stabn,A (a) ∼
= {a}×Mcts (Gn−1 , A)po /Stabn,A (a) ∼
= Orbn,A (a).
Kayvan Nejabati Zenouz
Knejabat@ed.ac.uk

Compiled on 5th Apr, 2018 at 00:27

4 G-Modules and The Second Main Theorem

37

The above shows Orbn,A (a) is a compact and Hausdorff space, it contains
a, so it is a nonempty, compact, and Hausdorff space.
Remark 4.6. A G-group homomorphism ϕ : A −→ A0 between G-groups
n
(G, A)po −→
A and A0 induces a continuous map, for each n ≥ 1, ϕ∗ : Zcts
n
0
n
n−1
Zcts (G, A )po , and for (a, f ) ∈ Z (G, A)po ×Mcts (G , A)po we have ϕ∗ cbn,A (a, f ) =
cbn,A0 (ϕ∗ (a), ϕ∗ (f )) i.e., ϕ induces the following commutative diagram
n
(G, A)po × Mcts (Gn−1 , A)po
Zcts

ϕ∗ ×ϕ∗

n
(G, A0 )po × Mcts (Gn−1 , A0 )po
Zcts
cbn,A0

cbn,A
ϕ∗

n
Zcts
(G, A)po

n
Zcts
(G, A0 )po ,

which follows from Lemma 2.1 and properties of ϕ.
Now the n-th continuous cohomology group of G with coefficients in
n
A can be defined as the quotient space of Zcts
(G, A)po , by the action of
n−1
Mcts (G , A)po i.e.,
def

n
n
Hcts
(G, A)po = Zcts
(G, A)po /Mcts (Gn−1 , A)po , for n ≥ 1.
n
(G, A)po is endowed with the quotient topology;
For each n ≥ 1, the set Hcts
in particular, the natural surjective homomorphism
n
n
πn,A : Zcts
(G, A)po −→ Hcts
(G, A)po ,
n
(G, A)po to its orbit under the action of
which sends an element a ∈ Zcts
n−1
n
(G, A)po
Mcts (G , A)po is a continuous open map cf. [Bre72, p. 37], also Hcts
is a topological group.
Remark 4.7. Fix n ≥ 1. Then by Remark 4.6 a G-group homomorphism
ϕ : A −→ A0 between G-groups A and A0 induces a commutative
n
Zcts
(G, A)po × Mcts (Gn−1 , A)po

ϕ∗ ×ϕ∗

n
Zcts
(G, A0 )po × Mcts (Gn−1 , A0 )po
cbn,A0

cbn,A
ϕ∗

n
Zcts
(G, A)po

n
Zcts
(G, A0 )po .

Therefore, ϕ induces a commutative diagram
n
Zcts
(G, A)po

ϕ∗

πn,A0

πn,A
n
Hcts
(G, A)po

Kayvan Nejabati Zenouz
Knejabat@ed.ac.uk

n
Zcts
(G, A0 )po

ϕ∗

n
Hcts
(G, A0 )po ,

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4 G-Modules and The Second Main Theorem

38

def

n
where for [a] ∈ Hcts
(G, A)po we set ϕ∗ ([a]) = [ϕ∗ (a)]. The map ϕ∗ is continuous, which one can check with similar calculation as done in Remark
2.22.
Similarly, if ψ : H −→ G is a continuous group homomorphism, one
obtains a commutative diagram of continuous maps
n
(G, A)po
Zcts

ψ∗

n
(H, A)po
Zcts

πn,A
n
Hcts
(G, A)po

πn,A
ψ

∗

∗

n
Hcts
(H, A)po ,
def

n
(G, A)po we set ψ ([a]) = [ψ ∗ (a)].
where for [a] ∈ Hcts

Lemma 4.8. Let A be a compact G-module, fix n ≥ 1 and assume A is
evenly continuous with respect to Gn−1 . Then the following facts hold:
n
i) The topological group Hcts
(G, A)po is Hausdorff.
n
(G, A)po is compact if and only if the topological
ii) The topological group Zcts
n
(G, A)po is compact; in particular if A is evenly continuous
groups Hcts
n
(G, A)po is a compact space.
with respect to Gn−1 , then Hcts

Proof. By Proposition 4.3 the cohomology map cbn is continuous. Since A
is compact G-module, which is evenly continuous with respect to Gn−1 , by
Lemma 2.3, i), and Lemma 2.6, the space Mcts (Gn−1 , A)po is a compact and
n
Hausdorff space, also Zcts
(G, A)po is a Hausdorff group. Now Mcts (Gn−1 , A)po
is a compact and Hausdorff topological group, which acts continuously on a
n
n
Hausdorff space Zcts
(G, A)po . Therefore, Zcts
(G, A)po is a Mcts (Gn−1 , A)po n−1
space with Mcts (G , A)po a compact space, which matches the definition of
[Bre72, p. 32, 1]. Now i) follows from [Bre72, p. 39, 3.1 Theorem, (1)], and
ii) follows from [Bre72, p. 39, 3.1 Theorem, (4)]; in particular, if A is evenly
n
continuous with respect to Gn−1 , then the space Zcts
(G, A)po is a compact
n
space, which implies that the space Hcts (G, A)po is compact.
Now we prove Theorem 4.1. The proof is similar the proof of Theorem 3.1,
and it is established using 3 lemmas; by showing there exists a well-defined
natural continuous bijection Θn .

Kayvan Nejabati Zenouz
Knejabat@ed.ac.uk

Compiled on 5th Apr, 2018 at 00:27

4 G-Modules and The Second Main Theorem

39

Lemma 4.9. Under the assumption of Theorem 4.1, there exists a welldefined continuous map
n
n
(G, Ar )po given by [a] 7−→ ([ϕr∗ (a)]),
(G, A)po −→ lim Hcts
Θn : Hcts
←−
r∈R

such that following diagram is commutative
n
Zcts
(G, A)po

θn

πn,A

n
limr∈R Zcts
(G, Ar )po
←−
lim

n
Hcts
(G, A)po

Θn

π

←
−r∈R n,Ar
n
limr∈R Hcts
(G, Ar )po .
←−

Proof. Since ϕrt : Ar −→ At for r ≥ t are continuous G-group homomorphisms, by Remark 4.7 they induce a commutative diagram of continuous
maps
ϕrt∗
n
n
Zcts
(G, Ar )po
Zcts
(G, At )po
π At

πn,Ar
n
Hcts
(G, Ar )po

ϕrt∗

n
Hcts
(G, At )po .

n
Note ϕrt are transition maps, and so {Hcts
(G, Ar )po , ϕrt∗ , R} is an inverse
system of topological groups.
Now for r ∈ R the continuous maps ϕr : A −→ Ar are G-group homomorphisms, so by Remark 4.7 they induces a commutative diagrams of
continuous maps
n
(G, A)po
Zcts

ϕr∗

πn,A

n
(G, Ar )po
Zcts
πn,Ar

n
Hcts
(G, A)po

ϕr∗

n
Hcts
(G, Ar )po ,

n
compatible with transition maps of the inverse systems {Zcts
(G, Ar )po , ϕrt∗ , R}
n
and {Hcts (G, Ar )po , ϕrt∗ , R}. Taking the inverse limit of the above diagram
we obtain a unique commutative diagram
n
Zcts
(G, A)po

θn

πn,A
n
Hcts
(G, A)po

Kayvan Nejabati Zenouz
Knejabat@ed.ac.uk

n
limr∈R Zcts
(G, Ar )po
←−
lim

Θn

π

←
−r∈R n,Ar
n
limr∈R Hcts
(G, Ar )po .
←−
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4 G-Modules and The Second Main Theorem

40

This shows that the map Θn is well-defined and is given by [a] 7−→ ([ϕr∗ (a)])
n
for [a] ∈ Hcts
(G, A)po . Now it follows from Lemma 2.4 that the map θn is
continuous (a homeomorphism in fact since A is compact), and limr∈R πn,Ar
←−
is a continuous map (being inverse limit of continuous maps), also since πn,A
is an open map, we find that Θn is a continuous map.
Lemma 4.10. Under the assumption of Theorem 4.1 the continuous map
n
n
(G, Ar )po ,
(G, A)po −→ lim Hcts
Θn : Hcts
←−
r∈R

defined in Lemma 4.9, is injective.
n
(G, A)po , and suppose
Proof. Let [a], [b] ∈ Hcts

Θn ([a]) = ([ϕr a]) = Θn ([b]) = ([ϕr b]).
Then for all r ∈ R there exists fr ∈ Mcts (Gn−1 , Ar )po such that ϕr a =
ϕr b + dn fr . Therefore, the set Sn,r = {fe ∈ Mcts (Gn−1 , Ar )po : ϕr a =
ϕr b + dn fe} is nonempty for all r ∈ R. Now Sn,r is equal to the coset
Stabn,Ar (ϕr b)+fr , where Stabn,Ar (ϕr b) is the stabiliser of ϕr b for the action of
n
(G, Ar )po . The set Stabn,Ar (ϕr b) ⊆ Mcts (Gn−1 , Ar )po
Mcts (Gn−1 , Ar )po on Zcts
is a closed, compact, and Hausdorff subgroup by Corollary 4.4. Therefore,
Sn,r = Stabn,Ar (ϕr b) + fr is nonempty, compact, and Hausdorff for each
r ∈ R. In particular, if r ≥ t and fe ∈ Sn,r , then
ϕt∗ (b)+dn ϕrt∗ (fe) = ϕrt∗ ϕr∗ (b)+dn ϕrt∗ (fe) = ϕrt∗ (ϕr∗ (b)+dn fe) = ϕrt∗ ϕr∗ (a) = ϕt∗ (a),
so ϕrt∗ (fe) ∈ Sn,t . Hence, the set {Sn,r , ϕrt∗ , R} is an inverse system of
nonempty, compact, and Hausdorff spaces. Therefore, limr∈R Sr is nonempty,
←−
compact, and Hausdorff cf. [RZ10, p. 4, Proposition 1.1.4]. Now for
f ∈ lim Sr ,→ lim Mcts (Gn−1 , Ar )po ∼
= Mcts (Gn−1 , A)po ,
←−
←−
r∈R

r∈R

one has
ϕr∗ (b + dn f ) = ϕr∗ (b) + dn ϕr∗ (f ) = ϕr∗ (a)
for all r ∈ R, so θn (b + dn f ) = θn (a), and since it follows from Lemma 2.4
that the map θn is injective, we must have b + dn f = a; therefore [a] = [b],
and this proves that the map Θn is injective.
Kayvan Nejabati Zenouz
Knejabat@ed.ac.uk

Compiled on 5th Apr, 2018 at 00:27

4 G-Modules and The Second Main Theorem

41

Lemma 4.11. Under the assumption of Theorem 4.1. Then the continuous
map
n
n
Θn : Hcts
(G, A)po −→ lim Hcts
(G, Ar )po ,
←−
r∈R

defined in Lemma 4.9, is surjective.
Proof. Recall we have surjective continuous maps
n
n
(G, Ar )po
πn,Ar : Zcts
(G, Ar )po −→ Hcts
n
(G, Ar )po , ϕrt∗ , R}
for r ∈ R, which are compatible with the transition maps of {Zcts
n
n
and {Hcts
(G, Ar )po , ϕrt∗ , R}. Let ([ar ]) ∈ limr∈R Hcts
(G, Ar )po . Then since
←−
Ar is compact and Hausdorff and evenly continuous with respect to Gn−1
−1
for each r ∈ R, the set πn,A
([ar ]) = Orbn,Ar (ar ) is a nonempty, compact,
r
and Hausdorff space for each r ∈ R by Corollary 4.5. In particular the set
{Orbn,Ar (ar ), ϕrt∗ , R} forms an inverse system of nonempty, compact, and
Hausdorff spaces. Therefore, limr∈R Orbn,Ar (ar ) is nonempty, compact, and
←−
Hausdorff cf. [RZ10, p. 4, Proposition 1.1.4]. Now for
n
a ∈ lim Orbn,Ar (ar ) ,→ lim Zcts
(G, Ar )po
←−
←−
r∈R

r∈R

n
we have πn,A θ−1 (a) ∈ Hcts
(G, A)po and Θn πn,A θn−1 (a) = ([ar ]); this proves
that the map Θn is surjective.

Now we provide the proof of the Theorem 4.1. We recall the statement
of Theorem 4.1 again.
Theorem 4.12. Let A a compact G-module and fix n ≥ 1. Assume A has
a presentation A = limr∈R Ar , where Ar is evenly continuous with respect to
←−
Gn−1 for all r ∈ R. Denote by ϕr : A −→ Ar the natural projections. Then
there exists a continuous bijection
n
n
Θn : Hcts
(G, A)po −→ lim Hcts
(G, Ar )po given by [a] 7−→ ([ϕr∗ (a)]).
←−
r∈R

In particular, if A is evenly continuous with respect to Gn , then Θn is a
homeomoprhism.

Kayvan Nejabati Zenouz
Knejabat@ed.ac.uk

Compiled on 5th Apr, 2018 at 00:27

4.1 One Application of The Second Main Theorem

42

Proof. By Lemma 4.9 there exists a commutative diagram of continuous
maps
Z n (G, A)co

θn

πn,A
n
(G, A)
Hcts

limr∈R Z n (G, Ar )co
←−
lim

Θn

π

←
−r∈R n,Ar
n
(G, Ar ),
limr∈R Hcts
←−

by Lemma 4.10 the map Θn is injective, and by Lemma 4.11 the map Θn
is surjective. Therefore, Θn is a continuous bijection. Using Lemma 4.8, i),
n
(G, Ar ) is a Hausdorff space for each r ∈ R. In
we deduce that limr∈R Hcts
←−
particular, if A is evenly continuous with respect to Gn , then using Lemma
n
(G, A) is a compact space; therefore,
2.6 and Lemma 4.8, ii), we have that Hcts
Θn is a continuous bijection from a compact space to a Hausdorff space, which
implies that Θn is a homeomorphism cf. [Wil04, p. 123, 17.14 Theorem].

4.1

One Application of The Second Main Theorem

One immediate application of the Theorem 4.1 is the following. Assume G
is a profinite group.
Corollary 4.13. Let A be a profinite G-module. Then A is an inverse limit
of finite discrete G-modules i.e., A has a presentation A ∼
A satis= lim
←−r∈R r
fying the assumptions of Theorem 4.1, and if Ar is evenly continuous with
respect to Gn−1 , then there exists a continuous bijection
n
n
Θn : Hcts
(G, A)po −→ lim Hcts
(G, Ar )po given by [a] 7−→ ([ϕr,∗ (a)]);
←−
r∈R

which is homeomorphism if A is evenly continuous with respect to Gn .
Proof. It follows from [RZ10, p. 170, Lemma 5.3.3, (c)] that A is an inverse
limit of finite discrete G-modules; therefore has a presentation A ∼
A
= lim
←−r∈R r
by finite discrete G-modules; and the corollary can be deduced from Theorem
4.1.
Recall the theorem [NSW08, p. 141, (2.7.5) Theorem], which states that
if G is a profinite group, and A is a compact G-module having a presentation
A = limn∈N An as a countable inverse limit of finite discrete G-modules An ,
←−
then there exist a natural exact sequence
1

lim1n∈N H i−1 (G, An )
←−

Kayvan Nejabati Zenouz
Knejabat@ed.ac.uk

i
Hcts
(G, A)

limn∈N H i (G, An )
←−

1.

Compiled on 5th Apr, 2018 at 00:27

REFERENCES

43

Therefore, Corollary 4.5 shows that if An is evenly continuous with respect
to Gi−1 for all n ∈ N, then lim1n∈N H i−1 (G, An ) = 1.
←−

References
[Bre72]

Glen E. Bredon. Introduction to compact transformation groups.
Academic Press, New York-London, 1972. Pure and Applied Mathematics, Vol. 46.

[Kel75]

John L. Kelley. General topology. Springer-Verlag, New YorkBerlin, 1975. Reprint of the 1955 edition [Van Nostrand, Toronto,
Ont.], Graduate Texts in Mathematics, No. 27.

[NSW08] Jürgen Neukirch, Alexander Schmidt, and Kay Wingberg. Cohomology of number fields, volume 323 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical
Sciences]. Springer-Verlag, Berlin, second edition, 2008.
[RZ10]

Luis Ribes and Pavel Zalesskii. Profinite groups, volume 40 of
3rd Series. A Series of Modern Surveys in Mathematics. SpringerVerlag, Berlin, second edition, 2010.

[Ser02]

Jean-Pierre Serre. Galois cohomology. Springer Monographs in
Mathematics. Springer-Verlag, Berlin, english edition, 2002. Translated from the French by Patrick Ion and revised by the author.

[Wil04]

Stephen Willard. General topology. Dover Publications, Inc., Mineola, NY, 2004. Reprint of the 1970 original [Addison-Wesley,
Reading, MA; MR0264581].

Kayvan Nejabati Zenouz
Knejabat@ed.ac.uk

Compiled on 5th Apr, 2018 at 00:27

