Nielsen coincidence, fixed point and root theories of n-valued maps

Nielsen coincidence, fixed point and roottheories of n-valued maps

Robert F. Brown and Kameran Kolahi

Abstract. Let (φ, ψ) be an (m,n)-valued pair of maps φ, ψ : X � Y ,where φ is an m-valued map and ψ is n-valued, on connected finitepolyhedra. A point x ∈ X is a coincidence point of φ and ψ if φ(x) ∩ψ(x) �= ∅. We define a Nielsen coincidence number N(φ : ψ) which is alower bound for the number of coincidence points of all (m,n)-valuedpairs of maps homotopic to (φ, ψ). We calculate N(φ : ψ) for all (m,n)-valued pairs of maps of the circle and show that N(φ : ψ) is a sharp lowerbound in that setting. Specifically, if φ is of degree a and ψ of degree b,then N(φ : ψ) = |an−bm|

〈m,n〉 , where 〈m,n〉 is the greatest common divisorof m and n. In order to carry out the calculation, we obtain results,of independent interest, for n-valued maps of compact connected Liegroups that relate the Nielsen fixed point number of Helga Schirmer tothe Nielsen root number of Michael Brown.

Mathematics Subject Classification. 55M20, 54C60.

Keywords. Nielsen number, (m,n)-valued pair of maps, Lie group, n-valued power map, determined u-valued map, Bezout’s identity.

1. Introduction

A multivalued function φ : X � Y is upper semicontinuous if, for each openset U ⊆ Y , the set {x ∈ X : φ(x) ⊆ U} is open in X and it is lower semi-continuous if {x ∈ X : φ(x) ∩ U �= ∅} is open in X. A function φ : X � Yis continuous if it is both upper and lower semicontinuous. A function φ isn-valued if φ(x) is an unordered set of n points of Y for each x ∈ X. Asin [12], a continuous n-valued function is called an n-valued map.

For a multivalued function φ : X � X, a point x ∈ X is a fixed pointof φ if x ∈ φ(x). The fixed point concept may be generalized as follows: letφ, ψ : X � Y be multivalued functions, then x ∈ X is a coincidence point of

J. Fixed Point Theory Appl. 17 (2007) 1–32DOI 10.1007/s11784-013-0143-2© Springer Basel 2013

Journal of Fixed Point Theoryand Applications

2 R. F. Brown and K. Kolahi JFPTA

φ and ψ if φ(x) ∩ ψ(x) �= ∅.1 Define the coincidence set Coin(φ : ψ) of φ andψ by

Coin(φ : ψ) = {x ∈ X : φ(x) ∩ ψ(x) �= ∅}.If φ, ψ : X � Y , where φ is an m-valued map and ψ is an n-valued

map, then we call the pair (φ, ψ) an (m,n)-valued pair of maps. An n-valuedhomotopy is an n-valued map Φ: X × I � Y , and we say that the n-valuedmaps φ0 and φ1, where φt(x) = Φ(x, t), are (n-valued) homotopic. Then(m,n)-valued pairs of maps (φ, ψ) and (φ′, ψ′) are ((m,n)-valued) homotopicif there is an m-valued homotopy Φ: X × I � Y and an n-valued homotopyΨ: X × I � Y such that φ0 = φ, φ1 = φ′, ψ0 = ψ and ψ1 = ψ′. Theminimum number for coincidences MC(φ : ψ) of an (m,n)-valued pair ofmaps (φ, ψ) is defined to be the minimum cardinality of Coin(φ′, ψ′) amongall (m,n)-valued pairs of maps (φ′, ψ′) homotopic to (φ, ψ).

The Nielsen fixed point theory of n-valued maps φ : X � X, where X isa connected finite polyhedron, was introduced by Schirmer in [19, 20, 21, 22]and has been developed further in [2, 3, 8, 9, 10, 11, 12]. There is an extensiveNielsen coincidence theory for single-valued maps that is described in thesurvey of Goncalves [15]. In this paper, we present the Nielsen coincidencetheory for (m,n)-valued pairs of maps. In [9] the Nielsen fixed point numbersof all n-valued maps of the circle were calculated and it was shown that, inthat setting, the Nielsen number calculates the minimum fixed point number.We will here determine the Nielsen coincidence theory of (m,n)-valued pairsof maps of the circle by calculating the Nielsen coincidence numbers of allsuch pairs, and we will demonstrate that it equals the minimum coincidencenumber.

In Section 2, we extend the definition of the Nielsen coincidence num-ber for single-valued maps, due to Brooks (see, for instance, [5]), to definethe Nielsen coincidence number N(φ : ψ) of an (m,n)-valued pair of mapsφ, ψ : X � Y . We show that N(φ : ψ) is a homotopy invariant and thereforeN(φ : ψ) ≤ MC(φ : ψ). We also describe in this section the Nielsen roottheory of M. Brown [8]. Let φ : X � Y be an n-valued map and let a ∈ Y .Then x ∈ X is a root of φ at a if a ∈ φ(x); write x ∈ φ−1(a). The Nielsenroot number N(φ, a), introduced in [8], is a lower bound for the number ofpoints in ψ−1(a) for all n-valued maps ψ homotopic to φ.

Section 3 is concerned with the fixed point theory of n-valued maps.We relate Schirmer’s theory in [20], based on algebraic topology, to a moregeometric theory that is similar to the coincidence theory of Section 2.

In order to study the coincidence theory of (m,n)-valued pairs of mapsof the circle, we first devote Section 4 to a more general topic of independentinterest. Let X be a compact connected Lie group with identity element e andlet φ, ψ : X � X be an (m,n)-valued pair of maps. If φ(x) = {w1, . . . , wm}1Although this definition of “coincidence” is that of [14], for instance, it is not the only onein the setting of multivalued functions (see [24]). Much of the literature concerns the specialcase in which φ = f is single valued and “coincidence” is defined to mean f(x) ∈ ψ(x),and that does correspond to the definition that we use.

Nielsen theories of n-valued maps 3

and ψ(x) = {z1, . . . , zn}, let φψ−1(x) = {wiz−1j : i = 1, . . . ,m; j = 1, . . . , n}.

If the number of distinct points of X in the set φψ−1(x) is the same num-ber u of points for all x ∈ X, then there is a u-valued map θ : X � Xdefined by letting θ(x) be the unordered set of distinct points in φψ−1(x).If ψ : X → X is the identity map, then for φ(x) = {x1, . . . , xm}, we haveθ(x) = {x1x

−1, . . . , xmx−1}. Then x is a fixed point of φ if and only ifit is a root of θ at e. The corresponding Nielsen theories agree, that is,N(θ, e) = N(φ), the Nielsen root number equals Schirmer’s Nielsen fixedpoint number of φ [20] (except, possibly, in the case that X is the 2-torus).

Section 5 specializes the previous results to n-valued maps of the cir-cle S1. Since the Nielsen fixed point numbers of n-valued maps of S1 werecalculated in [9], we can use the result of Section 4 to calculate the Nielsenroot numbers of such maps. By [11], an n-valued map of S1 induces a (single-valued) endomorphism ofH1(S

1), so the degree deg(φ) is defined as for single-valued maps. We prove that the Nielsen root number of an n-valued mapφ : S1 � S1 is N(φ, 1) = |deg(φ)|. We then consider the Nielsen coincidencetheory of (m,n)-valued pairs of maps φ, ψ : S1 � S1. Let 〈m,n〉 denote thegreatest common divisor of m and n. We prove that if the m-valued map φis of degree a and the n-valued map ψ is of degree b, then there is u-valuedmap θ : S1 � S1 of degree δ, where

u =mn

〈m,n〉 , δ =an− bm

〈m,n〉 ,

such that N(φ : ψ) = N(θ, 1) and therefore

N(φ : ψ) =|an− bm|〈m,n〉 .

We use this calculation to prove that N(φ : ψ) = MC(φ : ψ) for all (m,n)-valued pairs of maps of the circle to itself.

2. The Nielsen coincidence and root numbers

For an n-valued map φ : X � Y of compact connected metric spaces, ifφ(x) = {y1, . . . , yn}, we let γ(x, φ) denote the minimum of the distancesamong the yj . The gap [21] of φ, denoted γ(φ), is the infimum of the γ(x, φ)for all x ∈ X. The compactness of X implies that γ(φ) > 0.

The Splitting Lemma (see, for instance, [21, Lemma 1, page 74]) statesthat if φ : X � Y is an n-valued map of connected finite polyherdra and Xis simply connected, then φ is split. That is, there exist (single-valued) mapsf1, . . . , fn : X → Y such that φ(x) = {f1(x), . . . , fn(x)} for all x ∈ X.

Proposition 2.1. Let φ : X � Y be an n-valued map, where X and Y arecompact connected metric spaces. If φ is split, then the splitting is unique;


that is, if φ(x) = {f1(x), . . . , fn(x)} = {g1(x), . . . , gn(x)} for all x ∈ X, then{f1, . . . , fn} = {g1, . . . , gn} as unordered sets of single-valued functions.2

Proof. Let x0 ∈ X and write φ(x0) = {y1, . . . , yn}. We may assume thaty1 = f1(x0) = g1(x0), and it is sufficient to prove that f1(x) = g1(x) for allx ∈ X. Denote the metrics of X and Y by dX and dY . Let ε > 0 be suchthat ε < γ(φ)/2, where γ(φ) is the gap of φ. Since f1 and g1 are continuous,there exists δ > 0 such that dX(x, x0) < δ implies dY (f1(x), f1(x0)) < ε anddY (g1(x), g1(x0)) < ε. Suppose that dX(x, x0) < δ and f1(x) = gj(x) forsome j �= 1. Then

dY (gj(x), g1(x)) = dY (f1(x), g1(x))

≤ dY (f1(x), f1(x0)) + dY (f1(x0), g1(x))

= dY (f1(x), f1(x0)) + dY (g1(x0), g1(x))

< 2ε < γ(φ),

which contradicts the definition of the gap γ(φ), and therefore f1(x) = g1(x)for all x in a δ-neighborhood. Thus, the subset Coin(f1, g1) of X on whichf1 = g1 is open. But the same argument proves that Coin(f1, gj) is open forall gj with j �= 1, so Coin(f1, g1) is also closed. Since X is connected, weconclude that Coin(f1, g1) = X; that is, f1(x) = g1(x) for all x ∈ X. �

Let φ, ψ : X � Y be an (m,n)-valued pair of maps of connected fi-nite polyhedra. Define an equivalence relation on Coin(φ : ψ) by calling xand x′ equivalent coincidence points if there is a path p : I → X suchthat p(0) = x, p(1) = x′ and p has the following property: For the split-tings φp = {f1, . . . , fm} and ψp = {g1, . . . , gn}, there exist 1 ≤ j ≤ mand 1 ≤ k ≤ n such that fj(0) = gk(0), fj(1) = gk(1) and the pathsfj , gk : I → Y are homotopic relative to the endpoints. The correspondingequivalence classes are called the coincidence classes of φ and ψ, which arefinite in number because X is compact.

The next result corresponds to [20, Lemmas 6.2 and 6.3] in the settingof coincidences of (m,n)-valued pairs of maps. For Φ: X×I � Y an n-valuedhomotopy, let φt : X × {t} � Y be the restriction of Φ.

Lemma 2.1. Let Φ,Ψ: X × I � Y be an (m,n)-valued pair of homotopies.The intersection Ct of a coincidence class C of Φ and Ψ with X×{t} is eitherempty or a coincidence class of φt and ψt. Each coincidence class of φt andψt is contained in a unique coincidence class of Φ and Ψ.

Proof. Suppose (x, t) and (x′, t) are in a coincidence classC of Φ and Ψ. Thenthere is a path p : I → X×I such that p(0) = (x, t), p(1) = (x′, t) and from thesplittings Φp = {f1, . . . , fm} and Ψp = {g1, . . . , gn}, we may order the mapsso that f1(0) = g1(0) ∈ Φ(x, t) ∩ Ψ(x, t), f1(1) = g1(1) ∈ Φ(x′, t) ∩ Ψ(x′, t)and f1 and g1 are homotopic in Y relative to the endpoints. Let πt : X× I →2This simple observation has not appeared in the literature, and since [20] refers morethan once to “a splitting” rather than “the splitting,” we include it here to prevent anymisunderstanding.


X ×{t} be the projection map and define p = πtp : I → X ×{t}. Then thereis a homotopy H : I×I → X between p and p. The splittings of ΦH and ΨHrestrict to Φp = φtp = {f1, . . . , fm} and Ψp = ψtp = {g1, . . . , gn}, where,in particular, f1 is homotopic to f1 and g1 is homotopic to g1. Since thehomotopy H is constant at the endpoints, f1(0) = g1(0) and f1(1) = g1(1).Combining homotopies, we obtain a homotopy K : I × I → Y between f1and g1 relative to the endpoints that establishes the fact that (x, t) and(x′, t) are in the same coincidence class of φt and ψt. We have proved thatthe nonempty intersection Ct of C and X × {t} is a coincidence class of φt

and ψt. Conversely, suppose that (x, t) and (x′, t) are in a coincidence classCt of φt and ψt. A path p : I → X ×{t} that demonstrates their equivalenceas coincidence points of φt and ψt also shows that they are equivalent coin-cidence points of Φ and Ψ. Thus, they are in a coincidence class C such thatC ∩ (X × {t}) = Ct. �

A coincidence class C0 of an (m,n)-valued pair of maps φ, ψ : X � Yis inessential if there are homotopies Φ,Ψ: X × I � Y such that φ0 = φ,ψ0 = ψ and the coincidence class C of Φ and Ψ containing C0 has theproperty C∩(X×{1}) = ∅. Otherwise, the coincidence class is essential. TheNielsen coincidence number N(φ : ψ) is the number of essential coincidenceclasses.3

Theorem 2.1. The Nielsen coincidence number for (m,n)-valued pairs of mapsof connected finite polyhedra is a homotopy invariant. That is, if φ, φ′ : X �Y and ψ, ψ′ : X � Y are homotopic, respectively, then

N(φ : ψ) = N(φ′ : ψ′).

Therefore, N(φ : ψ) ≤ MC(φ : ψ).

Proof. Let Φ,Ψ: X × I � Y be homotopies such that φ0 = φ, φ1 = φ′,ψ0 = ψ and ψ1 = ψ′. Let C0 be an essential coincidence class of φ and ψand let C be the coincidence class of Φ and Ψ containing C0. Then C1 =C ∩ (X × {1}) is a coincidence class of φ′ and ψ′. If C1 was an inessentialcoincidence class of φ′ and ψ′, then there would be an (m,n)-valued pair ofhomotopies Φ′,Ψ′ : X × I � Y such that for C′, the coincidence class of Φ′

and Ψ′ containing C1, we would have C′ ∩ (X×{1}) = ∅. Define homotopiesΦ′′,Ψ′′ : X × I � Y by

Φ′′(x, t) =

{Φ(x, 2t) if 0 ≤ t ≤ 1

2 ,

Φ′(x, 1− 2t) if 12 ≤ t ≤ 1

and Ψ′′ similarly. The set C′′ = C ∪ C′ would be a coincidence class ofΦ′′,Ψ′′ containing C, an essential coincidence class, but C′′ ∩ (X ×{1}) = ∅.Therefore, C1 is also essential and the pair (Φ,Ψ) determines a one-to-onecorrespondence between the essential coincidence classes of φ and ψ and theessential coincidence classes of φ′ and ψ′. In the same way, the pair of (m,n)-valued homotopies Φ,Ψ: X × I � Y , defined by Φ(x, t) = Φ(x, 1 − t) and

3If m = n = 1, this definition agrees with that of the Δ-Nielsen number of [6].


Ψ(x, t) = Ψ(x, 1 − t), determines a one-to-one correspondence between theessential coincidence classes of φ′ and ψ′ and the essential coincidence classesof φ and ψ, so N(φ : ψ) = N(φ′ : ψ′). Since each (m,n)-valued pair (φ′, ψ′)homotopic to (φ, ψ) therefore has at least N(φ : ψ) coincidence classes, thenN(φ : ψ) ≤ MC(φ : ψ). �

Let φ : X � Y be an n-valued map. For a ∈ Y , a point x ∈ X is calleda root of φ at a if a ∈ φ(x). We denote the set of roots of φ at a by φ−1(a).Following [8], we define an equivalence relation on φ−1(a) by calling x0, x1

equivalent if there is a map p : I → X such that p(0) = x0, p(1) = x1 and forthe splitting φp = {f1, . . . , fn}, there is fi : I → Y that is a nullhomotopicloop at a. That is, there is a map H : I × I → Y such that H(s, 0) = fi(s),H(s, 1) = a for all s ∈ I and H(0, t) = H(1, t) = a for all t ∈ I. Anequivalence class R is called a root class of φ at a.

From Lemma 2.1, in the case that Ψ: X × I → Y is the constant mapat a, we obtain the following lemma.

Lemma 2.2. Let Φ: X × I � Y be an n-valued homotopy and let a ∈ Y . Theintersection Rt of a root class R of Φ at a with X ×{t} is either empty or aroot class of φt at a. Consequently, each root class Rt of φt at a is containedin a unique root class of Φ at a.

Let φ : X � Y be an n-valued map and let a ∈ Y . A root class R0

of φ at a is inessential if there is a homotopy Φ: X × I � Y such thatφ0 = φ and R, the root class of Φ at a that contains R0, has the propertyR ∩ (X × {1}) = ∅. Otherwise, R0 is essential. The Nielsen root numberN(φ, a) of φ at a is the number of essential root classes.4

3. Nielsen fixed point theories

We next review the Nielsen fixed point theory of Schirmer [20]. Given ann-valued map φ : X � X, where X is a connected finite polyhedron, definean equivalence relation on

Fix(φ) = {x ∈ X : x ∈ φ(x)}by calling x, x′ ∈ Fix(φ) equivalent if there exists a path p : I → X such thatp(0) = x, p(1) = x′ and, in the splitting φp = {f1, . . . , fn}, some fj : I → Xhas the properties fj(0) = x, fj(1) = x′ and fj is homotopic to p relativeto the endpoints. An equivalence class F is called a fixed point class. Thus,the definition of a fixed point class can be viewed as the special case of acoincidence class for an (m, 1)-valued pair of maps φ, ψ : X � X, whereψ : X → X is the identity map.

4The definition of an inessential root class, and thus of the Nielsen root number, in [8] isin terms of a generalization of the Hopf covering space (see [4, Definition 3.3]). However,it follows from [10, Theorem 3.1] that the definition in [8] is equivalent to the one givenhere.


In [20], Schirmer developed an integer-valued fixed point index theoryfor n-valued maps and called a fixed point class F essential if its index ind(F)is nonzero. We shall say that a class F is algebraically essential if ind(F) �= 0and algebraically inessential otherwise. Schirmer’s Nielsen fixed point numberN(φ) is the number of algebraically essential fixed point classes.

Let Φ: X × I � X be an n-valued homotopy. The corresponding fat

homotopy (see [16, Definition 6.1]) Φ : X × I � X × I is defined as follows:

if Φ(x, t) = {x1, . . . , xn}, then Φ(x, t) = {(x1, t), . . . , (xn, t)}. If we applyLemma 2.1 to the case of Y = X × I and Ψ the identity map of X × I, we

conclude that the intersection Ft of a fixed point class F of the fat homotopy

Φ and X × {t} is either empty or a fixed point class of φt. Consequently,

each fixed point class F of φt is contained in a unique fixed point class F

of the fat homotopy Φ. We call a fixed point class F of an n-valued mapφ : X � X geometrically inessential if there is a homotopy Φ: X × I � X

such that φ0 = φ and the fixed point class F of the fat homotopy Φ such that

F0 = F∩(X×{0}) = F has the property F1 = F∩(X×{1}) = ∅. Otherwise, Fis geometrically essential. We define the geometric Nielsen fixed point number

N(φ) of an n-valued map φ : X � X of a finite polyhedron to be the number

of geometrically essential fixed point classes of φ. Since N(φ) = N(φ : ψ),

where ψ is the identity map of X, it follows from Theorem 2.1 that N(φ) isa homotopy invariant.

A point x in a space X is a local cut point if there is a connected neigh-borhood U of x such that U \ x is disconnected.

Proposition 3.1. If X = S1 or if X is a connected finite polyhedron without

local cut points that is not a 2-manifold, then N(φ) = N(φ).

Proof. Let F be an algebraically essential fixed point class of φ and let Φ: X×I � X be a homotopy such that φ0 = φ. Let F be the fixed point class of the

fat homotopy Φ such that F0 = F. Then ind(F0) �= 0. By [20, Lemma 6.4],

ind(F0) = ind(F1), so F1 �= ∅ by [20, Corollary 4.7] and therefore, F isgeometrically essential. To prove the converse, suppose ind(F) = 0. By [9,Theorem 5.1] if X = S1 or by [2] otherwise, there is a homotopy Φ: X× I �X such that φ0 = φ and φ1 has exactly N(φ) fixed points. Thus, each fixed

point class of φ1 consists of a single point of nonzero index. Let F be the

fixed point class of the fat homotopy Φ corresponding to the homotopy of [9,

Theorem 5.1] or that of [2] such that F0 = F. If F1 were nonempty, by [20,

Lemma 6.4], F1 would be an algebraically inessential fixed point class of φ1.

But since φ1 has no such fixed point class, we conclude that F1 = ∅ andtherefore, F is geometrically inessential. �

Proposition 3.1 is false if X is a hyperbolic surface. The single fixedpoint class of the map f in the examples of [17] is algebraically inessential

but geometrically essential, so N(f) = 0 but N(f) = 1. The proposition is


true for single-valued maps of other surfaces, but it is not known whether

N(φ) = N(φ) for φ an n-valued map of a nonhyperbolic surface with n > 1.5

4. Nielsen theories on Lie groups

Let φ, ψ : W � X be an (m,n)-valued pair of maps, where W is a connectedfinite polyhedron and X is a compact connected Lie group with identityelement e. For each w ∈ W , define φψ−1(w) as follows: if we write φ(w) ={y1, . . . , ym} and ψ(w) = {z1, . . . , zn}, then

φψ−1(w) = {yiz−1j : i = 1, . . . ,m; j = 1, . . . , n}.

Therefore, φψ−1(w) is a point in the space Smn(X) of unordered subsets ofmn not necessarily distinct points of X. The function φψ−1 : W → Smn(X)defined in this way is continuous. In general, the number of distinct points ofX in the set φψ−1(w) will vary with w. However, if the number u of distinctpoints in φψ−1(w) is the same for all w ∈ W , we may define a u-valued mapθ : W � X by letting θ(w) be the unordered set of the distinct points ofφψ−1(w). We will call θ the u-valued map determined by φ and ψ. We notethat w is a coincidence point of φ and ψ if and only if it is a root of θ at e,that is, Coin(φ : ψ) = θ−1(e).

Lemma 4.1. Let φ, ψ : W � X be an (m,n)-valued pair of maps, where W isa connected finite polyhedron and X is a compact connected Lie group, suchthat φ and ψ determine a u-valued map θ : W � X. Let v : Z → W be asingle-valued map, where Z is a simply connected finite polyhedron. Then forthe splittings φv = {f1, . . . , fm}, ψv = {g1, . . . , gn} and θv = {h1, . . . , hu},for each i = 1, . . . , u there exist 1 ≤ ji ≤ m, 1 ≤ ki ≤ n such that hi = fjig

−1ki

.

Proof. To simplify notation, we denote by d the metrics of both Z and X.Since the maps fj , gk : Z → X and the group operations are continuous, thereexists δ > 0 such that if d(z, z′) < δ, then

d(fj(z)gk(z)

−1, fj(z′)gk(z′)−1

)<

γ(θ)

3

for all j and k, where γ(θ) is the gap of θ. Let 1 ≤ j, j′ ≤ m with j �= j′ and1 ≤ k, k′ ≤ n with k �= k′ and define

A ={z ∈ Z : fj(z)gk(z)

−1 �= fj′(z)gk′(z)−1},

B ={z ∈ Z : fj(z)gk(z)

−1 = fj′(z)gk′(z)−1}.

If z ∈ A and d(z, z′) < δ, then z′ ∈ A because

d(fj(z

′)gk(z′)−1, fj′(z′)gk′(z′)−1

)>

γ(θ)

3.

If z ∈ B and d(z, z′) < δ, then

d(fj(z

′)gk(z′)−1, fj′(z′)gk′(z′)−1

)<

2γ(θ)

3.

5Two, in general nonequivalent, approaches to coincidence theory were studied in [1].


But fj(z′)gk(z′)−1 and fj′(z

′)gk′(z′)−1 are in θ(z′), so then they must beequal and therefore z′ ∈ B. We have proved that both A and B are open inthe connected space Z, and thus one of them must be empty for any givenj, k, j′ and k′. So choose a basepoint z0 ∈ Z and write the distinct points ofφψ−1(z0) as {

fj1(z0)gk1(z0)−1, . . . , fju(z0)gku(z0)

−1}.

Since for all 1 ≤ i, i′ ≤ u with i �= i′ and all z ∈ Z we have fji(z)gki(z)−1 �=

fji′ (z)gki′ (z)−1, we conclude that

θv ={fj1g

−1k1

, . . . , fjug−1ku

}is the splitting of θv. �

Theorem 4.1. Let φ, ψ : W � X be an (m,n)-valued pair of maps, where Wis a connected finite polyhedron and X is a compact connected Lie group withidentity element e, such that φ and ψ determine the u-valued map θ : W � X.Then x0 and x1 are equivalent as coincidence points of φ and ψ if and onlyif they are equivalent roots of θ at e. Therefore, the coincidence classes of φand ψ are the root classes of θ at e.

Proof. Let p : I → W be a path, then there are splittings φp = {f1, . . . , fm}and ψp = {g1, . . . , gn} and, by Lemma 4.1,

θp = {h1, . . . , hu} ={fj1g

−1k1

, . . . , fjug−1ku

}.

Suppose x0, x1 ∈ W are equivalent coincidence points of φ and ψ. Thus, thereis a path p : I → W such that p(0) = x0, p(1) = x1 and, for the splittingsφp = {f1, . . . , fm} and ψp = {g1, . . . , gn}, we may assume that f1(0) = g1(0),f1(1) = g1(1) and f1 and g1 are homotopic relative to the endpoints. Thenh1 : I → X defined by h1(t) = f1(t)(g1(t))

−1 in the splitting of θp is a loopin X such that h1(0) = h1(1) = e. Now e = h1(0) ∈ θp(0) = θ(x0), sox0 ∈ θ−1(e), and similarly, e ∈ θ(x1), so x0 and x1 are roots of θ at e.Let H : I × I → X be a homotopy such that H(0, t) = f1(t), H(1, t) =g1(t), H(s, 0) = f1(0) = g1(0) and H(s, 1) = f1(1) = g1(1) for all s, t ∈ I.Define K : I × I → X by K(s, t) = H(s, t)(g1(t))

−1. Then K is a basepoint-preserving homotopy of the loop h1 to the constant loop at e, which provesthat x0 and x1 are equivalent roots of θ at e.

Conversely, suppose x0, x1 ∈ θ−1(e) are equivalent roots. Then thereis a path p : I → W such that p(0) = x0, p(1) = x1 and, in the splittingθp = {h1, . . . , hu}, the loop h1 is homotopic to the constant loop at e. ByLemma 4.1, we may number the maps in the splittings of φp and ψp sothat h1(t) = f1(t)(g1(t))

−1. Then h1(0) = e implies f1(0) = g1(0), wheref1(0) ∈ φp(0) = φ(x0) and g1(0) ∈ ψp(0) = ψ(x0), so φ(x0) ∩ ψ(x0) �= ∅and x0 is a coincidence point of φ and ψ; as is x1, for the same reason. LetK : I × I → X be a homotopy of h1 to the constant loop, so K(0, t) = h1(t)and K(1, t) = K(s, 0) = K(s, 1) = e. The homotopy H : I × I → X definedby H(s, t) = K(s, t)g1(t) shows that f1 and g1 are homotopic relative tothe endpoints and thus, x0 and x1 are equivalent coincidence points of φand ψ. �


If φ : X � X is an n-valued map and ψ : X → X is the identity map,then φ and ψ determine the n-valued map θ : X � X, where θ(x) = φ(x)x−1;that is, if φ(x) = {x1, . . . , xn}, then θ(x) = {x1x

−1, . . . , xnx−1}. The roots

of θ at e are thus the fixed points of φ.

Theorem 4.2. Let φ : X � X be an n-valued map of a compact connected Liegroup X with identity element e and let θ : X � X be the n-valued mapdetermined by φ and ψ, where ψ : X → X is the identity map. Then the fixedpoint classes of φ are the root classes of θ at e. Moreover, a subset of X isa geometrically essential fixed point class of φ if and only if it is an essential

root class of θ and therefore N(θ, e) = N(φ).

Proof. Let Φ: X×I � X be an n-valued homotopy such that φ0 = φ. DefineΘ: X × I � X by Θ(x, t) = Φ(x, t)x−1; that is, if Φ(x, t) = {x1t, . . . , xnt},then Θ(x, t) = {x1tx

−1, . . . , xntx−1}. Let F0 ⊆ X×{0} be a fixed point class

of φ. Then, by [20, Lemmas 6.2 and 6.3], there is a unique fixed point class

F of the fat homotopy Φ such that F ∩ (X × {0}) = F0. By Theorem 4.1,with W = X and ψ(x) = x, the set F0 is a root class of θ at e. Moreover, by

Lemma 2.2 and Theorem 4.1, with W = X × I and Ψ(x, t) = x, the set F isthe unique root class of Θ at e containing F0.

If F0 is a geometrically inessential fixed point class of φ, then there

exists an n-valued homotopy Φ: X × I � X such that F, the fixed point

class of Φ containing F0, has the property F ∩ (X × {1}) = ∅. Since, byTheorem 4.1, the set F is also the unique root class of Θ at e containing theroot class F0 of θ, we conclude that F0 is an inessential root class of θ at e.

Now suppose F0 is an inessential root class of θ at e. Then there exists ann-valued homotopy Θ: X×I � X such that θ0 = θ and F, the root class of Θat e containing F0, has the property F ∩ (X × {1}) = ∅. Define Φ: X × I �X as follows: if Θ(x, t) = {x1t, . . . , xnt}, then Φ(x, t) = {x1tx, . . . , xntx},that is, Φ(x, t) = Θ(x, t)x. Since Φ(x, t)x−1 = Θ(x, t), Theorem 4.1 implies

that F is the fixed point class of Φ that contains F0 and therefore, since

F ∩ (X × {1}) = ∅ , we conclude that F0 is a geometrically inessential fixed

point class of φ. We have proved that N(θ, e) = N(φ). �By Proposition 3.1, Theorem 4.2 implies that N(θ, e) = N(φ), the

Nielsen fixed point number defined by Schirmer in [20], except, possibly, if Xis the 2-torus.

5. (m,n)-Valued pairs of maps of the circle

We view the circle S1 as the quotient group S1 = R/Z. Thus, we representa point of S1 by s ∈ R, where s0 and s1 represent the same point of S1 ifand only if s0 − s1 ∈ Z. For δ, ν ∈ Z, where ν ≥ 1, define φν,δ : S

1 � S1, theν-valued power map of degree δ [9], by

φν,δ(s) =

{δs

ν,δs

ν+

1

ν, . . . ,

δs

ν+

ν − 1

ν

}for 0 ≤ s < 1.


Proposition 5.1. The Nielsen root number at 1 ∈ S1 of the ν-valued powermap φν,δ : S

1 � S1 is N(φν,δ, 1) = |δ|.Proof. Let φ = φν,δ+ν . Then for θ defined by θ(x) = φ(x)x−1, we haveθ = φν,δ. Therefore, by Theorem 4.2 and by [9, Theorem 4.1],

N(φν,δ, 1) = N(θ, 1) = N(φ) = |(δ + ν)− ν| = |δ|. �

Proposition 5.2. Let a, b,m, n ∈ Z, where 1 ≤ m ≤ n and an �= bm. Thepair of power maps φm,a, φn,b : S

1 � S1 determine the u-valued power mapθ = φu,δ of degree δ, where

u =mn

〈m,n〉 , δ =an− bm

〈m,n〉and 〈m,n〉 denotes the greatest common divisor of m and n.

Proof. The (m,n)-valued pair of power maps are defined by

φm,a(s) =

{as

m,as

m+

1

m, . . . ,

as

m+

m− 1

m

},

φn,b(s) =

{bs

n,bs

n+

1

n, . . . ,

bs

n+

n− 1

n

}for s ∈ [0, 1). Let φ = φm,a and ψ = φn,b. Then

φψ−1(s) =

{(a

m− b

n

)s+

j

m− k

n: j = 0, . . . ,m− 1; k = 0, . . . , n− 1

}.

Set

u =mn

〈m,n〉 .We will first show that (

a

m− b

n

)s+

u∈ φψ−1(s)

for all 0 ≤ < u by proving that there exist 0 ≤ j < m and 0 ≤ k < n suchthat (

j

m− k

n

)−

u∈ Z.

By Bezout’s identity, we may write 〈m,n〉 = pm + qn for some p, q ∈ Z.Choose j ≡ q (modm) and k ≡ −p (modn) so j−q = αm and k+p = βnfor some α, β ∈ Z, then

j

m− k

n=

q + αm

m− −p+ βn

n

=qn+ αmn+ pm− βmn

mn

=〈m,n〉mn

+ (α− β)

=

u+ (α− β).


Now suppose 0 ≤ j < m and 0 ≤ k < n are given and let 0 ≤ < u such that

≡ jn− km

〈m,n〉 (modu).

Then〈m,n〉 − (jn− km) = αmn

for some α ∈ Z and thus

u−(

j

m− k

n

)= α.

We have shown that

φψ−1(s) =

{(a

m− b

n

)s,

(a

m− b

n

)s+

1

u, . . . ,

(a

m− b

n

)s+

u− 1

u

},

which is the definition of the power map φν,δ, where

ν = u =mn

〈m,n〉and

δ

ν=

δ

u=

a

m− b

n.

Therefore,

δ =an− bm

〈m,n〉and we set θ(s) = φu,δ(s). �

As we remarked in Section 4, an (m,n)-valued pair of maps from a finitepolyhedron to a Lie group may not determine a u-valued map. Moreover, asthe following example demonstrates, even if the (m,n)-valued pair of mapsdoes determine a u-valued map, this property will not necessarily be preservedunder a homotopy of even one of the maps in the pair.

Example 5.1. By Proposition 5.2, the pair of power maps φ2,1, φ3,1 determinethe power map φ6,1. Let f(s) = s(1− s) and define φ : S1 � S1 = R/Z by

φ(s) =

{s

2,s

2+

1

2− f(s)

},

which is a 2-valued map because 0 ≤ f(s) ≤ 14 for 0 ≤ s < 1, and it is

2-valued homotopic to φ2,1. Let ψ = φ3,1, then

φψ−1(s) =

{s

6,s

6+

1

6− f(s),

s

6+

1

3,s

6+

1

2− f(s),

s

6+

2

3,s

6+

5

6− f(s)

},

so φψ−1(0) takes on six distinct values. But f(√

3+12√3

)= 1

6 , so there are only

three distinct values

φψ−1

(√3 + 1

2√3

)=

{√3 + 1

12√3,5√3 + 1

12√3

,9√3 + 1

12√3

}.

Therefore, the (2, 3)-valued pair of maps φ, ψ does not determine a u-valuedmap.


Proposition 5.3. If an �= bm, then the number of coincidence points of thepower maps φm,a and φn,b is

# Coin(φm,a : φn,b) =|an− bm|〈m,n〉 .

Proof. By Proposition 5.2, the power maps φm,a and φn,b determine thepower map θ defined by

θ(s) =

{(a

m− b

n

)s+

j

u; j = 0, 1, . . . , u− 1

},

where u = mn〈m,n〉 . Since a coincidence point of the power maps is a root at 0

of the power map that they determine, then s ∈ Coin(φm,a : φn,b) if and onlyif 0 ∈ θ(s); that is, for some j ∈ {0, 1, . . . , u− 1},(

a

m− b

n

)s+

j

u= 0,

so(an− bm)

mns = −〈m,n〉

mnj

and thus(an− bm)

〈m,n〉 s ∈ Z.

Since the number of solutions s in the interval 0 ≤ s < 1 is |an−bm|〈m,n〉 , we

conclude that

# Coin(φm,a : φn,b) = # θ−1(0) =|an− bm|〈m,n〉 . �

The degree deg(φ) of an n-valued map φ : S1 � S1 of the circle is definedin [9] in terms of the lift of the map to R. Alternatively, using the fact from [11,page 56] that an n-valued map φ of the circle induces a (single-valued) en-domorphism φ∗ : H1(S

1) → H1(S1) ∼= Z, then deg(φ) may be defined as for

single-valued maps.

Theorem 5.1. Let φ, ψ : S1 � S1 be an (m,n)-valued pair of maps, wheredeg(φ) = a, deg(ψ) = b. Then

N(φ : ψ) =|an− bm|〈m,n〉 .

Proof. By [9, Theorem 3.1], φ is homotopic to the power map φm,a and ψ ishomotopic to φn,b, so N(φ : ψ) = N(φm,a : φn,b) by Theorem 2.1.

Suppose an = bm. Given an n-valued map φ : S1 � S1 = R/Z andε > 0, define φε : S1 � S1 as follows: if φ(s) = {s1, . . . , sn}, then φε(s) ={s1+ε, . . . , sn+ε}. Then φε is n-valued homotopic to φ. Since an = bm, thenfor ε sufficiently small, Coin(φm,a : φε

n,b) = ∅ and therefore N(φ : ψ) = 0 inthis case.


For the rest of the proof, we assume that an �= bm. Let θ be the powermap determined in Proposition 5.2 by φm,a and φn,b. By Propositions 5.1and 5.2,

N(θ, 1) = | deg(θ)| = |an− bm|〈m,n〉 ,

which is the number of roots of θ at 1 since, by Proposition 5.3, that is thenumber of coincidences of φm,a and φn,b. Thus, by Theorem 4.1, each root ofθ at 1 is both a root class and a coincidence class. The local degrees (see [13,Definition 4.2, page 267]) of the roots of the power map θ at 1 are either all+1 or all −1. Let z0 ∈ S1 be a root of θ at 1 and let U be a simply connectedneighborhood of z0 that contains no other root. Let v : U → S1 be inclusion,so there are splittings φm,av = {f1, . . . , fm} and φn,bv = {g1, . . . , gn} and

thus, by Lemma 4.1, θv = {h1, . . . , hu}, where hi = fjig−1ki

. We renumber the

maps in the splittings so that f1g−11 (z0) = 1 and thus, the local degree at z0

is that of f1g−11 . Let ei2πs0 = z0, where 0 ≤ s0 < 1. Then the local degree of

f1g−11 at z0 equals the degree of f1 − g1 at s0. By definition, the coincidence

index in this setting is also the degree of f1−g1 at s0 (see [22, page 23]), so thecoincidence index of f1 and g1 at z0 is nonzero. Let Φ,Ψ: S1× I � S1 be an(m,n)-valued pair of homotopies such that φ0 = φm,a and ψ0 = φn,b. Let Cbe the coincidence class of Φ and Ψ that contains the coincidence class z0 andlet Ct = C ∩ (S1 × {t}). The hypothesis an �= bm implies that Coin(φt, ψt)is a proper subset of S1 for all t. Therefore, Coin(φt, ψt) is contained in asimply connected subset on which φt and ψt are split, so we may apply thecoincidence theory of single-valued maps. By the homotopy property of thecoincidence index (see [23, Lemma 6.8, page 180]), the coincidence index ofCt

is nonzero. In particular, [23, Lemma 6.2] therefore implies that C1 �= ∅ andthus, the coincidence class z0 is essential. We have demonstrated that eachcoincidence point of φm,a and φn,b is an essential coincidence class. Therefore,

N(φ : ψ) = N(φm,a : φn,b) =|an− bm|〈m,n〉 . �

From Theorem 5.1 and Proposition 5.3 we have the following.

Corollary 5.1. If φ, ψ : S1 � S1 is an (m,n)-valued pair of maps wheredeg(φ) = a, deg(ψ) = b, then

N(φ : ψ) = MC(φ : ψ) =|an− bm|〈m,n〉 .

Remark 5.1. The calculation of the Nielsen coincidence number of (m,n)-valued pairs of maps of the circle in Theorem 5.1 can be viewed as an ap-plication of the calculation in [9] of the Nielsen fixed point number of ann-valued map of the circle. The Lie group structure of the circle permits usto relate both fixed points and coincidences to roots.6 The Nielsen root num-ber of an n-valued map of the circle follows easily in Proposition 5.1 from the

6The relationship of fixed points and coincidences to roots, for single-valued maps in themore general context of homogenous spaces, was used by Brooks and Wong in [7].


fixed point calculation. The calculation of the Nielsen coincidence number isthen based on the root number of a map determined by the (m,n)-valuedpair of maps.7

Acknowledgment

We thank the referee for a helpful, and very prompt, referee’s report.

References

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[2] J. Better, A Wecken theorem for n-valued maps. Topology Appl. 159 (2012),3707–3715.

[3] J. Better, Equivariant Nielsen fixed point theory for n-valued maps. TopologyAppl. 157 (2010), 1804–1814.

[4] R. Brooks, Nielsen root theory. In: Handbook of Topological Fixed Point The-ory, Springer, Dordrecht, 2005, 375–431.

[5] R. Brooks, On the sharpness of the Δ2 and Δ1 Nielsen numbers. J. Reine Ang.Math. 259 (1973), 101–108.

[6] R. Brooks and R. Brown, A lower bound for the Δ-Nielsen number. Trans.Amer. Math. Soc. 143 (1969), 555–564.

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tori. Topology Appl. 157 (2010), 1990–1998.[13] A. Dold, Lectures on Algebraic Topology. 2nd ed., Springer, Berlin, 1980.[14] G. Gabor, L. Gorniewicz and M. Slosarski, Generalized topological essentiality

and coincidence points of multivalued maps. Set-Valued Var. Anal. 17 (2009),1–19.

[15] D. Goncalves, Coincidence theory. In: Handbook of Topological Fixed PointTheory, Springer, Dordrecht, 2005, 3–42.

[16] B. Jiang, A primer of Nielsen fixed point theory. In: Handbook of TopologicalFixed Point Theory, Springer, Dordrecht, 2005, 617–645.

[17] B. Jiang, Fixed points and braids, II. Math. Ann. 272 (1985), 249–256.[18] C. McCord, The three faces of Nielsen: Coincidences, intersections and preim-

ages. Topology Appl. 103 (2000), 155–177.

7The simultaneous consideration of different Nielsen theories, for single-valued maps, wasinvestigated by McCord in [18].


[19] H. Schirmer, A minimum theorem for n-valued multifunctions. Fund. Math.126 (1985), 83–92.

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[22] H. Schirmer, Mindestzahlen von Koinzidenzpunkten. J. Reine Ang. Math. 194(1955), 21–39.

[23] J. Vick, Homology Theory: An Introduction to Algebraic Topology. AcademicPress, New York, 1973.

[24] K. W�lodarczyk and D. Klim, Fixed-point and coincidence theorems for set-valued maps with nonconvex or noncompact domains in topological vectorspaces. Abstr. Appl. Anal. 2003 (2003), 1–18.

Robert F. BrownDepartment of MathematicsUniversity of CaliforniaLos Angeles, CA 90095-1555USAe-mail: [email protected]

Kameran KolahiDepartment of MathematicsUniversity of CaliforniaLos Angeles, CA 90095-1555USAe-mail: [email protected]

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Nielsen coincidence, fixed point and root theories of n-valued maps