- Research Article
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Minimal Nielsen Root Classes and Roots of Liftings
Fixed Point Theory and Applications volume 2009, Article number: 346519 (2009)
Abstract
Given a continuous map from a 2-dimensional CW complex into a closed surface, the Nielsen root number
and the minimal number of roots
of
satisfy
. But, there is a number
associated to each Nielsen root class of
and an important problem is to know when
. In addition to investigate this problem, we determine a relationship between
and
, when
is a lifting of
through a covering space, and we find a connection between this problems, with which we answer several questions related to them when the range of the maps is the projective plane.
1. Introduction
Let be a continuous map between Hausdorff, normal, connected, locally path connected, and semilocally simply connected spaces, and let
be a given base point. A root of
at
is a point
such that
. In root theory we are interested in finding a lower bound for the number of roots of
at
. We define the minimal number of roots of
at
to be the number
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F346519/MediaObjects/13663_2009_Article_1132_Equ1_HTML.gif)
When the range of
is a manifold, it is easy to prove that this number is independent of the selected point
, and, from [1, Propositions 2.10 and 2.12],
is a finite number, providing that
is a finite CW complex. So, in this case, there is no ambiguity in defining the minimal number of roots of
:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F346519/MediaObjects/13663_2009_Article_1132_Equ2_HTML.gif)
Definition 1.1.
If is a map homotopic to
and
is a point such that
, we say that the pair
provides
or that
is a pair providing
.
According to [2], two roots ,
of
at
are said to be Nielsen root
equivalent if there is a path
starting at
and ending at
such that the loop
in
at
is fixed-end-point homotopic to the constant path at
. This relation is easily seen to be an equivalence relation; the equivalence classes are called Nielsen root classes of
at
. Also a homotopy
between two maps
and
provides a correspondence between the Nielsen root classes of
at
and the Nielsen root classes of
at
. We say that such two classes under this correspondence are
-related. Following Brooks [2] we have the following definition.
Definition 1.2.
A Nielsen root class of a map
at
is essential if given any homotopy
starting at
, and the class
is
-related to a root class of
at
. The number of essential root classes of
at
is the Nielsen root number of
at
; it is denoted by
.
The number is a homotopy invariant, and it is independent of the selected point
, provid that
is a manifold. In this case, there is no danger of ambiguity in denot it by
.
In a similar way as in the previous definition, Gonçalves and Aniz in [3] define the minimal cardinality of Nielsen root classes.
Definition 1.3.
Let be a Nielsen root class of
. We define
to be the minimal cardinality among all Nielsen root classes
, of a map
,
-related to
, for
being a homotopy starting at
and ending at
:
Again in [3] was proved that if is a manifold, then the number
is independent of the Nielsen root class of
. Then, in this case, there is no danger of ambiguity in defining the minimal cardinality of Nielsen root classes of
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F346519/MediaObjects/13663_2009_Article_1132_Equ3_HTML.gif)
An important problem is to know when it is possible to deform a map to some map
with the property that all its Nielsen root classes have minimal cardinality. When the range
of
is a manifold, this question can be summarized in the following: when
?
Gonçalves and Aniz [3] answered this question for maps from CW complexes into closed manifolds, both of same dimension greater or equal to 3. Here, we study this problem for maps from -dimensional CW complexes into closed surfaces. In this context, we present several examples of maps having liftings through some covering space and not having all Nielsen root classes with minimal cardinality.
Another problem studied in this article is the following. Let be a
-fold covering. Suppose that
is a map having a lifting
through
. What is the relationship between the numbers
and
? We answer completely this question for the cases in which
is a connected, locally path connected and semilocally simply connected space, and
and
are manifolds either compact or triangulable. We show that
, and we find necessary and sufficient conditions to have the identity.
Related results for the Nielsen fixed point theory can be found in [4].
In Section 4, we find an interesting connection between the two problems presented. This whole section is devoted to the demonstration of this connection and other similar results.
In the last section of the paper, we answer several questions related to the two problems presented when the range of the considered maps is the projective plane.
Throughout the text, we simplify write is a map instead of
is a continuous map.
2. The Minimizing of the Nielsen Root Classes
In this section, we study the following question: given a map from a 2-dimensional CW complex into a closed surface, under what conditions we have
? In fact, we make a survey on the main results demonstrated by Aniz [5], where he studied this problem for dimensions greater or equal to 3. After this, we present several examples and a theorem to show that this problem has many pathologies in dimension two.
In [5] Aniz shows the following result.
Theorem 2.1.
Let be a map from an
-dimensional CW complex into a closed
-manifold, with
. If there is a map
homotopic to
such that one of its Nielsen root classes
has exactly
roots, each one of them belonging to the interior of
-cells of
, then
.
In this theorem, the assumption on the dimension of the complex and of the manifold is not superfluous; in fact, Xiaosong presents in [6, Section 4] a map from the bitorus into the torus with
and
.
In [3, Theorem 4.2], we have the following result.
Theorem 2.2.
For each , there is an
-dimensional CW complex
and a map
with
,
and
.
This theorem shows that, for each , there are maps
from
-dimensional CW complexes into closed
-manifolds with
. Here, we will show that maps with this property can be constructed also in dimension two. More precisely, we will construct three examples in this context for the cases in which the range-of the maps are, respectively, the closed surfaces
(the projective plane),
(the torus), and
(the Klein bottle). When the range is the sphere
, it is obvious that every map
satisfies
, since in this case there is a unique Nielsen root class.
Before constructing such examples, we present the main results that will be used.
Let be a map between connected, locally path connected, and semilocally simply connected spaces. Then
induces a homomorphism
between fundamental groups. Since the image
of
by
is a subgroup of
, there is a covering space
such that
. Thus,
has a lifting
through
. The map
is called a Hopf lift of
, and
is called a Hopf covering for
.
The next result corresponds to [2, Theorem 3.4].
Proposition 2.3.
The sets , for
, that are nonempty, are exactly the Nielsen root class of
at
and a class
is essential if and only if
is nonempty for every map
homotopic to
.
In [3], Gonçalves and Aniz exhibit an example which we adapt for dimension two and summarize now. Take the bouquet of copies of the sphere
, and let
be the map which restricted to each
is the natural double covering map. If
is at least 2, then
,
, and
.
Now, we present a little more complicated example of a map , for which we also have
. Its construction is based in [3, Theorem 4.2].
Example 2.4.
Let be the canonical double covering. We will construct a 2-dimensional CW complex
and a map
having a lifting
through
and satisfying:
(i)
(ii)
(iii)
(iv)
We start by constructing the 2-complex . Let
,
, and
be three copies of the 2-sphere regarded as the boundary of the standard 3-simplex
:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F346519/MediaObjects/13663_2009_Article_1132_Equ4_HTML.gif)
Let be the 2-dimensional (simplicial) complex obtained from the disjoint union
by identifying
and
. Thus, each
,
, is imbedded into
so that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F346519/MediaObjects/13663_2009_Article_1132_Equ5_HTML.gif)
Then, is a single point
. The (simplicial) 2-dimensional complex
is illustrated in Figure 1.
Two simplicial complexes and
are homeomorphic if there is a bijection
between the set of the vertices of
and of
such that
is a simplex of
if and only if
is a simplex of
(see [7, page 128]). Using this fact, we can construct homeomorphisms
and
such that
and
.
Let be any homeomorphism from
onto
. Define
and note that
for
. Now, define
and note that
for
. In particular,
. Thus,
,
, and
can be used to define a map
such that
for
.
Let be the composition
, where
is the canonical double covering. Note that
. Thus, we can use Proposition 2.3 to study the Nielsen root classes of
through the lifting
.
Let , and let
be the fiber of
over
.
Clearly, the homomorphism is surjective, with
and
. Hence, every map from
into
homotopic to
is surjective. It follows that, for every map
homotopic to
, we have
and
. By Proposition 2.3,
and
are the Nielsen root classes of
, and both are essential classes. Therefore,
.
Now, since , either
or
. Without loss of generality, suppose that
. Then, by the definition of
, we have
. Hence, one of the Nielsen root classes is unitary. Furthermore, since such class is essential, it follows that its minimal cardinality is equal to one. This proves that
.
In order to show that , note that since each restriction
is a homeomorphism and
is a double covering, for each map
homotopic to
, the equation
must have at least two roots in each
,
. By the decomposition of
this implies that
.
Moreover, it is very easy to see that , with the pair
providing
.
Now, we present a similar example where the range of the map is the torus
. Here, the complex
of the domain of
is a little bit more complicated.
Example 2.5.
Let be a double covering. We will construct a 2-dimensional CW complex
and a map
having a lifting
through
and satisfying the following:
(i)
(ii)
(iii)
(iv)
We start constructing the 2-complex . Consider three copies
,
, and
of the torus with minimal celular decomposition. Let
(resp.,
) be the longitudinal (resp., meridional) closed 1-cell of the torus
,
. Let
be the 2-dimensional CW complex obtained from the disjoint union
by identifying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F346519/MediaObjects/13663_2009_Article_1132_Equ6_HTML.gif)
That is, is obtained by attaching the tori
and
through the longitudinal closed 1-cell and, next, by attaching the longitudinal closed 1-cell of the torus
into the meridional closed 1-cell of the torus
.
Each torus is imbedded into
so that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F346519/MediaObjects/13663_2009_Article_1132_Equ7_HTML.gif)
where is the (unique) 0-cell of
, corresponding to 0-cells of
,
, and
through the identifications. The 2-dimensional CW complex
is illustrate, in Figure 2.
Henceforth, we write to denote the image of the original torus
into the 2-complexo
through the identifications above.
Certainly, there are homeomorphisms and
with
and
such that
carries
onto
, and
carries
onto
. Thus, given a point
we have
. We should use this fact later.
Let be an arbitrary homeomorphism carrying longitude into longitude and meridian into meridian. Define
and note that
for
. Now, define
and note that
for
. In particular,
. Thus,
,
, and
can be used to define a map
such that
for
.
Let be an arbitrary double covering. (We can consider, e.g., the longitudinal double covering
for each
.)
We define the map to be the composition
.
In order to use Proposition 2.3 to study the Nielsen root classes of using the information about
, we need to prove that
. Now, since
, it is sufficient to prove that
is an epimorphism. This is what we will do. Consider the composition
, where
is the obvious inclusion. This composition is exactly the homeomorphism
, and therefore the induced homomorphism
is an isomorphism. It follows that
is an epimorphism. Therefore, we can use Proposition 2.3.
Let , and let
be the fiber of
over
. (If
is the longitudinal double covering, as above, then if
, we have
.)
Clearly, the homomorphism is surjective, with
and
. Hence, every map from
into
homotopic to
is surjective. It follows that, for every map
homotopic to
, we have
and
. By Proposition 2.3,
and
are Nielsen root classes of
, and both are essential classes. Therefore,
.
Now, since , either
or
. Without loss of generality, suppose that
. Then, by the definition of
, we have
. Thus, one of the Nielsen root classes is unitary. Furthermore, since such class is essential, it follows that its minimal cardinality is equal to one. Therefore,
.
In order to prove that , note that since each restriction
is a homeomorphism and
is a double covering, for each map
homotopic to
, the equation
must have at least two roots in each
,
. By the decomposition of
, this implies that
. Now, let
be a point in
,
. As we have seen,
. Write
. By the definition of
, we have
. Denote
and
.
Let be a point, and let
be the fiber of
over
. Since
is a surface, there is a homeomorphism
homotopic to the identity map such that
and
. Let
be the composition
, and let
be the composition
. Then,
is homotopic to
and
. Since
, this implies that
.
Moreover, it is very easy to see that , with the pair
providing
.
Note that in this example, for every pair providing
(which is equal to 3), we have necessarily
with either
and
or
and
.
For the same complex of Example 2.5, we can construct a similar example with the range of
being the Klein bottle. The arguments here are similar to the previous example, and so we omit details.
Example 2.6.
Let be the orientable double covering. We will construct a 2-dimensional CW complex
and a map
having a lifting
through
and satisfying the following:
(i),
(ii)
(iii)
(iv)
We repeat the previous example replacing the double covering by the orientable double covering
. Also here, we have
, with the pair
providing
.
Small adjustments in the construction of the latter two examples are sufficient to prove the following theorem.
Theorem 2.7.
Let be the 2-dimensional CW complex of the previous two examples. For each positive integer
, there are cellular maps
and
satisfying the following:
(1),
and
.
(2),
and
.
Proof.
In order to prove item (1), let be as in Example 2.5. Let
be an
-fold covering (which certainly exists; e.g., for each
considered as a pair
, we can define
). Define
. Then, the same arguments of Example 2.5 can be repeated to prove the desired result.
In order to prove item (2), let be as in Example 2.6. Let
be an
-fold covering (e.g., as in the first item), and let
be the orientable double covering. Define
to be the composition
. Then
is a
-fold covering. Define
. Now proceed with the arguments of Example 2.6.
Observation.
It is obvious that if and
are different positive integers, then the maps
and
satisfying the previous theorem are such that
is not homotopic to
and
is not homotopic to
.
3. Roots of Liftings through Coverings
In the previous section, we saw several examples of maps from 2-dimensional CW complexes into closed surfaces having lifting through some covering space and not having all Nielsen root classes with minimal cardinality. In this section, we study the relationship between the minimal number of roots of a map and the minimal number of roots of one of its liftings through a covering space, when such lifting exists.
Throughout this section, and
are topological
-manifolds either compact or triangulable, and
denotes a compact, connected, locally path connected, and semilocally simply connected spaces All these assumptions are true, for example, if
is a finite and connected CW complex.
Lemma 3.1.
Let be a
-fold covering, and let
be a map having a lifting
through
. Let
be a point, and let
be the fiber of
over
. Then
.
Proof.
Let be a map homotopic to
such that
. Then, since
is a covering, we may lift
through
to a map
homotopic to
. It follows that
, with this union being disjoint, and certainly
for all
. Therefore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F346519/MediaObjects/13663_2009_Article_1132_Equ8_HTML.gif)
Theorem 3.2.
Let be a
-fold covering, and let
be a map having a lifting
through
. Then
. Moreover,
if and only if
.
Proof.
Let be an arbitrary point, and let
be the fiber of
over
. Since
and
are manifolds, we have
and
for all
. Hence, by the previous lemma,
. It follows that
if
. On the other hand, suppose that
. Then
and by [8, Theorem 2.3], there is a map
homotopic to
such that
, (where
is the dimension of
and
). Let
be the composition
. Then
is homotopic to
and
. Therefore
.
Note that if in the previous theorem we suppose that , then the covering
is a homeomorphism and
.
In Examples 2.4, 2.5, and 2.6 of the previous section, we presented maps from 2-dimensional CW complexes into closed surfaces (here
is the projective plane, the torus, and the Klein bottle, resp.) for which we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F346519/MediaObjects/13663_2009_Article_1132_Equ9_HTML.gif)
This shows that there are maps from 2-dimensional CW complexes into closed surfaces having liftings
through a double covering
and satisfying the strict inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F346519/MediaObjects/13663_2009_Article_1132_Equ10_HTML.gif)
Moreover, Theorem 2.7 shows that there is a 2-dimensional CW complex such that, for each integer
, there is a map
and a map
having liftings
through an
-fold covering
and
through a
-fold covering
, respectively, satisfying the relations
and
.
The proofs of the latter two theorems can be used to create a necessary and sufficient condition for the identity to be true. We show this after the following lemma.
Lemma 3.3.
Let be a
-fold covering, let
be different points of
, and let
be a point. Then, there is a
-fold covering
isomorphic and homotopic to
such that
.
Proof.
Let be the fiber of
over
. It can occur that some
is equal to some
. In this case, up to reordering, we can assume that
for
and
for
, for some
. If
for any
,
, then we put
. If
, then there is nothing to prove. Then, we suppose that
. For each
, let
be an open subset of
homeomorphic to an open
-ball, containing
and
and not containing any other point
and
. Let
be a homeomorphism homotopic to the identity map, being the identity map outside
and such that
. Let
be the homeomorphism
. Then
is homotopic to the identity map and
for each
. Let
be the composition
. Then
is a
-fold covering isomorphic and homotopic to
. Moreover,
.
Theorem 3.4.
Let be a
-fold covering, and let
be a map having a lifting
through
. Then
if and only if, for each pair
providing
, each pair
provides
, where
is a lifting of
homotopic to
and
.
Proof.
Let be a pair providing
, let
be the fiber of
over
, and let
be a lifting of
homotopic to
. Then
, with this union being disjoint. Hence
. Now,
for each
. Therefore,
if and only if
for each
, that is, each pair
provides
.
Theorem 3.5.
Let be a
-fold covering, and let
be a map having a lifting
through
. Then
if and only if, given
different points of
, say
, there is a map
such that, for each
: the pair
provides
.
Proof.
Let be a pair providing
, and let
be a covering isomorphic and homotopic to
, such that
, as in Lemma 3.3.
Suppose that . Let
be a lifting of
through
homotopic to
. Then, by the previous theorem,
provides
for each
.
On the other hand, suppose that there is a map such that, for each
, the pair
provides
. Let
be the composition
. Then
is a lifting of
through
homotopic to
and
But, by Theorem 3.2, we have
. Therefore
.
Theorem 3.6.
Let be a
-fold covering, and let
be a map having a lifting
through
. Then
if and only if, for every map
homotopic to
, there are at most
points in
whose preimage by
has exactly
points.
Proof.
From Theorem 3.2, if and only if
. Thus, a trivial argument shows that this theorem is equivalent to Theorem 3.5.
Example 3.7.
Let ,
and
be the maps of Examples 2.4, 2.5, or 2.6. Then, we have proved that
. (More precisely, in Examples 2.5 and 2.6 we have
.) Therefore, by Theorem 3.6, if
is a map providing
(which is equal to 1), then there is a unique point of
whose preimage by
is a single point.
Now, we present a proposition showing equivalences between the vanishing of the Nielsen numbers and the minimal number of roots of and its liftings
through a covering.
Proposition 3.8.
Let be a
-fold covering, and let
be a map having a lifting
through
. Then, the following statements are equivalent:
(i)
(ii)
(iii)
(iv)
Proof.
First, we should remember that, by Theorem 3.2, (iii)(iv). Also, since
for every map
, it follows that (iii)
(i) and (iv)
(ii). On the other hand, by [8, Theorem 2.1], we have that (i)
(iii) and (ii)
(iv). This completes the proof.
Until now, we have studied only the cases in which a given map has a lifting through a finite fold covering. When
has a lifting through an infinite fold covering, the problem is easily solved using the results of Gonçalves and Wong presented in [8].
Theorem 3.9.
Let be a map having a lifting
through an infinite fold covering
. Then the numbers
,
,
and
are all zero.
Proof.
Certainly, the subgroup has infinite index in the group
. Thus, by [8, Corollary 2.2],
and so
. Now, it is easy to check that also
and so
.
4. Minimal Classes versus Roots of Liftings
In this section we present some results relating the problems of Sections 2 and 3. We start remembering and proving general results which will be used in here.
Also in this section, is always a compact, connected, locally path connected and semilocally simply connected space and
and
are topological
-manifolds either compact or triangulable.
Let be a map with
having the same properties of
. We denote the Riedemeister number of
by
, which is defined to be the index of the subgroup
in the group
. In symbols,
. When
is a topological manifold (not necessarily compact), it follows from [2] that
. Thus, if
, then
.
Corollary 4.1.
Let be a map with
, let
be a
-fold covering and let
be a lifting of
through
. Then the following statements are equivalent:
(i)
(ii)
(iii)
(iv)
Proof.
The equivalences (i)(iii)
(iv) are proved in Proposition 3.8. The implication (ii)
(i) is trivial. For a proof that (i) implies (ii) see [2].
Theorem 4.2.
Let be a
-fold covering, and let
be a map having a lifting
. If
, then
.
Proof.
If , then all
,
, and
also are zero. In this case, there is nothing to prove. Now, suppose that
. Then, by Corollary 4.1,
and
and
are both nonzero. Thus, also
. Let
be a Nielsen root class of
, and let
be a homotopy starting at
and ending at
. Moreover, let
be the Nielsen root class of
that is
-related with
. Let
be a lifting of
through
homotopic to
. By Proposition 2.3,
for some point
over a specific point
of
. Thus, the cardinality
is minimal if and only if the cardinality
is minimal; that is,
if and only if
.
Theorem 4.3.
Let be a
-fold covering, and let
be a map having a lifting
through
. If
, then the following statements are equivalent:
(i)
(ii)
(iii)
Proof.
By the previous results, we have . Thus, if one of these numbers are zero, then the three statements are automatically equivalent. Now, if
, then
and, by Theorem 4.2,
. This proves the desired equivalences.
5. Maps into the Projective Plane
In this section, we use the capital letter to denote finite and connected 2-dimensional CW complexes, and we use
to denote closed surfaces.
In the next two lemmas, we consider the 2-sphere in the domain of with cellular decomposition
and the 2-sphere in the range of
with cellular decomposition
.
Lemma 5.1.
Let be a map with degree
, and let
be a point,
. Then, there is a cellular map
such that
and
.
Proof.
Without loss of generality, suppose that is the north pole and so
is the south pole.
There is a cellular map such that
and
. In fact, consider the domain sphere
fragmented in
southern tracks by meridians
chosen so that
is in
. Let
be a map defined so that each meridian
, for
, is carried homeomorphically onto a same distinguished meridian
of the range 2-sphere containing
, and each of the
tracks covers once the sphere
, always in the same direction, which is chosen according to the orientation of
, so that
is a map of degree
.
Since and
have the same degree, they are homotopic. Moreover,
and
, where
is the north pole of the domain 2-sphere, and so
is its south pole. Therefore, we have
. What we cannot guarantee immediately is that the homotopy between
and
is a homotopy relative to
.
Now, if is a homotopy starting at
and ending at
, then as in [9, Lemma 3.1], we can slightly modify
in a small closed neighborhood
of
, with
homeomorphic to a closed 2-disc and not containing
and
, to obtain a new homotopy
, which is relative to
. Let
be the end of this new homotopy, that is,
. Since
and
differ only on
and
and
do not belong to
, we have
and
.
This concludes the proof of this lemma.
Lemma 5.2.
Let be a map with zero degree and let
be the constant map at
. Then
. Moreover, if
,
, then
.
Proof.
This is [9, Lemma 3.2]. Also, it is an adaptation of the proof of the previous lemma.
Now, we insert an important definition about the type of maps which provides the minimal number of roots of a given map.
Definition 5.3.
Let be a map. We say that
is of type
if there is a pair
providing
such that
. Moreover, we say that
is of type
if in addition we can choose the map
being a cellular map.
Proposition 5.4.
Every map of type
is also of the type
.
Proof.
Let be a map and let
be a point such that
provides
and
. We can assume that
is in the interior of the unique 2-cell of
. (We consider
with a minimal cellular decomposition.) Let
be an open neighborhood of
in
homeomorphic to an open
-disc and such that the closure
of
in
is contained in
, where
is the 1-skeleton of
. Let
be the attaching map of the 2-cell of
, and let
be a homeomorphism, where
is the unitary closed
-disc.
Certainly, there is a retraction such that for each
we have
. Then, the maps
and
can be used to define a map
such that
and
. Now, it is easy to see that
is cellular and homotopic to the identity map
.
Let be the composition
and call
. Then,
is a cellular map homotopic to
and
. This concludes the proof.
Proposition 5.5.
Every map between closed surfaces is of type and so of type
.
Proof.
Let be a map between closed surfaces. Suppose that
, and let
be a pair providing
. Let
. If each
is in the interior of the 2-cell of
, then there is nothing to prove. Otherwise, let
be
different points of
belonging to its 2-cell. There is a homeomorphism
homotopic to the identity map
such that
for each
. Let
be the composition
. Then
is homotopic to
and
. Now, we use the previous proposition to complete the proof.
Theorem 5.6.
Let be a map having a lifting
through the double covering
. If
is of type
, then
.
Proof.
Since is of type
, then
is also of type
, by Proposition 5.4. Let
be a cellular map, and let
be a point different from
such that
and
. Let
be the 2-cells of
. For each
, we define the quotient map
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F346519/MediaObjects/13663_2009_Article_1132_Equ11_HTML.gif)
which collapses the complement of the interior of the 2-cell to a point
. The image
is naturally homeomorphic to a 2-sphere
which inherits from
a cellular decomposition
, where the interior of the 2-cell
corresponds homeomorphically to the image by
of the interior of the 2-cell
of the 2-complex
.
Since is a cellular map, the 1-skeleton
of
is carried by
into the 0-cell
of
. Moreover,
is carried by
(which is also a cellular map) into the 0-cell
of the sphere
, for all
. Then we can define, for each
, a unique cellular map
such that
In fact, for each
, we define
. Since
is a cellular map,
is well defined and is also a cellular map. Moreover, for each
, we have
.
Since , the set
is in one-to-one correspondence with the set
; in fact, we have
. Now, by the proof of Theorem 4.1 of [9], for each
, either
or
is homotopic to a constant map. Then, by Lemmas 5.1 and 5.2, for each
, there is a cellular map
such that
and
. Let
be such homotopies,
.
For each , choose once and for all an index
such that
. Then, define
by
. This map is clearly well defined and cellular. Moreover, the homotopies
,
, can be used to define a homotopy
starting at
and ending at
.
From this construction, we have . By Theorem 3.5, we have that
. Now, it is obvious that
. So, by Theorem 4.3,
.
Theorem 5.6 is not true, in general, when the map is not of the type
. We present an example to illustrate this fact.
Example 5.7.
Let be the bouquet of two 2 spheres with minimal cellular decomposition with one 0-cell
and two 2-cells
and
. Let
be a map which, restricted to each
,
, is homotopic to the identity map. Consider the sphere
with its minimal cellular decomposition
. Then, there is a cellular map
homotopic to
such that
. Thus, the pair
provides
(
, of course). Now, it is obvious that ,for every map
homotopic to
, the restrictions
,
, are surjective. Hence, for every such map
, the equation
has at least one root in each
,
, whatever the point
. Therefore, if
is a root of
belonging to the interior of one of the 2 cells of
, then the equation
must have a second root, which must belong to the closure of the other 2 cell of
. But in this case,
, and so the pair
do not provide
. This means that the map
is not of type
. Moreover, this shows that if
is a pair providing
, then necessarily
. Thus, for every map
homotopic to
, there is at most one point in
whose preimage by
is a set with
points. Now, let
be a double covering, and let
be the composition
. Then
is a lifting of
through
, and, by Theorem 3.6, we have
. More precisely,
. Moreover,
,
and
.
In the next theorem, denotes the absolute degree of the given map
(see [10] or [11]).
Theorem 5.8.
Let be a map inducing the trivial homomorphism on fundamental groups. Then,
if
and
if
.
Proof.
Since is trivial,
has a lifting
through the (universal) double covering
. By Proposition 5.5,
is of type
. Hence, by Theorem 5.6, we have
. Now, it is well known that
if
and
if
. (see, e.g., [11] or [9] or [3]). But, by the definition of absolute degree (see [11, page 371]) it is easy to check that
. This concludes the proof.
Theorem 5.8 is not true, in general, if the homomorphism is not the trivial homomorphism. To illustrate this, let
be the identity map. It is obvious that this map induces the identity isomorphism on fundamental groups and
.
In the next theorem, is a compact, connected, locally path connected, and semilocally simply connected space.
Theorem 5.9.
Let be a map. Then
if at least one of the following alternatives is true: < (i)
; (ii)
is a 2-dimensional CW complex, and
is of type
.
Proof.
Up to isomorphism, there are only two covering spaces for , namely, the identity covering
and the double covering
. Suppose that (i) is true. Then,
, and
is a covering corresponding to
. Thus, either
or
. Now, if
, then also
by Proposition 3.8. If
, then the result is obvious. Therefore, we have
. If, on the other hand, (ii) is true and (i) is false, then we use Theorem 5.6.
Example 5.7 shows that the assumptions in Theorem 5.9 are not superfluous.
References
Gonçalves DL: Coincidence theory. In Handbook of Topological Fixed Point Theory. Edited by: Brown RF, Furi M, Górniewicz L, Jiang B. Springer, Dordrecht, The Netherlands; 2005:3–42.
Brooks R: Nielsen root theory. In Handbook of Topological Fixed Point Theory. Edited by: Brown RF, Furi M, Górniewicz L, Jiang B. Springer, Dordrecht, The Netherlands; 2005:375–431.
Gonçalves DL, Aniz C: The minimizing of the Nielsen root classes. Central European Journal of Mathematics 2004,2(1):112–122. 10.2478/BF02475955
Jezierski J: Nielsen number of a covering map. Fixed Point Theory and Applications 2006, Article ID 37807, 2006:-11.
Aniz C: Raízes de funções de um Complexo em uma Variedade, M.S. thesis. ICMC, Universidade de São Paulo, São Carlos, Brazil; 2002.
Xiaosong L: On the root classes of a mapping. Acta Mathematica Sinica 1986,2(3):199–206. 10.1007/BF02582022
Armstrong MA: Basic Topology, Undergraduate Texts in Mathematics. Springer, New York, NY, USA; 1983:xii+251.
Gonçalves DL, Wong P: Wecken property for roots. Proceedings of the American Mathematical Society 2005,133(9):2779–2782. 10.1090/S0002-9939-05-07820-2
Fenille MC, Neto OM: Roots of maps from 2-complexes into the 2-sphere. preprint, 2009
Brown RF, Schirmer H: Nielsen root theory and Hopf degree theory. Pacific Journal of Mathematics 2001,198(1):49–80. 10.2140/pjm.2001.198.49
Epstein DBA: The degree of a map. Proceedings of the London Mathematical Society 1966,16(1):369–383. 10.1112/plms/s3-16.1.369
Acknowledgments
The authors would like to express their thanks to Daciberg Lima Gonçalves for his encouragement to the development of the project which led up to this article. This work is partially sponsored by FAPESP - Grant 2007/05843-5. They would like to thank the referee for his careful reading, comments, and suggestions which helped to improve the manuscript.
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Fenille, M.C., Neto, O.M. Minimal Nielsen Root Classes and Roots of Liftings. Fixed Point Theory Appl 2009, 346519 (2009). https://doi.org/10.1155/2009/346519
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DOI: https://doi.org/10.1155/2009/346519