Hindawi Publishing Corporation
Fixed Point Theory and Applications
Volume 2010, Article ID 327493, 9 pages
doi:10.1155/2010/327493
Research Article
Nielsen Type Numbers of Self-Maps on
the Real Projective Plane
Jiaoyun Wang
School of Mathematical Sciences and Institute of Mathematics and Interdisciplinary Science,
Capital Normal University, Beijing 100048, China
Correspondence should be addressed to Jiaoyun Wang,
Received 27 May 2010; Revised 26 July 2010; Accepted 23 September 2010
Academic Editor: Robert F. Brown
Copyright q 2010 Jiaoyun Wang. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
Employing the induced endomorphism of the fundamental group and using the homotopy
classification of self-maps of real projective plane RP
2
, we compute completely two Nielsen type
numbers, NP
n
f and NF
n
f , which estimate the number of periodic points of f and the number
of fixed points of the iterates of map f.
1. Introduction
Topological fixed point theory deals with the estimation of the number of fixed points of
maps. Readers are referred to 1 for a detailed treatment of this subject. The number of
essential fixed point classes of self-maps f of a compact polyhedron is called the Nielsen
number of f, denoted Nf. It is a lower bound for the number of fixed points of f.The
f
/
0}.
The purpose of this paper is to give a complete computation of the two Nielsen type
numbers NP
n
f and NF
n
f for all maps on the real projective plane RP
2
.
2. Preliminaries
We list some definitions and properties we need for our discussion. For the details see 1, 2, 7.
We consider a topological space X with universal covering p :
X → X. Assume f is a self-
map of X and let f
n
be its nth iterate. The nth iterate
f
n
of
f is a lifting of f
n
. We write D
X
for the covering transformation group and identify D
f
π
: D
X
−→ D
X
,
α −→
f
π
α
α
,
2.1
that is,
f ◦ α
f
f ◦ γ
−1
. Lifting classes are equivalence classes by
conjugacy, denoted by
f{γ ◦
f ◦ γ
−1
| γ ∈ D
X}, we will also call them fixed point classes
and denote their set by FPCf. We will call about these classes referring either to the fixed
point class
f or to the set p Fix
fNielsen class.
The restriction f :Fixf
n
→ Fixf
n
permutes Nielsen classes. We denote the
corresponding self-map of FPCf
n
by f
FPC
. This map can be described as follows. For a
given α
α
f
n
β
f
n
.
Let
f be a given lifting of f. Obviously, we have p Fix
f ⊂ p Fix
f
n
.
Fixed Point Theory and Applications 3
Definition 2.2. Let
f be a lifting class of f : X → X. Then the lifting class
f
n
of
f
n
is evidently independent of the choice of representative
−→ FPC
f
n
. 2.4
The next proposition shows that f
FPC
:FPCf
n
→ FPCf
n
is a built-in
automorphism. And the correspondence can help us to study the relations and properties
between the fixed point classes of f
n
.
Proposition 2.3 see 1,Proposition3.3. iLet
f
1
,
f
2
, ,
f
n
be liftings of f,thenf
n
◦···◦
f
2
◦
f
1
p Fix
f
1
◦
f
n
···◦
f
2
, thus the f-image of a fixed point class
of f
n
is again a fixed point class of f
n
.
iii indexf
n
,pFix
FPC
n
id :FPCf
n
→ FPCf
n
.
Proposition 2.4. Let
f :
X →
X be a lifting of f.Thenια ◦
fα
n
◦
f
n
,whereα
n
αf
π
α ···f
n−1
π
of period n is reducible to period m if it contains
some periodic point class ξ
f
m
of period m,thatisσ
f
n
ξ
f
m
n/m
,withσ, ξ ∈ D
X.Itis
irreducible if it is not reducible to any lower period.
We say that an orbit α∈Orb
n
f is reducible to m,withm | n, if there exists a β∈
Orb
m
f for some m | n, such that ιβα. We define the depth of α as the smallest
positive integer to which α is reducible, denoted by d dα.Ifα is not reducible to
any m | n with m
/
n, then that element is said to be irreducible.
From Proposition 2.4, we have a correspondence f
α
is essential and irreducible
. 2.5
Definition 2.8. A periodic orbit set S is said to be a representative of T if every orbit of T
reduces to an orbit of S. A finite set of orbits S is said to be a set of n-representatives if every
essential m-orbit β with m | n is reducible to some α∈S.
Definition 2.9. The full Nielsen-Jiang periodic number NF
n
f is defined as
NF
n
f
min
⎧
⎨
⎩
α
∈S
d
α
2
p
2
f
2
3.1
commutes. Assume
f is a lifting of f, then the other lifting of f is τ
f
n
, where τ is the nontrivial
element of π
1
RP
2
. Here we give the definition of the absolute degree see also 8.
Definition 3.1. Let f :RP
2
→ RP
2
be a self-map, and let
f : S
2
→ S
2
be a lifting of f.The
lifting degree of f is defined to be the absolute value of the degree of
. We define the mod 2 degree
deg
2
f ∈ Z
2
as
deg
2
fdegf
mod 2. The homotopy
classification of self-maps on real projective plane is as follows.
Proposition 3.2 see 9, Theorems III and II. Let f,g : RP
2
→ RP
2
be self-maps, they are
homotopic if and only if one of the cases is satisfied:
1 the endomorphism f
π
g
π
is the identity and
degf
degg;
2 the endomorphism f
2
were computed in 8,wegivethe
proposition here.
Proposition 3.3. Let f be a self-map of RP
2
with lifting degree
degf.Then
N
f
⎧
⎨
⎩
1, if
deg
f
0 or 1,
2, if
deg
f
> 1.
2
induced by f is
trivial.
Proof. Sufficiency is obvious. It remains to prove necessity.
For each n,ifp Fix
f
n
p Fixτ
f
n
, then we have τ
−1
τ
f
n
τ
f
n
,thatis
f
n
τ
f
n
f
n
and p Fixτ
f
n
of f for any period n.
Theorem 4.2. Let f : RP
2
→ RP
2
be a self-map, and let f
π
: π
1
RP
2
→ π
1
RP
2
be the
homomorphism induced by f.Let
f be a lifting of f. Then, for each n 2
s
· t with s ≥ 0 and odd
t,
1 if f
f
n
is reducible to p Fixτ
f and the orbit containing p Fixτ
f
n
has depth 1 if n is odd; is
reducible to p Fixτ
f
2
s
and the orbit containing p Fixτ
f
n
has depth 2
s
if n 2
s
· t with
odd t>1 and s>0; and is irreducible if n 2
s
with s>0.
Proof. We analyze the reducibility as follows.
Case 1 f
f
n
, hence, these two
periodic point classes lie in different orbits. It is easy to see that the class p Fix
f
n
is reducible
to p Fix
f. So the depth of this periodic point class orbit of f is 1. Determining whether the
periodic point class p Fixτ
f
n
is reducible or not is a little complicated because it depends
on the value of n.
Notice that τ
f
n
τ
f ◦ τ
f ···◦τ
f
f
n
τ
f
n
. The periodic point class
p Fixτ
f
n
is reducible to p Fixτ
f. We conclude that the depth of the periodic point class
orbit of f with period odd n is 1.
Subcase 2.2. If s>0andt 1, that is n 2
s
, then we have τ
f
n
/
τ
f
n
. The periodic point
class p Fixτ
f
k
0
p Fix
f
k
and F
k
τ
p Fixτ
f
k
. Thus, if the homomorphism f
π
induced by f is trivial, we find that the periodic point class orbit with period k is {F
k
0
};
whereas if f
π
is the identity, the two periodic point class orbits with period k are {F
k
0
}
and {F
k
τ
}. Moreover, for each k, whether f
π
f
2
, if deg
f
is odd,
1, if deg
f
is even.
4.1
Corollary 4.4. Let f : RP
2
→ RP
2
be a self-map, and let f
π
: π
1
RP
2
→ π
1
f > 0.
The above corollary is crucial to our theorem in the next two subsections.
Fixed Point Theory and Applications 7
Tabl e 1
n 1 n>1andn is odd n 2
s
, s>0 n 2
s
· t, s>0andt
/
1
degf ≤ 11 0 0 0
degf > 12 0 n 0
4.2. The Prime Nielsen-Jiang Periodic Number NP
n
f of RP
2
The number NP
n
f is a lower bound for the number of periodic points with least period n.
The computation of NP
n
f is somewhat difficult. We give a detailed computation of NP
n
f
of RP
2
f id
S
2
.Now
f
n
id
S
2
∈ Orb
n
f is reducible for n ≥ 2, while τ
f
n
τ ∈ Orb
n
f is inessential,
since Fixτ is empty. Thus, there is no essential irreducible class.
Case 3
degf > 1. We write F
k
0
p Fix
f
0
is reducible and essential; the periodic point class F
2
s
τ
is irreducible
and essential. The number of essential and irreducible periodic point class orbit of f with
period 2
s
is 1. Thus, NP
n
fn 2
s
.
Subcase 3.3 s>0andt>1.ByTheorem 4.2 2, the periodic point classes F
n
0
and F
n
τ
are
reducible. Thus, NP
n
f0.
4.3. The Full Nielsen-Jiang Periodic Number NF
n
f (See Definition 2.9)
Theorem 4.6. Let f : RP
2
f; we are interested in
the reducible orbits of f.
We discuss three cases, depending on the lifting degree of f.
Case 1
degf0.Iff
π
is trivial, then there is a single periodic point class for each n. For
each m | n, the periodic point class F
m
0
p Fix
f
m
is reducible to F
1
0
p Fix
f and by
Corollary 4.4 1, it is essential. We have that S {F
1
0
} is a set of n-representatives and
hS1. Thus, NF
n
f1.
Case 2
the periodic class F
m
0
reduces to the periodic point class F
1
0
p Fix
f. Also the periodic
class F
m
τ
reduces to F
1
τ
p Fixτ
f.Thus,S {F
1
0
, F
1
τ
} is a set of n-representatives
with minimal height 2. Thus, NF
n
f2.
Subcase 3.2 s>0andt 1, that is n 2
s
. For each m | n, m 2
τ
} is a set of n-representatives. By Theorem 4.2 2,
each F
2
k
τ
0 <k≤ s is irreducible, any n-representatives must contain each F
2
k
τ
. Therefore
we have NF
n
f1 1 2 2
2
··· 2
s
2
s1
2n.
Subcase 3.3 s>0andt>1. For each m | n, we write m 2
k
· q,with0 ≤ k ≤ s
and q | t.ByTheorem 4.2 2, the periodic point class F
m
0
reduces to F
, F
2
2
τ
, F
2
s
τ
} is a set of n-representatives. Since each F
2
k
τ
0 <k≤ s is irreducible, any n-representatives must contain each F
2
k
τ
. Therefore we have
NF
n
f1 1 2 2
2
··· 2
s
2
s1
.
9 P. Olum, “Mappings of manifolds and the notion of degree,” Annals of Mathematics, vol. 58, pp. 458–480,
1953.
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