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