Fixed Point Theory and Applications Volume 2009, Article ID 346519,16pages doi:10.1155/2009/346519
Research Article
Minimal Nielsen Root Classes and Roots of Liftings
Marcio Colombo Fenille and Oziride Manzoli Neto
Instituto de Ciˆencias Matem´aticas e de Computac¸˜ao, Universidade de S˜ao Paulo, Avenida Trabalhador S˜ao-Carlense, 400 Centro Caixa Postal 668, 13560-970 S˜ao Carlos, SP, Brazil
Correspondence should be addressed to Marcio Colombo Fenille,[email protected] Received 24 April 2009; Accepted 26 May 2009
Recommended by Robert Brown
Given a continuous mapf:K → Mfrom a 2-dimensional CW complex into a closed surface, the Nielsen root numberNfand the minimal number of rootsμfoffsatisfyNf≤μf. But, there is a numberμCfassociated to each Nielsen root class off, and an important problem is to know whenμf μCfNf. In addition to investigate this problem, we determine a relationship betweenμfandμf, whenfis a lifting off 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.
Copyrightq2009 M. C. Fenille and O. M. Neto. 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.
1. Introduction
Letf : X → Y be a continuous map between Hausdorff, normal, connected, locally path connected, and semilocally simply connected spaces, and leta∈Y be a given base point. A root offatais a pointx∈Xsuch thatfx a. In root theory we are interested in finding a lower bound for the number of roots offata. We define the minimal number of roots offata to be the number
μ f, a
min
#ϕ−1asuch thatϕis homotopic tof
. 1.1
When the rangeYoffis a manifold, it is easy to prove that this number is independent of the selected point a ∈ Y, and, from1, Propositions 2.10 and 2.12, μf, a is a finite number, providing thatX is a finite CW complex. So, in this case, there is no ambiguity in defining the minimal number of roots off:
μ f
:μ f, a
for somea∈Y. 1.2
Definition 1.1. Ifϕ :X → Y is a map homotopic tofanda∈ Y is a point such thatμf
#ϕ−1a, we say that the pairϕ, aprovidesμfor thatϕ, ais a pair providingμf.
According to 2, two roots x1, x2 of f at a are said to be Nielsen rootfequivalent if there is a pathγ : 0,1 → Xstarting atx1 and ending atx2 such that the loopf◦γ inY atais fixed-end-point homotopic to the constant path ata. This relation is easily seen to be an equivalence relation; the equivalence classes are called Nielsen root classes off ata. Also a homotopyHbetween two mapsfandfprovides a correspondence between the Nielsen root classes offataand the Nielsen root classes offata. We say that such two classes under this correspondence areH-related. Following Brooks2we have the following definition.
Definition 1.2. A Nielsen root class R of a map f at a is essential if given any homotopy H:f fstarting atf, and the classRisH-related to a root class offata. The number of essential root classes offatais the Nielsen root number off ata; it is denoted byNf, a.
The numberNf, ais a homotopy invariant, and it is independent of the selected pointa∈Y, provid thatYis a manifold. In this case, there is no danger of ambiguity in denot it byNf.
In a similar way as in the previous definition, Gonc¸alves and Aniz in3define the minimal cardinality of Nielsen root classes.
Definition 1.3. LetR be a Nielsen root class of f : X → Y. We define μCf,Rto be the minimal cardinality among all Nielsen root classesR, of a mapf,H-related toR, forH being a homotopy starting atfand ending atf:
Again in 3 was proved that if Y is a manifold, then the number μCf,R is independent of the Nielsen root class off :X → Y. Then, in this case, there is no danger of ambiguity in defining the minimal cardinality of Nielsen root classes off
μC f
:μCRfor some Nielsen rootclassR. 1.3
An important problem is to know when it is possible to deform a map f to some mapf with the property that all its Nielsen root classes have minimal cardinality. When the range Y of f is a manifold, this question can be summarized in the following: when μf μCfNf?
Gonc¸alves and Aniz3answered 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 2-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. Letpk :Y → Y be ak-fold covering. Suppose thatf :X → Y is a map having a liftingf:X → Y throughpk. What is the relationship between the numbersμfandμf? We answer completely this question for the cases in whichXis a connected, locally path connected and semilocally simply connected space, andYandYare manifolds either compact or triangulable. We show thatμf≥kμf, and we find necessary and sufficient conditions to have the identity.
Related results for the Nielsen fixed point theory can be found in4.
InSection 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 writefis a map instead offis a continuous map.
2. The Minimizing of the Nielsen Root Classes
In this section, we study the following question: given a map f : K → M from a 2- dimensional CW complex into a closed surface, under what conditions we have μf μCfNf? In fact, we make a survey on the main results demonstrated by Aniz5, 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.
In5Aniz shows the following result.
Theorem 2.1. Let f : K → M be a map from ann-dimensional CW complex into a closed n- manifold, withn ≥ 3. If there is a mapf : K → Mhomotopic to f such that one of its Nielsen root classesRhas exactlyμCfroots, each one of them belonging to the interior ofn-cells ofK, then μf μCfNf.
In this theorem, the assumption on the dimension of the complex and of the manifold is not superfluous; in fact, Xiaosong presents in6, Section 4a mapf :T2#T2 → T2 from the bitorus into the torus withμf 4 andμCfNf 3.
In3,Theorem 4.2, we have the following result.
Theorem 2.2. For eachn≥3, there is ann-dimensional CW complexKnand a mapfn:Kn → RPn withNfn 2,μCfn 1 andμfn≥3.
This theorem shows that, for eachn ≥ 3, there are maps f : Kn → Mn from n- dimensional CW complexes into closedn-manifolds withμf/μCfNf. 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 surfacesRP2the projective plane,T2the torus, andRP2#RP2 the Klein bottle. When the range is the sphereS2, it is obvious that every mapf :K → S2 satisfiesμf μCfNf, since in this case there is a unique Nielsen root class.
Before constructing such examples, we present the main results that will be used.
Letf :X → Y be a map between connected, locally path connected, and semilocally simply connected spaces. Thenf induces a homomorphismf# : π1X → π1Ybetween fundamental groups. Since the imagef#π1Xofπ1Xbyf#is a subgroup ofπ1Y, there is a covering spacep :Y → Ysuch thatp#π1Y f#π1X. Thus,fhas a liftingf :X → Y throughp . The mapf is called a Hopf lift off, andp : Y → Y is called a Hopf covering forf.
The next result corresponds to2, Theorem 3.4.
Proposition 2.3. The setsf −1ai, forai∈p −1a, that are nonempty, are exactly the Nielsen root class offataand a classf −1aiis essential if and only iff1−1aiis nonempty for every mapf1 :X → Y homotopic tof .
In3, Gonc¸alves and Aniz exhibit an example which we adapt for dimension two and summarize now. Take the bouquet ofmcopies of the sphereS2, and letf :∨mi1S2 → RP2be the map which restricted to eachS2is the natural double covering map. Ifmis at least 2, then Nf 2,μCf 1, andμf m 1.
Now, we present a little more complicated example of a mapf :K → RP2, for which we also haveμf/μCfNf. Its construction is based in3, Theorem 4.2.
Example 2.4. Let p2 : S2 → RP2 be the canonical double covering. We will construct a 2- dimensional CW complexKand a mapf:K → RP2having a liftingf:K → S2throughp2 and satisfying:
iNf 2, iiμCf 1, iiiμf≥3, ivμf 1.
We start by constructing the 2-complexK. Let S1,S2, andS3 be three copies of the 2-sphere regarded as the boundary of the standard 3-simplexΔ3:
S1∂x0, x1, x2, x3, S2∂y0, y1, y2, y3, S3∂z0, z1, z2, z3. 2.1 LetKbe the 2-dimensionalsimplicialcomplex obtained from the disjoint unionS1 S2 S3 by identifyingx0, x1 y0, y1andy0, y2 z0, z1. Thus, eachSi,i 1,2,3, is imbedded intoKso that
S1∩S2 x0, x1 y0, y1
, S2∩S3 y0, y2
z0, z1. 2.2 Then,S1∩S2∩S3is a single pointx0y0z0. Thesimplicial2-dimensional complex Kis illustrated inFigure 1.
Two simplicial complexesAandBare homeomorphic if there is a bijectionφbetween the set of the vertices ofAand of Bsuch that{v1, . . . , vs} is a simplex of Aif and only if {φv1, . . . , φvs} is a simplex of B see 7, page 128. Using this fact, we can construct homeomorphisms h21 : S2 → S1 and h32 : S3 → S2 such that h21|S1∩S2 identity map andh32|S2∩S3identity map.
Letf1:S1 → S2be any homeomorphism fromS1ontoS2. Definef2f1◦h21 :S2 → S2and note thatf2x f1xforx∈S1∩S2. Now, definef3f2◦h32 :S3 → S2and note thatf3x f2xforx∈S2∩S3. In particular,f1x0 f2x0 f3x0. Thus,f1,f2, andf3 can be used to define a mapf:K → S2such thatf|Si fifori1,2,3.
Letf:K → RP2be the compositionfp2◦f, where p2:S2 → RP2is the canonical double covering. Note thatf#π1K p2#π1S2. Thus, we can useProposition 2.3to study the Nielsen root classes offthrough the liftingf.
Letafx0∈RP2, and letp−12 a {a, −a} be the fiber ofp2overa.
Clearly, the homomorphismf∗ : H2K → H2S2is surjective, with H2K ≈ Z3 andH2S2 ≈ Z. Hence, every map fromK intoS2 homotopic tof is surjective. It follows that, for every map g : K → S2 homotopic tof, we haveg−1a/∅ andg−1−a/∅. By
x0y0z0 x3
z3
x2
z2
x1y1 z1y2
y3
S2
S1
S3
Figure 1: A simplicial 2-complex.
Proposition 2.3,f−1aand f−1−aare the Nielsen root classes off, and both are essential classes. Therefore,Nf 2.
Now, sincea fx0, eitherx0 ∈f−1aorx0 ∈ f−1−a. Without loss of generality, suppose thatx0∈f−1a. Then, by the definition off, we have f−1a {x0}. 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 thatμCf 1.
In order to show thatμf≥3, note that since each restrictionf|Siis a homeomorphism andp2:S2 → RP2is a double covering, for each mapghomotopic tof, the equationgx a must have at least two roots in eachSi,i1,2,3. By the decomposition ofKthis implies that μf≥3.
Moreover, it is very easy to see thatμf 1, with the pairf, fa 0providingμf.
Now, we present a similar example where the range of the mapfis the torusT2. Here, the complexKof the domain offis a little bit more complicated.
Example 2.5. Letp2 :T2 → T2 be a double covering. We will construct a 2-dimensional CW complexKand a mapf :K → T2having a liftingf:K → T2throughp2and satisfying the following:
iNf 2, iiμCf 1, iiiμf 3, ivμf 1.
We start constructing the 2-complex K. Consider three copies T1, T2, and T3 of the torus with minimal celular decomposition. Letαi resp.,βibe the longitudinalresp., meridionalclosed 1-cell of the torusTi,i1,2,3. LetKbe the 2-dimensional CW complex obtained from the disjoint unionT1T2T3by identifying
α1α2, α3β2. 2.3
e0 β1 β3
β2α3 T1
T2
T3
α1α2
Figure 2: A 2-complex obtained by attaching three tori.
That is,Kis obtained by attaching the toriT1andT2through the longitudinal closed 1-cell and, next, by attaching the longitudinal closed 1-cell of the torusT3into the meridional closed 1-cell of the torusT2.
Each torusTiis imbedded intoKso that
T1∩T2α1α2, T2∩T3α3β2, T1∩T3T1∩T2∩T3 e0
, 2.4
wheree0is theunique0-cell ofK, corresponding to 0-cells ofT1,T2, andT3through the identifications. The 2-dimensional CW complexKis illustrate, inFigure 2.
Henceforth, we write Ti to denote the image of the original torus Ti into the 2- complexoKthrough the identifications above.
Certainly, there are homeomorphisms h21 : T2 → T1 and h32 : T3 → T2 with h21|T1∩T2 identity map andh32|T2∩T3 identity mapsuch thath21 carriesβ2 ontoβ1, and h32carriesβ3ontoα2. Thus, given a pointx3 ∈β3we haveh32x3∈α1T1∩T2. We should use this fact later.
Letf1 :T1 → T2 be an arbitrary homeomorphism carrying longitude into longitude and meridian into meridian. Definef2 f1◦h21 :T2 → T2and note thatf2x f1xfor x∈T1∩T2. Now, definef3 f2◦h32 :T3 → T2and note thatf3x f2xforx∈T2∩T3. In particular, f1e0 f2e0 f3e0. Thus, f1,f2, and f3 can be used to define a map f:K → T2such thatf|Ti fifori1,2,3.
Let p2 : T2 → T2 be an arbitrary double covering. We can consider, e.g., the longitudinal double coveringp2z z21, z2for eachz z1, z2∈S1×S1∼T2.
We define the mapf:K → T2to be the compositionf p2◦f.
In order to use Proposition 2.3 to study the Nielsen root classes of f using the information aboutf, we need to prove that f#π1K p2#π1T2. Now, sincef# p2#◦f#, it is sufficient to prove that f# is an epimorphism. This is what we will do. Consider the compositionf◦l:T1 → T2, wherel :T1 →Kis the obvious inclusion. This composition is exactly the homeomorphismf1, and therefore the induced homomorphismf#◦l# f1#is an isomorphism. It follows thatf#is an epimorphism. Therefore, we can useProposition 2.3.
Let a fe0 ∈ T2, and let p−12 a {a,a} be the fiber of p2 overa.If p2 is the longitudinal double covering, as above, then ifa a1,a2, we havea −a1,a2.
Clearly, the homomorphismf∗:H2K → H2T2is surjective, withH2K≈Z3and H2T2≈Z. Hence, every map fromKintoT2homotopic tofis surjective. It follows that, for every mapg:K → T2homotopic tof, we have g−1a/∅andg−1a/∅. ByProposition 2.3,
f−1a and f−1a are Nielsen root classes of f, and both are essential classes. Therefore, Nf 2.
Now, sincea fe0, eithere0 ∈ f−1aore0 ∈ f−1a. Without loss of generality, suppose thate0 ∈ f−1a. Then, by the definition off, we have f−1a {e0}. 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,μCf 1.
In order to prove that μf 3, note that since each restriction f|Ti is a homeomorphism and p2 : T2 → T2 is a double covering, for each map g homotopic to f, the equation gx a must have at least two roots in each Ti, i 1,2,3. By the decomposition of K, this implies that μf ≥ 3. Now, let x3 be a point in β3, x3/e0. As we have seen,h32x3 ∈ α1 ⊂ T1∩T2. Writex12 h32x3. By the definition off, we have fx 12 fx3/fe0. Denotey0fe 0andy1 fx 12.
Leta∈ T2 be a point, and letp−12 a {a,a} be the fiber ofp2 overa. SinceT2 is a surface, there is a homeomorphismh : T2 → T2 homotopic to the identity map such that hy0 aandhy1 a. Letq2:T2 → T2be the compositionq2p2◦h, and letϕ:K → T2 be the composition ϕ q2 ◦f. Then, ϕ is homotopic tof andϕ−1a {e0, x12, x3}. Since μf≥3, this implies thatμf 3.
Moreover, it is very easy to see thatμf 1, with the pairf,fe 0providingμf. Note that in this example, for every pairϕ, aprovidingμf which is equal to 3, we have necessarilyϕ−1a {e0, x1, x2}with eitherx1∈α1andx2∈β3orx1∈β1andx2 ∈β2.
For the same complexKofExample 2.5, we can construct a similar example with the range off being the Klein bottle. The arguments here are similar to the previous example, and so we omit details.
Example 2.6. Letp2 :T2 → RP2#RP2 be the orientable double covering. We will construct a 2-dimensional CW complexK and a mapf : K → RP2#RP2having a liftingf: K → T2 throughp2and satisfying the following:
iNf 2, iiμCf 1, iiiμf 3, ivμf 1.
We repeat the previous example replacing the double coveringp2 : T2 → T2 by the orientable double coveringp2 :T2 → RP2#RP2. Also here, we haveμf 1, with the pair f,fe0providingμf.
Small adjustments in the construction of the latter two examples are sufficient to prove the following theorem.
Theorem 2.7. LetKbe the 2-dimensional CW complex of the previous two examples. For each positive integern, there are cellular mapsfn:K → T2andgn :K → RP2RP2satisfying the following:
1Nfn n,μCfn 1 andμfn 2n−1.
2Ngn 2n,μCgn 1 andμgn 4n−1.
Proof. In order to prove item1, let f : K → T2 be as in Example 2.5. Letpn : T2 → T2 be ann-fold coveringwhich certainly exists; e.g., for eachz ∈ T2considered as a pairz z1, z2∈S1×S1, we can definepnz zn1, z2. Definefn pn◦f:K → T2. Then, the same arguments ofExample 2.5can be repeated to prove the desired result.
In order to prove item2, letf:K → T2 be as inExample 2.6. Letpn :T2 → T2be ann-fold coveringe.g., as in the first item, and letp2 : T2 → RP2#RP2be the orientable double covering. Defineq2n:T2 → RP2#RP2to be the compositionq2np2◦pn. Thenq2nis a 2n-fold covering. Definefnq2n◦f:K → RP2#RP2. Now proceed with the arguments of Example 2.6.
Observation 2.8. It is obvious that ifmand nare different positive integers, then the maps fm, fnandgm, gnsatisfying the previous theorem are such thatfmis not homotopic tofnand gmis not homotopic togn.
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, M and N are topological n-manifolds either compact or triangulable, andX denotes a compact, connected, locally path connected, and semilocally simply connected spaces All these assumptions are true, for example, if X is a finite and connected CW complex.
Lemma 3.1. Letpk : Y → Y be ak-fold covering, and letf :X → Y be a map having a lifting f:X → Y throughpk. Leta∈Y be a point, and letp−1k a {a1, . . . , ak}be the fiber ofpkovera.
Thenμf, a≥ ki1μf, a i.
Proof. Letϕ : X → Y be a map homotopic tof such that #ϕ−1a μf, a. Then, sincepk
is a covering, we may liftϕthroughpkto a mapϕ :X → Y homotopic tof. It follows that ϕ−1a ∪ki1ϕ−1ai, with this union being disjoint, and certainly #ϕ−1ai ≥ μf, a ifor all 1≤i≤k. Therefore,
μ f, a
# k
i1
ϕ−1ai
≥k
i1
μ f, a i
. 3.1
Theorem 3.2. Letpk :M → Nbe ak-fold covering, and letf:X → Nbe a map having a lifting f:X → Mthroughpk. Thenμf≥kμf. Moreover, μf 0 if and only ifμf 0.
Proof. Leta∈Nbe an arbitrary point, and letp−1k a {a1, . . . , ak}be the fiber ofpkovera.
SinceMandN are manifolds, we haveμf μf, aandμf μf, a ifor all 1≤ i≤k.
Hence, by the previous lemma,μf ≥ kμf. It follows that μf 0 ifμf 0. On the other hand, suppose thatμf 0. ThenNf 0 and by8, Theorem 2.3, there is a map
g : X → Mhomotopic tofsuch that dimgX ≤ n−1, wherenis the dimension of M and N. Letϕ : X → N be the composition ϕ pk◦ϕ. Then ϕ is homotopic tof and dimϕX≤n−1. Thereforeμf 0.
Note that if in the previous theorem we suppose thatk 1, then the coveringpk : M → Nis a homeomorphism andμf μf.
In Examples2.4,2.5, and2.6of the previous section, we presented mapsf :K → N from 2-dimensional CW complexes into closed surfaceshereNis the projective plane, the torus, and the Klein bottle, resp.for which we have
μ f
≥3>22μ f
. 3.2
This shows that there are mapsf :K → Nfrom 2-dimensional CW complexes into closed surfaces having liftingsf: K → Mthrough a double coveringp2 : M → Nand satisfying the strict inequality
μ f
>2μ f
. 3.3
Moreover,Theorem 2.7shows that there is a 2-dimensional CW complexKsuch that, for each integern > 1, there is a mapfn : K → T2and a mapgn :K → RP2#RP2 having liftingsfn :K → T2through ann-fold coveringpn :T2 → T2 andgn :K → T2through a 2n-fold coveringq2n :T2 → RP2#RP2, respectively, satisfying the relationsμfn 2n−1 >
nnμfnandμgn 4n−1<2n2nμgn.
The proofs of the latter two theorems can be used to create a necessary and sufficient condition for the identityμf kμfto be true. We show this after the following lemma.
Lemma 3.3. Letpk :M → Nbe ak-fold covering, leta1, . . . , akbe different points ofM, and let a∈Nbe a point. Then, there is ak-fold coveringqk:M → Nisomorphic and homotopic topksuch thatqk−1a {a1, . . . , ak}.
Proof. Letpk−1a {b1, . . . , bk}be the fiber ofpkovera. It can occur that someaiis equal to somebj. In this case, up to reordering, we can assume thatai bifor 1≤i≤randai/bifor i > r, for some 1≤r≤k. Ifai/bjfor anyi,j, then we putr 0. Ifrk, then there is nothing to prove. Then, we suppose thatr /k. For eachi r 1, . . . , k, letUi be an open subset of Mhomeomorphic to an openn-ball, containingaiandbiand not containing any other point ajandbj. Lethi :M → Mbe a homeomorphism homotopic to the identity map, being the identity map outsideUiand such thathiai bi. Leth:M → Mbe the homeomorphism hhk◦ · · · ◦hr 1. Thenhis homotopic to the identity map andhai bifor each 1≤i≤k.
Letqk:M → Nbe the compositionqkpk◦h. Thenqkis ak-fold covering isomorphic and homotopic topk. Moreover,q−1k a {a1, . . . , ak}.
Theorem 3.4. Letpk :M → Nbe ak-fold covering, and letf:X → Nbe a map having a lifting f:X → Mthroughpk. Thenμf kμf if and only if, for each pairϕ, aprovidingμf, each pairϕ, a iprovidesμf, whereϕis a lifting ofϕhomotopic tofandp−1k a {a1, . . . , ak}.
Proof. Letϕ, abe a pair providingμf, letp−1k a {a1, . . . , ak}be the fiber ofpkovera, and letϕbe a lifting ofϕhomotopic tof. Then ϕ−1a ∪ki1ϕ−1ai, with this union being disjoint.
Henceμf ki1#ϕ−1ai. Now, #ϕ−1ai≥μffor each 1≤i≤k. Therefore,μf kμf if and only if #ϕ−1ai μf for each 1≤i≤k, that is, each pairϕ, a iprovidesμf.
Theorem 3.5. Letpk : M → N be ak-fold covering, and letf : X → N be a map having a liftingf:X → Mthroughpk. Thenμf kμf if and only if, givenkdifferent points ofM, say a1, . . . , ak, there is a mapϕ:X → Msuch that, for each 1≤i≤k: the pairϕ, a iprovidesμf.
Proof. Letϕ, abe a pair providingμf, and letqk:M → Nbe a covering isomorphic and homotopic topk, such thatq−1k a {a1, . . . , ak}, as inLemma 3.3.
Suppose thatμf kμf. Let ϕ:X → Mbe a lifting ofϕthroughqkhomotopic to f. Then, by the previous theorem, ϕ, a iprovidesμf for each 1≤i≤k.
On the other hand, suppose that there is a mapϕ:X → Msuch that, for each 1≤i≤k, the pairϕ, a iprovides μf. Let ϕ : X → N be the compositionϕ qk◦ϕ. Then ϕ is a lifting ofϕthroughqkhomotopic tofandμf ≤#ϕ−1a ki1#ϕ−1ai kμf.But, by Theorem 3.2, we haveμf≥kμf. Therefore μf kμf.
Theorem 3.6. Letpk :M → Nbe ak-fold covering, and letf:X → Nbe a map having a lifting f:X → Mthroughpk. Thenμf> kμfif and only if, for every mapϕ:X → Mhomotopic to f, there are at most k−1 points inMwhose preimage byϕhas exactlyμf points.
Proof. FromTheorem 3.2,μf/kμf if and only ifμf > kμf. Thus, a trivial argument shows that this theorem is equivalent toTheorem 3.5.
Example 3.7. Letf : K → N,p2 : M → Nandf: K → Mbe the maps of Examples2.4, 2.5, or2.6. Then, we have proved thatμf ≥ 3 > 2 2μf. More precisely, in Examples 2.5and2.6we haveμf 3.Therefore, byTheorem 3.6, ifϕ:K → Mis a map providing μf which is equal to 1, then there is a unique point of Mwhose preimage byϕis a single point.
Now, we present a proposition showing equivalences between the vanishing of the Nielsen numbers and the minimal number of roots offand its liftingsfthrough a covering.
Proposition 3.8. Letpk :M → Nbe ak-fold covering, and letf :X → Mbe a map having a liftingf:X → Mthroughpk. Then, the following statements are equivalent:
iNf 0, iiNf 0, iiiμf 0, ivμf 0.
Proof. First, we should remember that, byTheorem 3.2,iii⇔iv. Also, sinceNg ≤ μg for every mapg, it follows thatiii⇒iandiv⇒ii. On the other hand, by8, Theorem 2.1, we have thati⇒iiiandii⇒iv. This completes the proof.
Until now, we have studied only the cases in which a given mapfhas a lifting through a finite fold covering. Whenfhas a lifting through an infinite fold covering, the problem is easily solved using the results of Gonc¸alves and Wong presented in8.
Theorem 3.9. Letf : X → N be a map having a liftingf: X → Mthrough an infinite fold coveringp∞:M → N. Then the numbersNf,Nf, μfandμf are all zero.
Proof. Certainly, the subgroup f#π1X has infinite index in the groupπ1N. Thus, by8, Corollary 2.2,μf 0 and soNf 0. Now, it is easy to check that alsoμf 0 and so Nf 0.
4. Minimal Classes versus Roots of Liftings
In this section we present some results relating the problems of Sections 2and 3. We start remembering and proving general results which will be used in here.
Also in this section,X is always a compact, connected, locally path connected and semilocally simply connected space andMandNare topologicaln-manifolds either compact or triangulable.
Letf : X → Y be a map with Y having the same properties of X. We denote the Riedemeister number offbyRf, which is defined to be the index of the subgroupf#π1Xin the groupπ1M. In symbols,Rf |π1M:f#π1X|. WhenY is a topological manifold not necessarily compact, it follows from 2thatNf >0 ⇒Nf Rf <∞. Thus, if Rf ∞, thenNf 0.
Corollary 4.1. Letf:X → Nbe a map withRf k, letpk:M → Nbe ak-fold covering and letf:X → Mbe a lifting offthroughpk. Then the following statements are equivalent:
iNf/0, iiNf k, iiiμf/0, ivμf /0.
Proof. The equivalencesi⇔iii⇔ivare proved inProposition 3.8. The implicationii⇒i is trivial. For a proof thatiimpliesiisee2.
Theorem 4.2. Letpk :M → Nbe ak-fold covering, and letf:X → Nbe a map having a lifting f:X → M. IfRf k, thenμf μCf.
Proof. IfNf 0, then allμf,μf, and μCfalso are zero. In this case, there is nothing to prove. Now, suppose thatNf/0. Then, byCorollary 4.1,Nf kandμfandμf are both nonzero. Thus, alsoμCf/0. LetRbe a Nielsen root class off, and letH :f f1be a homotopy starting atfand ending atf1. Moreover, letR1be the Nielsen root class off1that isH-related withR. Letf1 be a lifting off1 throughpkhomotopic tof. ByProposition 2.3, R1 f1−1a for some pointa ∈Mover a specific pointaofN. Thus, the cardinality #R1is minimal if and only if the cardinality #f1−1ais minimal; that is, #R1 μCfif and only if
#f1a μf.
Theorem 4.3. Letpk :M → Nbe ak-fold covering, and letf:X → Nbe a map having a lifting f:X → Mthroughpk. IfRf k, then the following statements are equivalent:
iμf μCfNf, iiμf kμf, iiiμf μfNf.
Proof. By the previous results, we haveNf 0 ⇔ μf 0 ⇔ μf 0. Thus, if one of these numbers are zero, then the three statements are automatically equivalent. Now, if Nf/0, thenNf Rf kand, byTheorem 4.2,μf μCf. This proves the desired equivalences.
5. Maps into the Projective Plane
In this section, we use the capital letterKto denote finite and connected 2-dimensional CW complexes, and we useMto denote closed surfaces.
In the next two lemmas, we consider the 2-sphere in the domain off with cellular decompositionS2 e0∪e2and the 2-sphere in the range off with cellular decomposition S2e0∗∪e2∗.
Lemma 5.1. Letf :S2 → S2 be a map with degreed /0, and leta∈S2 be a point,a /e0∗. Then, there is a cellular mapϕ:S2 → S2such thatfϕrel{e0}and #ϕ−1a 1#ϕ−1−a.
Proof. Without loss of generality, suppose thatais the north pole and so−ais the south pole.
There is a cellular mapg:S2 → S2such thatgfand #g−1a 1#g−1−a. In fact, consider the domain sphereS2 fragmented in|d|southern tracks by meridiansm1, . . . ,m|d|
chosen so thate0 is inm1. Letg : S2 → S2 be a map defined so that each meridianmi, for 1≤i≤ |d|, is carried homeomorphically onto a same distinguished meridianmof the range 2- sphere containinge0∗, and each of the|d|tracks covers once the sphereS2, always in the same direction, which is chosen according to the orientation ofS2, so thatgis a map of degreed.
Sincefandghave the same degree, they are homotopic. Moreover,g−1a {b}and g−1−a {−b}, wherebis the north pole of the domain 2-sphere, and so−bis its south pole.
Therefore, we have #g−1a 1 #g−1−a. What we cannot guarantee immediately is that the homotopy betweenfandgis a homotopy relative to{e0}.
Now, ifH : S2×I → S2is a homotopy starting atf and ending atg, then as in9, Lemma 3.1, we can slightly modifyHin a small closed neighborhoodV ×Iofe0×I, with V homeomorphic to a closed 2-disc and not containingaand−a, to obtain a new homotopy H:S2×I → S2, which is relative toe0. Letϕ:S2 → S2be the end of this new homotopy, that is,ϕH·, 1. SinceHandHdiffer only onV ×Iandaand−ado not belong toV, we haveϕ−1a {b}andϕ−1−a {−b}.
This concludes the proof of this lemma.
Lemma 5.2. Letf :S2 → S2be a map with zero degree and letκ0 :S2 → S2be the constant map ate0∗. Thenfκ0rel{e0}. Moreover, ifa∈S2,a /e0∗, thenκ0−1a ∅ κ0−1−a.
Proof. This is9, 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. Letf :K → Mbe a map. We say thatf is of type∇2 if there is a pairϕ, a providingμfsuch thatϕ−1a⊂K\K1. Moreover, we say thatfis of type∇3if in addition we can choose the mapϕbeing a cellular map.
Proposition 5.4. Every mapf :K → Mof type∇2is also of the type∇3.
Proof. Letϕ : K → Mbe a map and leta ∈ Mbe a point such thatϕ, aprovides μf andϕ−1a⊂ K\K1. We can assume thatais in the interior of the unique 2-cell ofM.We considerMwith a minimal cellular decomposition.LetV be an open neighborhood ofain Mhomeomorphic to an open 2-disc and such that the closureV ofV inMis contained in M\M1, whereM1is the 1-skeleton ofM. Letχ:D2 → Mbe the attaching map of the 2-cell ofM, and leth:V → D2be a homeomorphism, whereD2is the unitary closed 2-disc.
Certainly, there is a retractionr : M\V → M1such that for eachx ∈ ∂V we have rx χ◦hx. Then, the mapsrandχ◦hcan be used to define a mapg :M → Msuch thatg|M\V r andg|V χ◦h. Now, it is easy to see thatgis cellular and homotopic to the identity map id :M → M.
Letψ :K → Mbe the compositionψ g◦ϕand calla ga. Then,ψis a cellular map homotopic tofandψ−1a ϕ−1a⊂K\K1. This concludes the proof.
Proposition 5.5. Every map between closed surfaces is of type∇2and so of type∇3.
Proof. Letf : M → N be a map between closed surfaces. Suppose thatn μf, and let ϕ, abe a pair providingμf. Letϕ−1a {x1, . . . , xn}. If each xj is in the interior of the 2-cell ofM, then there is nothing to prove. Otherwise, lety1, . . . , yn bendifferent points of Mbelonging to its 2-cell. There is a homeomorphismh:M → Mhomotopic to the identity map id :M → Msuch thathyj xjfor each 1≤j ≤n. Letψ :M → Nbe the composition ψ ϕ◦h. Thenψis homotopic tof andψ−1a {y1, . . . , yn} ⊂M\M1. Now, we use the previous proposition to complete the proof.
Theorem 5.6. Letf:K → RP2be a map having a liftingf:K → S2through the double covering p2:S2 → RP2. Iffis of type∇2, then 2μf μf μCfNf.
Proof. Sincefis of type∇2, thenfis also of type∇3, byProposition 5.4. Letϕ:K → S2be a cellular map, and leta∈S2e0∗∪e∗2be a point different frome0∗such that #ϕ−1a μf and
ϕ−1a⊂K\K1. Lete12, . . . , e2mbe the 2-cells ofK. For each 1≤i≤m, we define the quotient map
ωi:K K
K\e2i, 5.1
which collapses the complement of the interior of the 2-celle2i to a pointci0. The imageωiK K/K \e2i is naturally homeomorphic to a 2-sphere S2i which inherits from K a cellular decompositionS2i c0i ∪ci2, where the interior of the 2-cellc2i corresponds homeomorphically to the image byωiof the interior of the 2-celle2i of the 2-complexK.
Sinceϕ : K → S2 is a cellular map, the 1-skeletonK1 ofK is carried byϕinto the 0-celle0∗ofS2. Moreover,K1is carried byωiwhich is also a cellular mapinto the 0-cellc0i of the sphereS2i, for all 1≤i≤m. Then we can define, for each 1≤i≤m, a unique cellular map
ϕi:S2i → S2such thatϕ|e2i ϕi◦ωi|e2i.In fact, for eachxωix∈S2i, we defineϕix ϕx.
Sinceϕis a cellular map, ϕi is well defined and is also a cellular map. Moreover, for each x∈ei2, we haveϕx ϕi◦ωix.
Sinceϕ−1a ⊂ K \K1, the set ϕ−1ais in one-to-one correspondence with the set
∪mi1ϕ−1i a; in fact, we haveϕ−1a ∪mi1ϕi ◦ωi−1a. Now, by the proof of Theorem 4.1 of9, for each 1 ≤ i ≤ m, either #ϕ−1i a 1 or ϕis homotopic to a constant map. Then, by Lemmas5.1and5.2, for each 1 ≤ i≤ m, there is a cellular mapψi : S2i → S2such that
ϕi ψirel{c0i}and #ψi−1a #ϕ−1i a #ψi−1−a. LetHi :ϕiψirel{c0i}be such homotopies, 1≤i≤m.
For eachx∈K, choose once and for all an indexix∈ {1, . . . , m}such thatx∈eix. Then, defineψ : K → S2 byψx ψixωixx. This map is clearly well defined and cellular. Moreover, the homotopies Hi, 1 ≤ i ≤ m, can be used to define a homotopy H starting atϕand ending atψ.
From this construction, we have #ψ−1a μf #ψ−1−a. ByTheorem 3.5, we have thatμf 2μf. Now, it is obvious that Rf 2. So, byTheorem 4.3,μf μCfNf.
Theorem 5.6is not true, in general, when the mapf is not of the type∇2. We present an example to illustrate this fact.
Example 5.7. Let K S21 ∨ S22 be the bouquet of two 2 spheres with minimal cellular decomposition with one 0-celle0and two 2-cellse12ande22. Letf:K → S2be a map which, restricted to eachS2i,i 1,2, is homotopic to the identity map. Consider the sphereS2with its minimal cellular decompositionS2 e0∗∪e2∗. Then, there is a cellular mapϕ : K → S2 homotopic tofsuch thatϕ−1e0∗ {e0}. Thus, the pairϕ, e 0∗providesμf 1, of course.
Now, it is obvious that ,for every map g homotopic tof, the restrictions g|S2i, i 1,2, are surjective. Hence, for every such mapg, the equationgx ahas at least one root in each S2i,i1,2, whatever the pointa∈S2. Therefore, ifx0is a root ofgx abelonging to the interior of one of the 2 cells ofK, then the equationgx amust have a second root, which must belong to the closure of the other 2 cell ofK. But in this case, #g−1a ≥ 2, and so the pairg, ado not provideμf. This means that the map fis not of type∇2. Moreover, this shows that ifϕ, a is a pair providingμf, then necessarilyϕ−1a {e0}. Thus, for every mapϕ :K → S2homotopic tof, there is at most one point in S2 whose preimage byϕis a set withμfpoints. Now, letp2 :S2 → RP2 be a double covering, and letf :K → RP2 be the compositionf p2◦f. Thenfis a lifting off throughp2, and, byTheorem 3.6, we haveμf > 2μf. More precisely,μf 3. Moreover,μCf 1,Nf 2 and μf/ μCfNf.
In the next theorem,Afdenotes the absolute degree of the given mapfsee10or 11.
Theorem 5.8. Letf : M → RP2 be a map inducing the trivial homomorphism on fundamental groups. Then,μf 0 ifAf 0 andμf 2 ifAf/0.
