REIDEMEISTER NUMBER IN LEFSCHETZ FIXED POINT
THEORY
T. Mubeena
Assistant Professor, Department of Mathematics, University of Calicut. Malappuram (India).
E-mail: mubeenatc@uoc.ac.in, mubeenatc@gmail.com
ORCID:https://orcid.org/0000-0002-2493-5893
Reception: 21/08/2022 Acceptance: 05/09/2022 Publication: 29/12/2022
Suggested citation:
T. Mubeena (2022). Reidemeister Number in Lefschetz Fixed point theory. 3C TIC. Cuadernos de desarrollo aplicados a
las TIC, 11 (2), 61-70. https://doi.org/10.17993/3ctic.2022.112.61-70
https://doi.org/10.17993/3ctic.2022.112.61-70
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed. 41 Vol. 11 N.º 2 August - December 2022
61
ABSTRACT
Several interesting numbers such as the homotopy invariant numbers the Lefschets number
L
(
f
), the
Nielsen number
N
(
f
), fixed point index
i
(
X, f, U
)and the Reidemeister number
R
(
f
)play important
roles in the study of fixed point theorems. The Nielsen number gives more geometric information about
fixed points than other numbers. However the Nielsen number is hard to compute in general. To compute
the Nielsen number, Jiang related it to the Reidemeister number
R
(
fπ
)of the induced homomorphism
fπ
:
π1
(
X
)
→π1
(
X
)when
X
is a lens space or an H-space (Jian type space). For such spaces, either
N
(
f
)=0or
N
(
f
)=
R
(
f
)the Reidemeister number of
fπ
and if
R
(
f
)=
∞
then
N
(
f
)=0which
implies that
f
is homotopic to a fixed point free map. This is a review article to discuss how these
numbers are related in fixed point theory.
KEYWORDS
Twisted conjugacy, Reidemiester number, Lefschetz number, Nielsen number, Jiang space
https://doi.org/10.17993/3ctic.2022.112.61-70
1 INTRODUCTION
Let φ:G→Gbe an endomorphism of an infinite group G. One has an equivalence relation ∼φon G
defined as
x∼φy
if there exists a
g∈G
such that
y
=
gxφ
(
g
)
−1
. The equivalence classes are called the
Reidemeister classes of
φ
or
φ
-conjugacy classes. When
φ
is the identity, the Reidemeister classes of
φ
are the usual conjugacy classes. The Reidemeister classes of
φ
are the orbits of the action of
G
on itself
defined as
g.x
=
gxφ
(
g−1
). The Reidemeister classes of
φ
containing
x∈G
is denoted [
x
]
φ
or simply [
x
]
when
φ
is clear from the context. The set of all Reidemeister classes of
φ
is denoted by
R
(
φ
). We denote
by
R
(
φ
)the cardinality of
R
(
φ
)if it is finite and if it is infinite we set
R
(
φ
) :=
∞
and
R
(
φ
)is called
the Reidemeister number of
φ
on
G
. We say that
G
has the
R∞
-property if the Reidemeister number
of φis infinite for every automorphism φof G. If Ghas the R∞-property, we call Gan R∞-group.
The notion of Reidemeister number originated in the Nielsen–Reidemeister fixed point theory. See [
?
]
and the references therein. The problem of determining which classes of groups have
R∞
-property
is an area of active research. Many mathematicians have been trying to determine which class of
groups have the
R∞
-property using the internal structure of the class, such as Lie group structure,
C∗
-algebra structure or purely algebraic properties of the class. There is no particular way to solve
this problem, which makes it more difficult and interesting. If
X
is an H-space or a lens space, their
fundamental groups are abelian. The Reidemeister number of an endomorphism of an abelian group is
easily computable in many cases. In fact, if
G
is an abelian group,
R
(
φ
)is an abelian group under the
well defined operation [x][y]:=[xy], x, y ∈G.
The
R∞
-property does not behave well with respect to finite index subgroups and quotients as the
D∞
and any free group of rank
n>
1has the
R∞
-property although the infinite cyclic group and
finitely generated free abelian groups, which are quotients of free groups, do not (ref. [5], [4]). Thus
the
R∞
-property is not invariant under quasi isometry, that is it is not geometric among the class
of all finitely generated groups. The works of Levitt and Lustig [5] and Felshtyn [2] show that this
property is geometric in the class of non-elementary hyperbolic groups. It is been proved in [7] that
the
R∞
-property is geometric for the class of all finitely generated groups that are quasi-isometric
to irreducible lattices in real semisimple Lie groups with finite centre and finitely many connected
components. The R∞-property for irreducible lattices was proved in [6].
We have stated some results without proofs. For proofs and further readings, we refer the reader [1].
2 THE LEFSCHETZ NUMBER
Let
X
be a connected compact ANR and
f
:
X→X
a continuous map. We have seen fixed point
theorems like Brouwer fixed point theorem that states "Any map
f
:
Dn→Dn
has a fixed point"where
Dn
is the closed disk in
Rn
and the traditional Lefschetz fixed point theorem that states “If
L
(
f
)
=0
then
f
has a fixed point”, where
L
(
f
)is the Lefschetz number with respect to the rational homology.
Our statement of the Lefschetz fixed point theorem differs from the traditional one. We will prove
the theorem for
L
(
f,F
), where
F
is any field, because it is often easier to compute
L
(
f,F
)if the field
is chosen properly than it is to compute
L
(
f
), and the conclusion is for all maps homotopic to
f
rather than just for the map
f
. An important reason, however, was that the converse of the traditional
statement is -“If
L
(
f
)=0
,
then
f
is fixed point free”- and this is trivially false (we will see an example).
On the other hand, the converse of our statement is -“If
L
(
f,F
)=0for all fields
F
, then there is a fixed
point free map ghomotopic to f”. This is true. A proof can be seen in [1].
To define the Lefschetz number we need the following definitions. A subset
A
of a space
X
is called
aneighbourhood retract of
X
if there exists an open subset
U
of
X
containing
A
and a retraction of
U
onto
A
, i.e., a map
r
:
U→A
such that the restriction of
r
to
A
is the identity map. A space
X
is an
absolute neighbourhood retract (
ANR
) if it has the following property: If
X
imbeds into a separable
metric space
Y
, then
X
is a neighbourhood retract of
Y
. The ANR property is a topological invariant.
A compact space
X
is a compact
ANR
if and only if there exists an imbedding
i
:
X→I∞
such that
https://doi.org/10.17993/3ctic.2022.112.61-70
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed. 41 Vol. 11 N.º 2 August - December 2022
62
ABSTRACT
Several interesting numbers such as the homotopy invariant numbers the Lefschets number
L
(
f
), the
Nielsen number
N
(
f
), fixed point index
i
(
X, f, U
)and the Reidemeister number
R
(
f
)play important
roles in the study of fixed point theorems. The Nielsen number gives more geometric information about
fixed points than other numbers. However the Nielsen number is hard to compute in general. To compute
the Nielsen number, Jiang related it to the Reidemeister number
R
(
fπ
)of the induced homomorphism
fπ
:
π1
(
X
)
→π1
(
X
)when
X
is a lens space or an H-space (Jian type space). For such spaces, either
N
(
f
)=0or
N
(
f
)=
R
(
f
)the Reidemeister number of
fπ
and if
R
(
f
)=
∞
then
N
(
f
)=0which
implies that
f
is homotopic to a fixed point free map. This is a review article to discuss how these
numbers are related in fixed point theory.
KEYWORDS
Twisted conjugacy, Reidemiester number, Lefschetz number, Nielsen number, Jiang space
https://doi.org/10.17993/3ctic.2022.112.61-70
1 INTRODUCTION
Let φ:G→Gbe an endomorphism of an infinite group G. One has an equivalence relation ∼φon G
defined as
x∼φy
if there exists a
g∈G
such that
y
=
gxφ
(
g
)
−1
. The equivalence classes are called the
Reidemeister classes of
φ
or
φ
-conjugacy classes. When
φ
is the identity, the Reidemeister classes of
φ
are the usual conjugacy classes. The Reidemeister classes of
φ
are the orbits of the action of
G
on itself
defined as
g.x
=
gxφ
(
g−1
). The Reidemeister classes of
φ
containing
x∈G
is denoted [
x
]
φ
or simply [
x
]
when
φ
is clear from the context. The set of all Reidemeister classes of
φ
is denoted by
R
(
φ
). We denote
by
R
(
φ
)the cardinality of
R
(
φ
)if it is finite and if it is infinite we set
R
(
φ
) :=
∞
and
R
(
φ
)is called
the Reidemeister number of
φ
on
G
. We say that
G
has the
R∞
-property if the Reidemeister number
of φis infinite for every automorphism φof G. If Ghas the R∞-property, we call Gan R∞-group.
The notion of Reidemeister number originated in the Nielsen–Reidemeister fixed point theory. See [
?
]
and the references therein. The problem of determining which classes of groups have
R∞
-property
is an area of active research. Many mathematicians have been trying to determine which class of
groups have the
R∞
-property using the internal structure of the class, such as Lie group structure,
C∗
-algebra structure or purely algebraic properties of the class. There is no particular way to solve
this problem, which makes it more difficult and interesting. If
X
is an H-space or a lens space, their
fundamental groups are abelian. The Reidemeister number of an endomorphism of an abelian group is
easily computable in many cases. In fact, if
G
is an abelian group,
R
(
φ
)is an abelian group under the
well defined operation [x][y]:=[xy], x, y ∈G.
The
R∞
-property does not behave well with respect to finite index subgroups and quotients as the
D∞
and any free group of rank
n>
1has the
R∞
-property although the infinite cyclic group and
finitely generated free abelian groups, which are quotients of free groups, do not (ref. [5], [4]). Thus
the
R∞
-property is not invariant under quasi isometry, that is it is not geometric among the class
of all finitely generated groups. The works of Levitt and Lustig [5] and Felshtyn [2] show that this
property is geometric in the class of non-elementary hyperbolic groups. It is been proved in [7] that
the
R∞
-property is geometric for the class of all finitely generated groups that are quasi-isometric
to irreducible lattices in real semisimple Lie groups with finite centre and finitely many connected
components. The R∞-property for irreducible lattices was proved in [6].
We have stated some results without proofs. For proofs and further readings, we refer the reader [1].
2 THE LEFSCHETZ NUMBER
Let
X
be a connected compact ANR and
f
:
X→X
a continuous map. We have seen fixed point
theorems like Brouwer fixed point theorem that states "Any map
f
:
Dn→Dn
has a fixed point"where
Dn
is the closed disk in
Rn
and the traditional Lefschetz fixed point theorem that states “If
L
(
f
)
=0
then
f
has a fixed point”, where
L
(
f
)is the Lefschetz number with respect to the rational homology.
Our statement of the Lefschetz fixed point theorem differs from the traditional one. We will prove
the theorem for
L
(
f,F
), where
F
is any field, because it is often easier to compute
L
(
f,F
)if the field
is chosen properly than it is to compute
L
(
f
), and the conclusion is for all maps homotopic to
f
rather than just for the map
f
. An important reason, however, was that the converse of the traditional
statement is -“If
L
(
f
)=0
,
then
f
is fixed point free”- and this is trivially false (we will see an example).
On the other hand, the converse of our statement is -“If
L
(
f,F
)=0for all fields
F
, then there is a fixed
point free map ghomotopic to f”. This is true. A proof can be seen in [1].
To define the Lefschetz number we need the following definitions. A subset
A
of a space
X
is called
aneighbourhood retract of
X
if there exists an open subset
U
of
X
containing
A
and a retraction of
U
onto
A
, i.e., a map
r
:
U→A
such that the restriction of
r
to
A
is the identity map. A space
X
is an
absolute neighbourhood retract (
ANR
) if it has the following property: If
X
imbeds into a separable
metric space
Y
, then
X
is a neighbourhood retract of
Y
. The ANR property is a topological invariant.
A compact space
X
is a compact
ANR
if and only if there exists an imbedding
i
:
X→I∞
such that
https://doi.org/10.17993/3ctic.2022.112.61-70
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed. 41 Vol. 11 N.º 2 August - December 2022
63
i(X)is a neighbourhood retract of I∞, where
I∞=
n∈N[−1
n,1
n]
is the infinite Hilbert cube with the metric
d
(
xn,y
n
)=(
|xn2−yn2|
)
1/2
.
I∞
itself is an ANR by
definition. Since
i
:
Dn→I∞
defined by (
x1, ...xn
)
→
(
x1,x
2/
2
, ...xn/n,
0
,
0
....
)is an imbedding such
that
i
(
Dn
)is a retraction of
I∞
, the n-cell
Dn
is an ANR. Any locally finite polyhedron is an ANR.
Open subsets of an ANR and a neighbourhood retract of an ANR are also an ANR.
Throughout this note we will assume
X
is a connected compact ANR. Note that any compact
ANR has only countably many connected components with each is open and an ANR. For a compact
ANR, the homology
H∗
(
X, F
)is a finitary graded
F
-vector space and any map
f
:
X→X
induces a
morphism
f∗
:
H∗
(
X, F
)
→H∗
(
X, F
)between the homology groups which is a morphism of finitary
graded vector space over F.
Definition 1.
Let
f
:
X→X
be a map on
X
,
F
be a field. The Lefschetz number of
f
over
F
is defined
to be the number:
L(f,F) := (−1)qTr(fq)
We denote L(f)=L(f,Q).
We state the Lefschets fixed point theorem without proof.
Theorem 1
(Lefschetz Fixed Point Theorem ( [1]))
.
If
X
is a compact ANR and
f
:
X→X
is a map
such that L(f,F)=0for some field F, then every map homotopic to fhas a fixed point.
For any field
F
the homology and cohomology are isomorphic and the induced morphism is the
transpose of f∗so we can define the Lefschetz number using the cohomolgy too.
Observe that: (1) for the identity map 1Xof X, the Lefschetz number
L(1X)=
q
(−1)qTr(1q)=(−1)qdim(Hq(X, q)) = (−1)qbq=χ(X)
where
χ
(
X
)is the Euler characteristic and
bq
is the
qth
Betti number of
X
. (2) Since homotopic maps
induce the same homomorphism on the homology groups, if
f
:
X→X
any continuous map and if
g
is
homotopic to fthen L(g, F)=L(f,F)for all field F.
A space
X
is said to have the fixed point property if every continuous self map on
X
has a fixed point.
Thus a contractible compact ANR
X
has fixed point property since
Hq
(
X, Q
)=0
,∀q
=0and
H0
=
Q
,
thus
L
(
f
)=1(since every map on a path connected space induces the identity morphism on
H0
)
implies
f
has a fixed point by the Lefschetz fixed point theorem. For
X
=
Sn
, the
n−
sphere,
χ
(
X
)=0
whenever
n
is odd. Thus the converse of the traditional fixed point theorem is false. Brouwer Fixed
point theorem is an immediate consequence of the Lefschetz fixed point theorme, for; let
f
:
Dn→Dn
is any map. Since
I∞
is a contractible compact ANR and since retract of a space with fixed point
property has the fixed point property, fhas a fixed point.
3 Index for ANRs
For
X
, a compact ANR, a map
f
:
X→X
, and an open set
U
of
X
without fixed points of
f
on its
boundary it is possible to associate a number
i
(
X, f, U
), the index of
f
on
U
. We define the index for
such triples.
3.1 The axioms for an index
Let
CA
denote the collection of all connected compact ANR spaces
X
where the Lefschetz number is
defined since H∗(X, q)is finitary. We define index for triples i(X, f, U)with the following propeties:
https://doi.org/10.17993/3ctic.2022.112.61-70
1. X∈C
A,
2. f:X→Xis a map,
3. Uis open in X,
4. there are no fixed points of fon the boundary of U.
The collection of such triples (
X, f, U
)is denoted by
C
. Observe that (
X,f,X
)
,
(
X, f, φ
)satisfy these
properties.
A(fixed point)index on CAis a function i:C→Qwhich satisfies the following axioms:
1.
(Localization). If (
X, f, U
)
∈C
and
g
:
X→X
is a map such that
g
(
x
)=
f
(
x
)for all
x∈¯
U
(the
closure of U), then
i(X, f, U)=i(X,g,U).
2.
(Homotopy). For
X∈C
A
and
H
:
X×I→X
a homotopy, define
ft
:
X→X
by
ft
(
x
)=
H
(
x, t
).
If (X, ft,U)∈Cfor all t∈I, then
i(X, f0,U)=i(X, f1,U).
3.
(Additivity). If (
X, f, U
)
∈C
and
U1, ...Un
is a set of mutually disjoint open subsets of
U
such
that f(x)=xfor all
x∈U−
n
j=1
Uj,
then
i(X, f, U)=
n
j=1
i(X, f, Uj).
4. (Normalization). If X∈C
Aand f:X→Xis a map, then
i(X,f,X)=L(f).
5.
(Commutativity) If
X, Y ∈C
A
and
f
:
X→Y,g
:
Y→X
are maps such that (
X,gf,U
)
∈C
,
then
i(X,gf,U)=i(Y,fg,g
−1(U)).
The localization axiom 1 obviously makes the definition of the index “local” in the sense that
i
(
X,f,U
)
is not affected by the behaviuor of
f
outside of
¯
U
. The normalization axiom 4 connects the index to
Lefschetz theory. The homotopy and commutativity axioms are generalizations of properties of the
Lefschetz number.
Lemma 1.
If there is an index
i
on
CA
and if (
X, f, U
)
∈C,
such that
i
(
X, f, U
)
=0, then
f
has a
fixed point in U.
Proof. Note that
i
(
X, f, φ
)=
i
(
X, f, φ
)+
i
(
X, f, φ
)by additivity 3 (
U
=
U1
=
U2
=
φ
). Thus
i
(
X, f, φ
)=0since it is rational. Suppose
f
(
x
)
=
x
on
U
. Then we can apply additivity 3 for the given
open set
U
and
U1
=
φ
, and we get
i
(
X, f, U
)=
i
(
X, f, φ
)=0. Which is a contradiction. Hence
f
has
a fixed point in U.
Lemma 2.
Assume there is an index on
C
and if
X∈C
A,f
:
X→X
a map such that
L
(
f
)
=0then
every map homotopic to fhas a fixed point.
Proof. Let
g
be any map homotopic to
f
; then
L
(
g
)=
L
(
f
)
=0. By the normalization axiom,
i(X,g,X)=L(g)=0, so ghas a fixed point in Xby 1.
https://doi.org/10.17993/3ctic.2022.112.61-70
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed. 41 Vol. 11 N.º 2 August - December 2022
64
i(X)is a neighbourhood retract of I∞, where
I∞=
n∈N[−1
n,1
n]
is the infinite Hilbert cube with the metric
d
(
xn,y
n
)=(
|xn2−yn2|
)
1/2
.
I∞
itself is an ANR by
definition. Since
i
:
Dn→I∞
defined by (
x1, ...xn
)
→
(
x1,x
2/
2
, ...xn/n,
0
,
0
....
)is an imbedding such
that
i
(
Dn
)is a retraction of
I∞
, the n-cell
Dn
is an ANR. Any locally finite polyhedron is an ANR.
Open subsets of an ANR and a neighbourhood retract of an ANR are also an ANR.
Throughout this note we will assume
X
is a connected compact ANR. Note that any compact
ANR has only countably many connected components with each is open and an ANR. For a compact
ANR, the homology
H∗
(
X, F
)is a finitary graded
F
-vector space and any map
f
:
X→X
induces a
morphism
f∗
:
H∗
(
X, F
)
→H∗
(
X, F
)between the homology groups which is a morphism of finitary
graded vector space over F.
Definition 1.
Let
f
:
X→X
be a map on
X
,
F
be a field. The Lefschetz number of
f
over
F
is defined
to be the number:
L(f,F) := (−1)qTr(fq)
We denote L(f)=L(f,Q).
We state the Lefschets fixed point theorem without proof.
Theorem 1
(Lefschetz Fixed Point Theorem ( [1]))
.
If
X
is a compact ANR and
f
:
X→X
is a map
such that L(f,F)=0for some field F, then every map homotopic to fhas a fixed point.
For any field
F
the homology and cohomology are isomorphic and the induced morphism is the
transpose of f∗so we can define the Lefschetz number using the cohomolgy too.
Observe that: (1) for the identity map 1Xof X, the Lefschetz number
L(1X)=
q
(−1)qTr(1q)=(−1)qdim(Hq(X, q)) = (−1)qbq=χ(X)
where
χ
(
X
)is the Euler characteristic and
bq
is the
qth
Betti number of
X
. (2) Since homotopic maps
induce the same homomorphism on the homology groups, if
f
:
X→X
any continuous map and if
g
is
homotopic to fthen L(g, F)=L(f,F)for all field F.
A space
X
is said to have the fixed point property if every continuous self map on
X
has a fixed point.
Thus a contractible compact ANR
X
has fixed point property since
Hq
(
X, Q
)=0
,∀q
=0and
H0
=
Q
,
thus
L
(
f
)=1(since every map on a path connected space induces the identity morphism on
H0
)
implies
f
has a fixed point by the Lefschetz fixed point theorem. For
X
=
Sn
, the
n−
sphere,
χ
(
X
)=0
whenever
n
is odd. Thus the converse of the traditional fixed point theorem is false. Brouwer Fixed
point theorem is an immediate consequence of the Lefschetz fixed point theorme, for; let
f
:
Dn→Dn
is any map. Since
I∞
is a contractible compact ANR and since retract of a space with fixed point
property has the fixed point property, fhas a fixed point.
3 Index for ANRs
For
X
, a compact ANR, a map
f
:
X→X
, and an open set
U
of
X
without fixed points of
f
on its
boundary it is possible to associate a number
i
(
X, f, U
), the index of
f
on
U
. We define the index for
such triples.
3.1 The axioms for an index
Let
CA
denote the collection of all connected compact ANR spaces
X
where the Lefschetz number is
defined since H∗(X, q)is finitary. We define index for triples i(X, f, U)with the following propeties:
https://doi.org/10.17993/3ctic.2022.112.61-70
1. X∈C
A,
2. f:X→Xis a map,
3. Uis open in X,
4. there are no fixed points of fon the boundary of U.
The collection of such triples (
X, f, U
)is denoted by
C
. Observe that (
X,f,X
)
,
(
X, f, φ
)satisfy these
properties.
A(fixed point)index on CAis a function i:C→Qwhich satisfies the following axioms:
1.
(Localization). If (
X, f, U
)
∈C
and
g
:
X→X
is a map such that
g
(
x
)=
f
(
x
)for all
x∈¯
U
(the
closure of U), then
i(X, f, U)=i(X,g,U).
2.
(Homotopy). For
X∈C
A
and
H
:
X×I→X
a homotopy, define
ft
:
X→X
by
ft
(
x
)=
H
(
x, t
).
If (X, ft,U)∈Cfor all t∈I, then
i(X, f0,U)=i(X, f1,U).
3.
(Additivity). If (
X, f, U
)
∈C
and
U1, ...Un
is a set of mutually disjoint open subsets of
U
such
that f(x)=xfor all
x∈U−
n
j=1
Uj,
then
i(X, f, U)=
n
j=1
i(X, f, Uj).
4. (Normalization). If X∈C
Aand f:X→Xis a map, then
i(X,f,X)=L(f).
5.
(Commutativity) If
X, Y ∈C
A
and
f
:
X→Y,g
:
Y→X
are maps such that (
X,gf,U
)
∈C
,
then
i(X,gf,U)=i(Y,fg,g
−1(U)).
The localization axiom 1 obviously makes the definition of the index “local” in the sense that
i
(
X,f,U
)
is not affected by the behaviuor of
f
outside of
¯
U
. The normalization axiom 4 connects the index to
Lefschetz theory. The homotopy and commutativity axioms are generalizations of properties of the
Lefschetz number.
Lemma 1.
If there is an index
i
on
CA
and if (
X, f, U
)
∈C,
such that
i
(
X, f, U
)
=0, then
f
has a
fixed point in U.
Proof. Note that
i
(
X, f, φ
)=
i
(
X, f, φ
)+
i
(
X, f, φ
)by additivity 3 (
U
=
U1
=
U2
=
φ
). Thus
i
(
X, f, φ
)=0since it is rational. Suppose
f
(
x
)
=
x
on
U
. Then we can apply additivity 3 for the given
open set
U
and
U1
=
φ
, and we get
i
(
X, f, U
)=
i
(
X, f, φ
)=0. Which is a contradiction. Hence
f
has
a fixed point in U.
Lemma 2.
Assume there is an index on
C
and if
X∈C
A,f
:
X→X
a map such that
L
(
f
)
=0then
every map homotopic to fhas a fixed point.
Proof. Let
g
be any map homotopic to
f
; then
L
(
g
)=
L
(
f
)
=0. By the normalization axiom,
i(X,g,X)=L(g)=0, so ghas a fixed point in Xby 1.
https://doi.org/10.17993/3ctic.2022.112.61-70
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed. 41 Vol. 11 N.º 2 August - December 2022
65
The last Lemma 1 makes the important point that Index Theory is more powerful than Lefschetz
Theory in the sense that the existance of a function on
CA
satisfying just two of the axioms of an index,
namely additivity 3 and normalization 4, is enough to imply that the Lefschetz Fixed Point Theorem 1
is true for all maps on spaces in a collection Con which an index is defined.
Example 1.
Let
X
be a compact connected ANR and (
X, f, U
)
∈C
where
f
is a constant map say
f(x)=x0,∀x∈X. Then
i(X, f, U)=0if x0/∈U
1if x0∈U
For, if
x0/∈U
. Then by additivity 3 for the given
U
and
U1
=
φ
,
i
(
X, f, U
)=
i
(
X, f, φ
)=0.
Now suppose
x0∈U
. Let
Y
=
{x0}
the singleton space and
g
:
X→Y,h
:
Y→X
be the maps
x→ x0,h
=1
Y
respectively. Then
i
(
X, f, U
)=
i
(
X,hg,U
)=
i
(
Y, gh, Y
)=
L
(
gh
)=
L
(1
Y
)by the
commutativity 5 and normalization 4 axioms. Since any map
f
on a path connected space induces the
identity on the homology group
H0
and since
Y
is path connected and higher homology groups are
trivial,
L
(1
Y
)=1. Hence
i
(
X,f,U
)=1in this case. The following theorem tells us that such an index
exists on CA. Details can be seen in chapters IV and V of [1].
Theorem 2.
For the collection
CA
of all connected compact ANR, there is a unique index defined on it
satisfying all the five axioms.
Now we are ready to define an index on the Nielsen classes of a map f:X→X.
4 The Nielsen Number
For
X
, a compact ANR, and a map
f
:
X→X
, we shall define a non-negative integer
N
(
f
), called the
Nielsen number of f. The Nielsen number is a lower bound for the number of fixed points of f.
4.1 Nielsen Classes
Assume that the set
F ixf
of all fixed points of
f
is non-empty. Two points
x0,x
1∈F ixf
are
f
-
equivalent if there is a path
c
:
I→X
from
x0
to
x1
such that
c
and
f◦c
are homotopic with respect
to the end points. This relation defines an equivalence relation on
F ixf
. The equivalence classes are
called Nielsen classes or fixed point classes of
f
. It is known that the set of Nielsen classes of a map
f
on a connected, compact, ANR Xis finite.
Theorem 3. A map f:X→Xon a compact ANR has only finitely many Nielsen classes.
Hence we will denote the set of fixed point classes by N(f) := {F1,F
2, ..., Fn}.
4.2 Nielsen Number
Let
f
:
X→X
be a map with fixed point classes
F1, ..., Fn
. Then for each
j
=1
, .., n
, there is an open
set
Uj⊂X
such that
Fj⊂Uj
and
¯
Uj∩F ixf
=
Fj
. Let
i
be the index on
CA
. Note that (
X,f,U
j
)
∈C
.
Define the index
i
(
Fj
)of the fixed point class
Fj
by
i
(
Fj
)=
i
(
X, f, Uj
). This definition is independent
of the choice of the open set
Uj⊆X
such that
Fj⊆Uj
and
¯
Uj∩F ixf
=
Fj
because, suppose
U, V
are two such open sets. If
x∈U−U∩V
then since
x
belongs to
U, x /∈Fk
for
k
=
j
, while
x/∈V
,
so
x/∈Fj
. Thus
x/∈F ixf
. By the additivity axiom,
i
(
X, f, U
)=
i
(
X, f, U ∩V
). The same reasoning
implies
i
(
X, f, V
)=
i
(
X,f,U ∩V
). A fixed point class
F
of
f
is said to be essential if
i
(
F
)
=0and
inessential otherwise. The Nielsen number
N
(
f
)of the map
f
is defined to be the number of fixed
point classes of fthat are essential.
A fixed point theorem with this number is that:
Theorem 4. Any continuous map fon a connected compact ANR has at least N(f)fixed points.
Theorem 5. Let f, g :X→Xbe homotopic maps; then N(f)=N(g).
https://doi.org/10.17993/3ctic.2022.112.61-70
Thus any continuous function ghomotopic to fhas at least N(f)fixed points.
The example 1 is an example with only one fixed point class and it is essential, i.e, N(f)=1.
Computing
N
(
f
)is difficult in general, in some cases it can be computed via the Reidemeister
number
R
(
f
)by knowing
L
(
f
)and
fπ
, the induced homomorphism on the fundamental group of
X
.
The main tool to compute the Nielsen number is the Jiang subgroup of the fundamental group. Before
going to the computation method of Nielsen number we will see how it is related to the Reidemeister
number R(f).
5 The Reidemeister Number
Let
f
:
X→X
be a map on a connected compact ANR and let
F ixf
=
{x∈X
:
f
(
x
)=
x}
. Let
p
:
X→X
be the universal covering of
X
and
˜
f
:
X→
X
be a lifting of
f
, ie.
p◦˜
f
=
f◦p
. Two
liftings
˜
f
and
˜
f
are called conjugate if there is a
γ∈
Γ=
π1
(
X
)such that
˜
f
=
γ˜
fγ−1
. Note that if
˜
f
is a lift of
f
and
γ∈
Γ, then
p
(
F ix ˜
f
)=
p
(
F ixγ ˜
fγ−1
)and
p
(
F ix ˜
f
)=
p
(
F ix ˜
f
), then
˜
f
=
γ˜
fγ−1
for some
γ∈
Γ. This is an equivalence relation on the set
F ixf
=
[˜
f]p
(
F ix ˜
f
), ie.
F ixf
is a disjoint
union of projections of fixed points of lifts from distinct lifting classes. The subset
p
(
F ix ˜
f
)
⊆F ixf
is called the fixed point class of
f
determined by the lifting class [
˜
f
]. Note that for any point
y∈
X
,
since (
f◦p
)
π
(
π1
(
X
))
⊂pπ
(
π1
(
X
)), there is a unique map
˜
f
such that
p◦˜
f
=
f◦p
and
˜
f
(
y
)=
y
. In
particular, any fixed point of fis a projection of a fixed point of some lift ˜
fof f.
Now we define the Reidemeister number of a group homomorphism
ϕ
:
G→G
. Let
ϕ
:
G→G
be a
group homomorphism on a group
G
. One has an action of
G
on itself given by
g.x
=
gxϕ
(
g
)
−1
. Two
elements
x, y ∈G
are said to be
ϕ−
twisted conjugate,
x∼ϕy
, if they are in the same orbit of this
action. The orbits are called the
ϕ
-twisted conjugacy classes or the Reidemeister classes and
R
(
ϕ
), the
number of Reidemeister classes is called the Reidemeister number of
ϕ
. If
ϕ
=
Id
then
R
(
ϕ
)coincides
with the number of conjugacy classes of G.
By fixing a lift
˜
f
of
f
and
α∈
Γ
π1
(
X
)
,˜x∈
X
, we obtain a unique element
˜α∈
Γsuch that
˜α˜
f
(
˜x
)=
˜
f
(
α
(
˜x
)). Thus we obtain a homomorphism
φ
:
π1
(
X
)
→π1
(
X
)such that
˜
fα
=
φ
(
α
)
˜
f,∀α∈
Γ.
Also once we fix a lift ˜
fof f, then every lift is of the form α˜
ffor some α∈Γ. Let α, β ∈Γ. Then
[α˜
f]=[β˜
f]⇐⇒ β˜
f=γα ˜
fγ−1
for some γ∈Γ. ie.
⇐⇒ β˜
f=γαφ(γ−1)˜
f
By the uniqueness of lifts, we have [
α˜
f
]=[
β˜
f
]
⇐⇒ β
=
γαφ
(
γ−1
)for some
γ∈
Γ. Furthermore,
by choosing appropriate base point,
φ
can be identified with the induced homomorphism
fπ
on Γ.
From now on, we write
R
(
f
)for
R
(
fπ
). It follows from the above definition that there is a one-one
correspondence between the lifting classes of fand the Reidemeister classes of fπ.
Remark 1.
If we choose a different lifting
˜
f
and thus a different homomorphism
φ
, we get a bijection
between the
φ
-Reidemeister classes and the
φ
-Reidemeister classes so that the cardianality of such sets
is a constant.
5.1 Relationship with Nielsen number
Suppose
x1,x
2∈F ixf
are in the same Nielsen class, ie. there exists a path
c
:
I→X
from
x1
to
x2
such
that
f◦c
and
c
are homotopic relative to the end points. Let
˜
f
be a lift of
f
and
˜x1∈F ix ˜
f
such that
p
(
˜x1
)=
x1
. Lift
c
to a path
˜c
starting from
˜x1
and ending at some
˜x2
in
˜
X
. Then
˜
f◦˜c
projects onto
f◦c
which is homotopic to
c
. Thus
˜
f◦˜c
also ends at
˜x2
. Hence
˜
f
(
˜x2
)=
˜x2
. In otherwords, they belong to
the same lifting class. Conversely, let
˜x1,˜x2∈F ix ˜
f
such that
p
(
˜x1
)=
x1
=
x2
=
p
(
˜x2
). Let
˜c
:
I→
X
be a path from
˜x1
to
˜x2
. Then
c
=
p◦˜c
is a path from
x1
to
x2
in
X
and
p
(
˜
f◦˜c
)=
f◦p◦˜c
=
f◦c
,
ie.
˜
f◦˜c
projects onto
f◦c
. In fact, the loop
˜c
(
˜
f◦˜c
)
−1
projects to the loop
c
(
f◦c
)
−1
. Since
X
is
simply-connected, the former loop is trivial in
π1
(
˜
X
)and thus the later loop is homotopic to the trivial
loop, ie.
c∼f◦c
. That is
x1
and
x2
are in the same Nielsen class. This shows that there is a one-to-one
https://doi.org/10.17993/3ctic.2022.112.61-70
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed. 41 Vol. 11 N.º 2 August - December 2022
66
The last Lemma 1 makes the important point that Index Theory is more powerful than Lefschetz
Theory in the sense that the existance of a function on
CA
satisfying just two of the axioms of an index,
namely additivity 3 and normalization 4, is enough to imply that the Lefschetz Fixed Point Theorem 1
is true for all maps on spaces in a collection Con which an index is defined.
Example 1.
Let
X
be a compact connected ANR and (
X, f, U
)
∈C
where
f
is a constant map say
f(x)=x0,∀x∈X. Then
i(X, f, U)=0if x0/∈U
1if x0∈U
For, if
x0/∈U
. Then by additivity 3 for the given
U
and
U1
=
φ
,
i
(
X, f, U
)=
i
(
X, f, φ
)=0.
Now suppose
x0∈U
. Let
Y
=
{x0}
the singleton space and
g
:
X→Y,h
:
Y→X
be the maps
x→ x0,h
=1
Y
respectively. Then
i
(
X, f, U
)=
i
(
X,hg,U
)=
i
(
Y, gh, Y
)=
L
(
gh
)=
L
(1
Y
)by the
commutativity 5 and normalization 4 axioms. Since any map
f
on a path connected space induces the
identity on the homology group
H0
and since
Y
is path connected and higher homology groups are
trivial,
L
(1
Y
)=1. Hence
i
(
X,f,U
)=1in this case. The following theorem tells us that such an index
exists on CA. Details can be seen in chapters IV and V of [1].
Theorem 2.
For the collection
CA
of all connected compact ANR, there is a unique index defined on it
satisfying all the five axioms.
Now we are ready to define an index on the Nielsen classes of a map f:X→X.
4 The Nielsen Number
For
X
, a compact ANR, and a map
f
:
X→X
, we shall define a non-negative integer
N
(
f
), called the
Nielsen number of f. The Nielsen number is a lower bound for the number of fixed points of f.
4.1 Nielsen Classes
Assume that the set
F ixf
of all fixed points of
f
is non-empty. Two points
x0,x
1∈F ixf
are
f
-
equivalent if there is a path
c
:
I→X
from
x0
to
x1
such that
c
and
f◦c
are homotopic with respect
to the end points. This relation defines an equivalence relation on
F ixf
. The equivalence classes are
called Nielsen classes or fixed point classes of
f
. It is known that the set of Nielsen classes of a map
f
on a connected, compact, ANR Xis finite.
Theorem 3. A map f:X→Xon a compact ANR has only finitely many Nielsen classes.
Hence we will denote the set of fixed point classes by N(f) := {F1,F
2, ..., Fn}.
4.2 Nielsen Number
Let
f
:
X→X
be a map with fixed point classes
F1, ..., Fn
. Then for each
j
=1
, .., n
, there is an open
set
Uj⊂X
such that
Fj⊂Uj
and
¯
Uj∩F ixf
=
Fj
. Let
i
be the index on
CA
. Note that (
X,f,U
j
)
∈C
.
Define the index
i
(
Fj
)of the fixed point class
Fj
by
i
(
Fj
)=
i
(
X, f, Uj
). This definition is independent
of the choice of the open set
Uj⊆X
such that
Fj⊆Uj
and
¯
Uj∩F ixf
=
Fj
because, suppose
U, V
are two such open sets. If
x∈U−U∩V
then since
x
belongs to
U, x /∈Fk
for
k
=
j
, while
x/∈V
,
so
x/∈Fj
. Thus
x/∈F ixf
. By the additivity axiom,
i
(
X, f, U
)=
i
(
X, f, U ∩V
). The same reasoning
implies
i
(
X, f, V
)=
i
(
X,f,U ∩V
). A fixed point class
F
of
f
is said to be essential if
i
(
F
)
=0and
inessential otherwise. The Nielsen number
N
(
f
)of the map
f
is defined to be the number of fixed
point classes of fthat are essential.
A fixed point theorem with this number is that:
Theorem 4. Any continuous map fon a connected compact ANR has at least N(f)fixed points.
Theorem 5. Let f, g :X→Xbe homotopic maps; then N(f)=N(g).
https://doi.org/10.17993/3ctic.2022.112.61-70
Thus any continuous function ghomotopic to fhas at least N(f)fixed points.
The example 1 is an example with only one fixed point class and it is essential, i.e, N(f)=1.
Computing
N
(
f
)is difficult in general, in some cases it can be computed via the Reidemeister
number
R
(
f
)by knowing
L
(
f
)and
fπ
, the induced homomorphism on the fundamental group of
X
.
The main tool to compute the Nielsen number is the Jiang subgroup of the fundamental group. Before
going to the computation method of Nielsen number we will see how it is related to the Reidemeister
number R(f).
5 The Reidemeister Number
Let
f
:
X→X
be a map on a connected compact ANR and let
F ixf
=
{x∈X
:
f
(
x
)=
x}
. Let
p
:
X→X
be the universal covering of
X
and
˜
f
:
X→
X
be a lifting of
f
, ie.
p◦˜
f
=
f◦p
. Two
liftings
˜
f
and
˜
f
are called conjugate if there is a
γ∈
Γ=
π1
(
X
)such that
˜
f
=
γ˜
fγ−1
. Note that if
˜
f
is a lift of
f
and
γ∈
Γ, then
p
(
F ix ˜
f
)=
p
(
F ixγ ˜
fγ−1
)and
p
(
F ix ˜
f
)=
p
(
F ix ˜
f
), then
˜
f
=
γ˜
fγ−1
for some
γ∈
Γ. This is an equivalence relation on the set
F ixf
=
[˜
f]p
(
F ix ˜
f
), ie.
F ixf
is a disjoint
union of projections of fixed points of lifts from distinct lifting classes. The subset
p
(
F ix ˜
f
)
⊆F ixf
is called the fixed point class of
f
determined by the lifting class [
˜
f
]. Note that for any point
y∈
X
,
since (
f◦p
)
π
(
π1
(
X
))
⊂pπ
(
π1
(
X
)), there is a unique map
˜
f
such that
p◦˜
f
=
f◦p
and
˜
f
(
y
)=
y
. In
particular, any fixed point of fis a projection of a fixed point of some lift ˜
fof f.
Now we define the Reidemeister number of a group homomorphism
ϕ
:
G→G
. Let
ϕ
:
G→G
be a
group homomorphism on a group
G
. One has an action of
G
on itself given by
g.x
=
gxϕ
(
g
)
−1
. Two
elements
x, y ∈G
are said to be
ϕ−
twisted conjugate,
x∼ϕy
, if they are in the same orbit of this
action. The orbits are called the
ϕ
-twisted conjugacy classes or the Reidemeister classes and
R
(
ϕ
), the
number of Reidemeister classes is called the Reidemeister number of
ϕ
. If
ϕ
=
Id
then
R
(
ϕ
)coincides
with the number of conjugacy classes of G.
By fixing a lift
˜
f
of
f
and
α∈
Γ
π1
(
X
)
,˜x∈
X
, we obtain a unique element
˜α∈
Γsuch that
˜α˜
f
(
˜x
)=
˜
f
(
α
(
˜x
)). Thus we obtain a homomorphism
φ
:
π1
(
X
)
→π1
(
X
)such that
˜
fα
=
φ
(
α
)
˜
f,∀α∈
Γ.
Also once we fix a lift ˜
fof f, then every lift is of the form α˜
ffor some α∈Γ. Let α, β ∈Γ. Then
[α˜
f]=[β˜
f]⇐⇒ β˜
f=γα ˜
fγ−1
for some γ∈Γ. ie.
⇐⇒ β˜
f=γαφ(γ−1)˜
f
By the uniqueness of lifts, we have [
α˜
f
]=[
β˜
f
]
⇐⇒ β
=
γαφ
(
γ−1
)for some
γ∈
Γ. Furthermore,
by choosing appropriate base point,
φ
can be identified with the induced homomorphism
fπ
on Γ.
From now on, we write
R
(
f
)for
R
(
fπ
). It follows from the above definition that there is a one-one
correspondence between the lifting classes of fand the Reidemeister classes of fπ.
Remark 1.
If we choose a different lifting
˜
f
and thus a different homomorphism
φ
, we get a bijection
between the
φ
-Reidemeister classes and the
φ
-Reidemeister classes so that the cardianality of such sets
is a constant.
5.1 Relationship with Nielsen number
Suppose
x1,x
2∈F ixf
are in the same Nielsen class, ie. there exists a path
c
:
I→X
from
x1
to
x2
such
that
f◦c
and
c
are homotopic relative to the end points. Let
˜
f
be a lift of
f
and
˜x1∈F ix ˜
f
such that
p
(
˜x1
)=
x1
. Lift
c
to a path
˜c
starting from
˜x1
and ending at some
˜x2
in
˜
X
. Then
˜
f◦˜c
projects onto
f◦c
which is homotopic to
c
. Thus
˜
f◦˜c
also ends at
˜x2
. Hence
˜
f
(
˜x2
)=
˜x2
. In otherwords, they belong to
the same lifting class. Conversely, let
˜x1,˜x2∈F ix ˜
f
such that
p
(
˜x1
)=
x1
=
x2
=
p
(
˜x2
). Let
˜c
:
I→
X
be a path from
˜x1
to
˜x2
. Then
c
=
p◦˜c
is a path from
x1
to
x2
in
X
and
p
(
˜
f◦˜c
)=
f◦p◦˜c
=
f◦c
,
ie.
˜
f◦˜c
projects onto
f◦c
. In fact, the loop
˜c
(
˜
f◦˜c
)
−1
projects to the loop
c
(
f◦c
)
−1
. Since
X
is
simply-connected, the former loop is trivial in
π1
(
˜
X
)and thus the later loop is homotopic to the trivial
loop, ie.
c∼f◦c
. That is
x1
and
x2
are in the same Nielsen class. This shows that there is a one-to-one
https://doi.org/10.17993/3ctic.2022.112.61-70
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed. 41 Vol. 11 N.º 2 August - December 2022
67
map, say
ψ
, from the set of Nielsen classes to the set of Reidemeister classes and which implies that
N
(
f
)
≤R
(
f
). Note that a lifting class
p
(
F ix ˜
f
)might be empty, but Nielsen classes are non-empty.
Also
R
(
f
)need not be finite while
N
(
f
)is always finite. For example, if
f
=1
X
then any two points
are Nielsen equivalent, thus
N
(
f
)
≤
1while
R
(
f
)is the number of conjugacy classes in
π1
(
X
). In
particular, if π1(X)is abelian then R(f)=|π1(X)|.
6 Computing Nielsen Number
First, let us consider a simple example: For a simply connected space
X
, there is only one Nielsen class
for any self map
f
of
X
, so
N
(
f
)
≤
1. In this case
L
(
f
)=0
⇒N
(
f
)=0or
L
(
f
)
=0
⇒N
(
f
)=1.
N(f)does not give more information than L(f).
The main tool to calculate
N
(
f
)is the Jiang subgroup
T
(
f
)
≤π1
(
X
)introduced by B. Jiang(1963).
6.1 The Jiang Subgroup
Fix a point
x0
in a compact connected ANR
X
and a self map
f
on
X
. We denote by
Map
(
X
)the
set of all maps from
X
to itself with the supremum metric
d
(
f,g
)=
sup{d
(
f
(
x
)
,g
(
x
))
|x∈X}
,
then it is a complete metric space. Let
p
:
Map
(
X
)
→X
be the map given by
p
(
g
)=
g
(
x0
). Then
p
induces a homomorphism
pπ
:
π1
(
Map
(
X
)
,f
)
→π1
(
X, f
(
x0
)). The Jiang subgroup
T
(
f,x0
)is the
image of the homomorphism
pπ
. Equivalently, an element
α∈π1
(
X, f
(
x0
)) is said to be in the Jiang
subgroup
T
(
f,x0
)of
f
if there is a loop
H
in
Map
(
X
)based at
f
such that the loop
c
in
X
defined by
c(t)=H(t)(x0)is homotopic to α.
Lemma 3.
The Jiang subgroup is independent of the base point, ie.
T
(
f,x0
)
T
(
f,x1
)for any
x0,x
1∈X.
Theorem 6.
If
f
is such that
T
(
f,x0
)
π1
(
X, x0
). Then all the fixed point classes have the same
index. If
f
:
X→X
is such that
T
(
f,x0
)=
π1
(
X, x0
), then
L
(
f
)=0implies
N
(
f
)=0.Proof.If
F ixf
=
φ
, then certainly
N
(
f
)=0. Otherwise, let
{F1,F
2, ..., Fn}
be the different fixed point classes of
f
, and assume
x0∈F1
(Lemma 3). By Theorem 6
i
(
Fj
)=
i
(
F1
)for every
j
; so, by additivity (3) and
normalization (4) axioms,
0=L(f)=
j
i(Fj)=ni(F1)
Thus i(F1)=0⇒i(Fj)=0for every j, which implies N(f)=0.
Lemma 4. If fand gare homotopic, then T(f, x0)T(g, x0).
Lemma 5. f
:
X→X, x0,x
1∈X
. Then there exists a map
g
:
X→X
such that both
f−1
(
x0
)
,x
1∈
g−1(x0).
This lemma implies that, given
f,x0
as above, there is a map
g∼f
such that
g
(
x0
)=
x0
. Hence we
can choose
x0∈F ixf
(Lemmas 3, 4, 5). We will drop the base point from the fundamental group and the
Jiang subgroup. The Jiang subgroup of the identity map on
X
is denoted by
T
(
X
)and
T
(
f
)=
T
(
f,x0
).
Theorem 7.
For any map
f
:
X→X
,
T
(
X
)
⊆T
(
f
).Proof.Let
α∈T
(
X
)
≤π1
(
X
). Then there is a
loop [
H
]
∈π1
(
Map
(
X
)
,
1
X
)based at the identity map such that [
pH
]=
α
. Define a loop
H
in
Map
(
X
)
(based at
f
) by
H
(
t
)(
x
)=
H
(
t
)(
f
(
x
)). Then, since
f
(
x0
)=
x0
, it follows that
H
(
t
)(
x0
)=
H
(
t
)(
x0
),
which proves that α=[pH]=[pH]∈T(f). An ANR is an H-space if there is an element e∈X
and a map
µ
:
X×X→X
such that
µ
(
x, e
)=
µ
(
e, x
)=
x, ∀x∈X
. (The fundamental group of an
H-spaces is abelian), (
S0,S1,S3,S7
are the only spheres which are H-spaces). An important property of
an H-space is:
Theorem 8.
If
X
is a H-space, then
T
(
X
)=
π1
(
X
).Proof.We use
e
as the base point. Let
c
be any
loop in Xat eand define H: [0,1]aMap(X)byH(t)(x)=µ(c(t),x). Thus [c]∈T(X).
Note that for any H-space, L(f) = 0 implies N(f) = 0.
https://doi.org/10.17993/3ctic.2022.112.61-70
Now on we will work with Xa connected polyhedron and will fix a triangulation (K, τ)on X.
A space Xis aspherical if πn(X)=1, for all n≥2.
Theorem 9. Let Xbe a connected aspherical polyhedron and f:XaX.T henZ(fπ(π1(X))) ⊆T(f).
Note that, if
fπ
(
π1
(
X
))
⊆Z
(
π1
(
X
)), then
Z
(
fπ
(
π1
(
X
)))
⊆π1
(
X
). If
fπ
(
π1
(
X
))
⊆Z
(
π1
(
X
)), then
T(f)=π1(X).Proof.fπ(π1(X)) ⊆Z(π1(X))Z(fπ(π1(X))) ⊆T(f)T(f)=π1(X). Now on
we assume
L
(
f
)
=0(then there is at least one essential fixed point class, ie.
L
(
f
)
=0
N
(
f
)
≥
1(by
additivity 3), and
X
a compact ANR. If we apply the equivalence relation of
fπ
-equivalence (twisted
action) to the Jiang subgroup
T
(
f
)
⊆π1
(
X
), then the set of equivalence classes is denoted by
T
(
f
).
Let
J
(
f
)be the cardinality of
T
(
f
). In other words,
J
(
f
)is the number of
fπ
- twisted classes in
π1
(
X
)
which contain elements of T(f).
Theorem 10.
If
α∈T
(
f
), then there is an essential fixed point class
F
of
f
such that
ψ
(
F
)=[
α
], the
Reidemeister class containing
α
, where
ψ
is the map from the set of all Nielsen classes to the set of all
Reidemeister classes of
f
discussed in section 5.1. It follows that
J
(
f
)
≤N
(
f
). If
T
(
f
)=
π1
(
X
), then
N
(
f
)=
R
(
f
).Proof.
T
(
f
)=
π1
(
X
)implies that
J
(
f
)=
R
(
f
)by definition. We know that
N
(
f
)
≤R
(
f
).
Now the result follows from theorem 10, it states J(f)≤N(f).
Example 2.
Let
X
=
S1
, the circle, an aspherical H-space with
π1
(
X
)=. Then
T
(
X
)=
T
(1
X
)=
π1
(
S1
)=. If
f
:
S1a
S
1
be any map, then
T
(
X
)
⊆T
(
f
)
T
(
f
)=
π1
(
S1
). Now
L
(
f
)=0
⇐⇒ fπ
is
the identity isomorphism (since
H0
(
S1
) ==
H1
). Thus
L
(
f
)=0
N
(
f
)=0. If
fπ
is not the identity
isomorphism, say
fπ
(1) =
q
=1, then
T
(
f
)=
π1
(
S1
)
N
(
f
)=
R
(
f
)=#
Coker
(1
−fπ
)=
|
1
−q|
since for
an abelan group
G
and a given homomorphism
φ
:
Ga
G
, theReidemeisternumber
R(
φ
)=#
Coker
(1
−φ
).
Example 3.
Let
f
:
S2a
S
2
be a rotation by an angle
θ
. Let
p, n ∈S2
be the south and north poles and
are the only fixed points of
f
. Since
S2
is simply connected, there is exactly one Nielsen class
F
and hence
N
(
f
)
≤
1. Note that
f
is homotopic to the identity map. Thus
L
(
f
)=
L
(1) =
χ
(
S2
)=2
=0
N
(
f
)
≥
1.
Hence N(f)=1and i(F)=1.
Example 4.
Let
f
:
Sna
S
n
be the map
f
(
x
)=
−x
for all
x∈Sn
. Then
L
(
f
)=1
−deg
(
f
), where
degree of fis deg(f)=(−1)n+1.
ACKNOWLEDGMENT
The author acknowledges University of Calicut, "Seed Money"(U.O. No. 11733/2021/Admn; Dated:
11.10.2021), INDIA for financial support. The author is thankful to the referee for their valuable
suggestions.
REFERENCES
[1] R.F. Brown, (1971). The Lefschetz Fixed Point Theorem. Scott. Foresman.
[2] A.L. Fel’shtyn
, (2001). The Reidemeister number of any automorphism of a Gromov hyperbolic
group is infinite. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 279(Geom. i
Topol.6), 250, pp. 229–240.
[3] A.L. Fel’shtyn
, (2010). New directions in Nielsen-Reidemeister theory.Topology Appl., 157(10-11),
pp. 1724-1735.
[4] Daciberg Gonçalves
and
Peter Wong
(year). Twisted conjugacy classes in nilpotent groups. J.
Reine Angew. Math., 633, pp. 11-27.
[5] Gilbert Levitt
and
Martin Lustig
(2000). Most automorphisms of a hyperbolic group have very
simple dynamics.Ann. Sci. École Norm. Sup.(4), 33, 4, pp.507–517.
https://doi.org/10.17993/3ctic.2022.112.61-70
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed. 41 Vol. 11 N.º 2 August - December 2022
68
map, say
ψ
, from the set of Nielsen classes to the set of Reidemeister classes and which implies that
N
(
f
)
≤R
(
f
). Note that a lifting class
p
(
F ix ˜
f
)might be empty, but Nielsen classes are non-empty.
Also
R
(
f
)need not be finite while
N
(
f
)is always finite. For example, if
f
=1
X
then any two points
are Nielsen equivalent, thus
N
(
f
)
≤
1while
R
(
f
)is the number of conjugacy classes in
π1
(
X
). In
particular, if π1(X)is abelian then R(f)=|π1(X)|.
6 Computing Nielsen Number
First, let us consider a simple example: For a simply connected space
X
, there is only one Nielsen class
for any self map
f
of
X
, so
N
(
f
)
≤
1. In this case
L
(
f
)=0
⇒N
(
f
)=0or
L
(
f
)
=0
⇒N
(
f
)=1.
N(f)does not give more information than L(f).
The main tool to calculate
N
(
f
)is the Jiang subgroup
T
(
f
)
≤π1
(
X
)introduced by B. Jiang(1963).
6.1 The Jiang Subgroup
Fix a point
x0
in a compact connected ANR
X
and a self map
f
on
X
. We denote by
Map
(
X
)the
set of all maps from
X
to itself with the supremum metric
d
(
f,g
)=
sup{d
(
f
(
x
)
,g
(
x
))
|x∈X}
,
then it is a complete metric space. Let
p
:
Map
(
X
)
→X
be the map given by
p
(
g
)=
g
(
x0
). Then
p
induces a homomorphism
pπ
:
π1
(
Map
(
X
)
,f
)
→π1
(
X, f
(
x0
)). The Jiang subgroup
T
(
f,x0
)is the
image of the homomorphism
pπ
. Equivalently, an element
α∈π1
(
X, f
(
x0
)) is said to be in the Jiang
subgroup
T
(
f,x0
)of
f
if there is a loop
H
in
Map
(
X
)based at
f
such that the loop
c
in
X
defined by
c(t)=H(t)(x0)is homotopic to α.
Lemma 3.
The Jiang subgroup is independent of the base point, ie.
T
(
f,x0
)
T
(
f,x1
)for any
x0,x
1∈X.
Theorem 6.
If
f
is such that
T
(
f,x0
)
π1
(
X, x0
). Then all the fixed point classes have the same
index. If
f
:
X→X
is such that
T
(
f,x0
)=
π1
(
X, x0
), then
L
(
f
)=0implies
N
(
f
)=0.Proof.If
F ixf
=
φ
, then certainly
N
(
f
)=0. Otherwise, let
{F1,F
2, ..., Fn}
be the different fixed point classes of
f
, and assume
x0∈F1
(Lemma 3). By Theorem 6
i
(
Fj
)=
i
(
F1
)for every
j
; so, by additivity (3) and
normalization (4) axioms,
0=L(f)=
j
i(Fj)=ni(F1)
Thus i(F1)=0⇒i(Fj)=0for every j, which implies N(f)=0.
Lemma 4. If fand gare homotopic, then T(f, x0)T(g, x0).
Lemma 5. f
:
X→X, x0,x
1∈X
. Then there exists a map
g
:
X→X
such that both
f−1
(
x0
)
,x
1∈
g−1(x0).
This lemma implies that, given
f,x0
as above, there is a map
g∼f
such that
g
(
x0
)=
x0
. Hence we
can choose
x0∈F ixf
(Lemmas 3, 4, 5). We will drop the base point from the fundamental group and the
Jiang subgroup. The Jiang subgroup of the identity map on
X
is denoted by
T
(
X
)and
T
(
f
)=
T
(
f,x0
).
Theorem 7.
For any map
f
:
X→X
,
T
(
X
)
⊆T
(
f
).Proof.Let
α∈T
(
X
)
≤π1
(
X
). Then there is a
loop [
H
]
∈π1
(
Map
(
X
)
,
1
X
)based at the identity map such that [
pH
]=
α
. Define a loop
H
in
Map
(
X
)
(based at
f
) by
H
(
t
)(
x
)=
H
(
t
)(
f
(
x
)). Then, since
f
(
x0
)=
x0
, it follows that
H
(
t
)(
x0
)=
H
(
t
)(
x0
),
which proves that α=[pH]=[pH]∈T(f). An ANR is an H-space if there is an element e∈X
and a map
µ
:
X×X→X
such that
µ
(
x, e
)=
µ
(
e, x
)=
x, ∀x∈X
. (The fundamental group of an
H-spaces is abelian), (
S0,S1,S3,S7
are the only spheres which are H-spaces). An important property of
an H-space is:
Theorem 8.
If
X
is a H-space, then
T
(
X
)=
π1
(
X
).Proof.We use
e
as the base point. Let
c
be any
loop in Xat eand define H: [0,1]aMap(X)byH(t)(x)=µ(c(t),x). Thus [c]∈T(X).
Note that for any H-space, L(f) = 0 implies N(f) = 0.
https://doi.org/10.17993/3ctic.2022.112.61-70
Now on we will work with Xa connected polyhedron and will fix a triangulation (K, τ)on X.
A space Xis aspherical if πn(X)=1, for all n≥2.
Theorem 9. Let Xbe a connected aspherical polyhedron and f:XaX.T henZ(fπ(π1(X))) ⊆T(f).
Note that, if
fπ
(
π1
(
X
))
⊆Z
(
π1
(
X
)), then
Z
(
fπ
(
π1
(
X
)))
⊆π1
(
X
). If
fπ
(
π1
(
X
))
⊆Z
(
π1
(
X
)), then
T(f)=π1(X).Proof.fπ(π1(X)) ⊆Z(π1(X))Z(fπ(π1(X))) ⊆T(f)T(f)=π1(X). Now on
we assume
L
(
f
)
=0(then there is at least one essential fixed point class, ie.
L
(
f
)
=0
N
(
f
)
≥
1(by
additivity 3), and
X
a compact ANR. If we apply the equivalence relation of
fπ
-equivalence (twisted
action) to the Jiang subgroup
T
(
f
)
⊆π1
(
X
), then the set of equivalence classes is denoted by
T
(
f
).
Let
J
(
f
)be the cardinality of
T
(
f
). In other words,
J
(
f
)is the number of
fπ
- twisted classes in
π1
(
X
)
which contain elements of T(f).
Theorem 10.
If
α∈T
(
f
), then there is an essential fixed point class
F
of
f
such that
ψ
(
F
)=[
α
], the
Reidemeister class containing
α
, where
ψ
is the map from the set of all Nielsen classes to the set of all
Reidemeister classes of
f
discussed in section 5.1. It follows that
J
(
f
)
≤N
(
f
). If
T
(
f
)=
π1
(
X
), then
N
(
f
)=
R
(
f
).Proof.
T
(
f
)=
π1
(
X
)implies that
J
(
f
)=
R
(
f
)by definition. We know that
N
(
f
)
≤R
(
f
).
Now the result follows from theorem 10, it states J(f)≤N(f).
Example 2.
Let
X
=
S1
, the circle, an aspherical H-space with
π1
(
X
)=. Then
T
(
X
)=
T
(1
X
)=
π1
(
S1
)=. If
f
:
S1a
S
1
be any map, then
T
(
X
)
⊆T
(
f
)
T
(
f
)=
π1
(
S1
). Now
L
(
f
)=0
⇐⇒ fπ
is
the identity isomorphism (since
H0
(
S1
) ==
H1
). Thus
L
(
f
)=0
N
(
f
)=0. If
fπ
is not the identity
isomorphism, say
fπ
(1) =
q
=1, then
T
(
f
)=
π1
(
S1
)
N
(
f
)=
R
(
f
)=#
Coker
(1
−fπ
)=
|
1
−q|
since for
an abelan group
G
and a given homomorphism
φ
:
Ga
G
, theReidemeisternumber
R(
φ
)=#
Coker
(1
−φ
).
Example 3.
Let
f
:
S2a
S
2
be a rotation by an angle
θ
. Let
p, n ∈S2
be the south and north poles and
are the only fixed points of
f
. Since
S2
is simply connected, there is exactly one Nielsen class
F
and hence
N
(
f
)
≤
1. Note that
f
is homotopic to the identity map. Thus
L
(
f
)=
L
(1) =
χ
(
S2
)=2
=0
N
(
f
)
≥
1.
Hence N(f)=1and i(F)=1.
Example 4.
Let
f
:
Sna
S
n
be the map
f
(
x
)=
−x
for all
x∈Sn
. Then
L
(
f
)=1
−deg
(
f
), where
degree of fis deg(f)=(−1)n+1.
ACKNOWLEDGMENT
The author acknowledges University of Calicut, "Seed Money"(U.O. No. 11733/2021/Admn; Dated:
11.10.2021), INDIA for financial support. The author is thankful to the referee for their valuable
suggestions.
REFERENCES
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[2] A.L. Fel’shtyn
, (2001). The Reidemeister number of any automorphism of a Gromov hyperbolic
group is infinite. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 279(Geom. i
Topol.6), 250, pp. 229–240.
[3] A.L. Fel’shtyn
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[4] Daciberg Gonçalves
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Peter Wong
(year). Twisted conjugacy classes in nilpotent groups. J.
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and
Martin Lustig
(2000). Most automorphisms of a hyperbolic group have very
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Ed. 41 Vol. 11 N.º 2 August - December 2022
70
