Minimal Nielsen Root Classes and Roots of Liftings
© M. C. Fenille and O. M. Neto. 2009
Received: 24 April 2009
Accepted: 26 May 2009
Published: 16 June 2009
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.
According to , 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  we have the following definition.
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 .
In a similar way as in the previous definition, Gonçalves and Aniz in  define the minimal cardinality of Nielsen root classes.
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  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 .
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.
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 , 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  Aniz shows the following result.
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.
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].
In , 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].
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 .
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 .
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 .
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 .
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.
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 .
Small adjustments in the construction of the latter two examples are sufficient to prove the following theorem.
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.
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.
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 .
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 .
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, .
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 .
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 .
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.
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 .
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.
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  that . Thus, if , then .
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 .
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 .
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
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.
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.
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 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.
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 , 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 .
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 .
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.,  or  or ). 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 .
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.
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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