aa r X i v : . [ m a t h . GN ] J a n Partitions of n -valued maps P. Christopher StaeckerJanuary 26, 2021
Abstract An n -valued map is a set-valued continuous function f such that f ( x )has cardinality n for every x . Some n -valued maps will “split” into aunion of n single-valued maps. Characterizations of splittings has been amajor theme in the topological theory of n -valued maps.In this paper we consider the more general notion of “partitions” ofan n -valued map, in which a given map is decomposed into a union ofother maps which may not be single-valued. We generalize several split-ting characterizations which will describe partitions in terms of mixedconfiguration spaces and mixed braid groups, and connected componentsof the graph of f . We demonstrate the ideas with some examples on tori.We also discuss the fixed point theory of n -valued maps and theirpartitions, and make some connections to the theory of finite-valued mapsdue to Crabb. Given sets X and Y and a positive integer n , an n -valued function from X to Y is a set-valued function f on X such that f ( x ) ⊆ Y has cardinality exactly n for every x ∈ X . Equivalently, an n -valued function on X is a single-valuedfunction f : X → D n ( Y ), where D n ( Y ) is the unordered configuration space of n points in Y , defined as: D n ( Y ) = {{ y , . . . , y n } | y i ∈ Y, y i = y j for i = j } . When Y is a topological space, we give D n ( Y ) a topology as follows: beginwith the product topology on Y n , then consider the subspace F n ( Y ) of tuples( y , . . . , y n ) ∈ Y n with y i = y j for i = j . This is the ordered configuration space .Then D n ( Y ) is the quotient of F n ( Y ) up to ordering, and so its topology isgiven by the quotient topology. When f : X → D n ( Y ) is continuous, we call itan n -valued map from X to Y . Continuity of n -valued maps can also be definedin terms of lower- and upper-semicontinuity. These approaches are equivalent:see [5].An n -valued map f : X → D n ( Y ) is split if there are n single valued con-tinuous functions f , . . . , f n : X → Y with f ( x ) = { f ( x ) , . . . , f n ( x ) } for every1 ∈ X . In this case we write f = { f , . . . , f n } , and we say that { f , . . . , f n } isa splitting of f . Not all n -valued maps are split.In this paper we wish to generalize the notion of splitting to the case wherethe various f i are not necessarily single-valued. If k ≤ n and g : X → D k ( X ) isa k -valued map with g ( x ) ⊆ f ( x ) for every x , then we say g is a submap of f .When k < n we say g is a proper submap . If f has no proper submap, then wesay f is irreducible .Let f be an n -valued map and let k , . . . , k l be a partition of n , that is,positive integers which sum to n , and assume that there is a set of l maps f , . . . , f l , where f i is a k i -valued map and f ( x ) = f ( x ) ∪ · · · ∪ f l ( x ) for each x . Then we say { f , . . . , f l } is a partition of f , and we write f = { f , . . . , f l } .If each k i = 1, then this is a splitting. When each map f i is irreducible, we say { f , . . . , f l } is a partition of f into irreducibles .Consider a simple example: Example 1.1.
Let f : S → D n ( S ) be the 4-valued circle map given by f ( x ) = (cid:26) x , x , x , x (cid:27) . where S is parameterized as real numbers read modulo 1. In the terminologyof [1], f is the linear 4-valued circle map of degree 2. The graph of f , definedby { ( x, y ) | y ∈ f ( x ) } , looks like:In this example, f partitions into two 2-valued maps: f ( x ) = (cid:26) x , x (cid:27) , f ( x ) = (cid:26) x , x (cid:27) , and this is a partition into irreducibles.We will discuss partitions of n -valued maps from several viewpoints: InSection 2 we recall existing work which relates partitions to the connected com-ponents of the graph of f . In Section 3 we relate partitions to liftings of themap to mixed configuration spaces. In Section 4 we show how partitions relateto the theory of lifting classes in [4], and in Section 5 we show some results con-cerning partitions of linear n -valued maps on tori. In Section 6 we extend somework from [4] concerning the Jiang property for n -valued maps. We conclude inSection 7 by pointing out some connections to the work of Crabb on the fixedpoint theory of finite-valued maps. f In this paper all spaces will be assumed to be finite polyhedra. For any function f : X → D n ( Y ), let Γ f ⊂ X × Y be the graph of f , defined by:Γ f = { ( x, y ) ∈ X × X | y ∈ f ( x ) } . n -valued maps, which he calls“ w -splittings”. Brown’s Proposition 2.1 states that when Γ f has k path com-ponents, then f has a partition into k maps. In fact Brown’s proof implies thefollowing more specific statement: Theorem 2.1 ([2], Proposition 2.1) . For any map f : X → D n ( Y ) , there is anirreducible partition f = { f , . . . , f k } if and only if Γ f has k path components.In this case, the components of Γ f are exactly the sets Γ f k . It is well known that splittings, when they exist, are unique. Since Γ f has a unique decomposition into its path components, we obtain the followinggeneralization: Corollary 2.2.
Any map f : X → D n ( Y ) , has a unique irreducible partitionas f = { f , . . . , f k } , where perhaps k = 1 and the partition is unique up to theordering of the f i . A basic criterion for the existence of a splitting involves the covering of the or-dered configuration space over the unordered configuration space. As discussedin [8, Section 2.1], the map r : F n ( Y ) → D n ( Y ) which forgets the ordering ofan n -element configuration is a covering map, and a map f : X → D n ( Y ) splitsif and only if it has a lifting ˆ f : X → F n ( Y ). In this section we generalize thisresult for partitions.As in [9], p.131, if ( d , . . . , d k ) is a partition of n (that is, the d i are positiveintegers summing to n ), let D d ,...,d k ( Y ) be the quotient of F n ( Y ) by Σ d ×· · · × Σ d k , where Σ d is the symmetric group on d elements. This is the “mixedconfiguration space” of n points, in which certain subsets of the n points areordered. In particular when k = 2 and n = m + l we have D m,l ( Y ) = { ( { y , . . . , y m } , { y m +1 , . . . , y n } ) | y i = y j for i = j } Proposition 3.1.
For m + l = n , there are coverings s m,l : F n ( Y ) → D m,l ( Y ) and t m,l : D m,l ( Y ) → D n ( Y ) with t m,l ◦ s m,l = r .Proof. The required coverings are simply: s m,l ( y , . . . , y n ) = ( { y , . . . , y m } , { y m +1 , . . . , y n } ) t m,l ( { y , . . . , y m } , { y m +1 , . . . , y n } ) = { y , . . . , y n } . The following theorem generalizes the splitting criterion discussed above.
Theorem 3.2.
Let f : X → D n ( Y ) be an n -valued map. Then f has a properpartition if and only if there are positive integers m and l with m + l = n and f lifts to a map ˆ f : X → D m,l ( Y ) . roof. First assume that f has a proper partition, say f = g ∪ h where g : X → D m ( Y ) and h : X → D l ( Y ). Then let ˆ f be defined by ˆ f ( x ) = ( g ( x ) , h ( x )).Since f = g ∪ h , the sets g ( x ) and h ( x ) are disjoint for all x , and thus ˆ f is thedesired lift ˆ f : X → D m,l ( Y ).For the converse, assume that f has a lift ˆ f : X → D m,l ( Y ). This ˆ f will bea pair of functions of the form f ( x ) = ( g ( x ) , h ( x )) where g : X → D m ( Y ) and h : X → D l ( Y ), and this g and h provide the desired partition.The fundamental group π ( D n ( X )) is the n -strand braid group B n ( X ).When ( d , . . . , d k ) is a partition of n , let B d ,...,d k ( Y ) = π ( D d ,...,d k ( Y )). Thisis the mixed braid group on Y first defined in [11], the subgroup of B n ( Y ) inwhich the strands starting in the first d positions must end in the first d po-sitions, the strands starting in the next d positions must end in the next d positions, etc. When all d i = 1, we have B ,..., ( Y ) = P n ( Y ) = π ( F n ( Y )), thepure braid group on Y . When k = 1 and thus d = n , we obtain simply B n ( Y ),the full braid group on Y .Our next theorem uses the “lifting criterion” from covering space theory:Let p : ¯ Y → Y be a covering space and f : X → Y be a map with X pathconnected and locally path connected. Then a lift ¯ f : X → ¯ Y exists if and onlyif f ( π ( X )) ⊆ p ( π ( ¯ Y )). See [10], Proposition 1.33.The following is a generalization of the “Splitting Characterization Theo-rem”, Theorem 3.1 of [5], which states that f splits if and only if f ( π ( X ))is a subgroup of P n ( Y ), where t m,l : D m,l ( Y ) → D n ( Y ) whichforgets the ordering in D m,l ( Y ). Its induced homomorphism in fundamentalgroups is t m,l : B m,l ( Y ) → B n ( Y ), which is the inclusion. Theorem 3.3.
Let f : X → D n ( Y ) be an n -valued map. Then f has a properpartition if and only if there are positive integers m and l with m + l = n suchthat f ( π ( X )) is a subgroup of t m,l ( B m,l ( Y )) .Proof. By Theorem 3.2, f has a proper partition if and only if there are m, l with m + l = n and f lifts to some ˆ f : X → D m,l ( Y ). By the lifting criterion,this is equivalent to f ( π ( X )) ⊆ t m,l ( B m,l ( Y )). A fixed point of such a map is a point x ∈ X with x ∈ f ( x ), and we denotethe set of fixed points of f by Fix( f ) ⊂ X . Topological fixed point theory of n -valued maps was first studied by Schirmer in [12, 13].For the rest of the paper we will focus on selfmaps f : X → D n ( X ). First webriefly review the theory of lifting classes and Reidemeister classes for n -valuedmaps which is developed in [4]. Given the universal covering p : ˜ X → X , the orbit configuration space with respect to this cover is: F n ( ˜ X, π ) = { (˜ x , . . . , ˜ x n ) ∈ ˜ X n | p (˜ x i ) = p (˜ x j ) for i = j } , p n : F n ( ˜ X, π ) → D n ( X ) given by applying p to each coordinate isa covering map.Given a map f : X → D n ( X ) and some choices of basepoints for ˜ X and F n ( ˜ X, π ), there is a well-defined basic lifting ¯ f : ˜ X → F n ( ˜ X, π ) so that thediagram commutes: ˜
X F n ( ˜ X, π ) X D n ( X ) ¯ fp p n f This basic lifting ¯ f is a lifting of f with respect to the covering p n : F n ( ˜ X, π ) → D n ( X ). The covering group of F n ( ˜ X, π ) is π ( X ) n ⋊ Σ n , where Σ n is thesymmetric group on n elements. Given α ∈ π ( X ), we have the basic lift ¯ f , andviewing α as a covering transformation on ˜ X gives ¯ f ◦ α , which is some otherlift of f .Lemma 2.5 of [4] describes how any n -valued map f determines a homomor-phism ψ f : π ( X ) → π ( X ) n ⋊ Σ n by the formula: ψ f ( α ) ◦ ¯ f = ¯ f ◦ α. Writing ψ f in coordinates as ψ f ( α ) = ( φ ( α ) , . . . , φ n ( α ); σ ( α )), we obtainfunctions φ i : π ( X ) → π ( X ) and σ : π ( X ) → Σ n such that (writing σ ( α ) = σ α ): ¯ f i ( α ˜ x ) = φ i ( α ) ¯ f σ − α ( i ) (˜ x ) . The functions φ i : π ( X ) → π ( X ) are not necessarily homomorphisms, but σ is. Given some i, j ∈ { , . . . , n } and α, β ∈ π ( X ), the pairs ( α, i ) and ( β, j )are Reidemeister equivalent , and we write [( α, i )] = [( β, j )], when there is some γ ∈ π ( X ) with σ γ ( j ) = i and α = γβφ i ( γ − ) . This is an equivalence relation, and the number of these equivalence classes isthe
Reidemeister number R ( f ). Any set of the form p (Fix( α ¯ f i )) for α ∈ π ( X )is called a fixed point class , and by [4, Theorem 2.9] two nonempty fixed pointclasses p (Fix( α ¯ f i )) and p (Fix( β ¯ f j )) are equal if and only if [( α, i )] = [( β, j )],and are disjoint when [( α, i )] = [( β, j )].For i, j ∈ { , . . . , n } , we will write i ∼ j when there is some γ ∈ π ( X ) with σ γ ( j ) = i . Proposition 4.1.
The relation ∼ is an equivalence relation.Proof. The required properties will follow from the fact that σ is a homomor-phism.For reflexivity, since σ is a homomorphism we have σ ( i ) = i and thus i ∼ i .For symmetry, let i ∼ j with σ α ( j ) = i . Then σ α − ( i ) = σ − α ( i ) = j and so j ∼ i . 5or transitivity, let i ∼ j and j ∼ k with σ α ( j ) = i and σ β ( k ) = j . Then: σ αβ ( k ) = σ α ( σ β ( k )) = σ α ( j ) = i and so i ∼ k .The relation ∼ divides the basic lift ¯ f = { ¯ f , . . . , ¯ f n } into equivalence classeswhere ¯ f i and ¯ f j are in the same class when i ∼ j . These classes are called the σ -classes of the basic lift.The next two results relate partitions of f to the structure of the σ -classes. Theorem 4.2.
Let f : X → D n ( X ) be an n -valued map, and let ¯ f : ˜ X → F n ( ˜ X, π ) be the basic lift. Let ¯ g = { ¯ f i , . . . , ¯ f i k } ⊂ { ¯ f , . . . , ¯ f n } = ¯ f be a σ -class. Then there is a k -valued submap of f which lifts to ¯ g .Proof. We define g : X → D k ( X ) by g ( x ) = p k (¯ g (˜ x )), where ˜ x ∈ ˜ X is somepoint with p (˜ x ) = x . This g is a k -valued submap of f which lifts to ¯ g . We needonly show that g is well-defined with respect to the choice of point ˜ x ∈ p − ( x ).That is, we must show that p k (¯ g (˜ x )) = p k (¯ g ( α ˜ x )) for any α ∈ π ( X ).Since { ¯ f i , . . . , ¯ f i k } is a σ -class, we will have σ α ( { i , . . . , i k } ) ⊆ { i , . . . , i k } .And since σ α is a permutation and these are both sets of k elements, we willhave σ α ( { i , . . . , i k } ) = { i , . . . , i k } . Thus we have p k (¯ g ( α ˜ x )) = { p ( ¯ f i ( α ˜ x )) , . . . , p ( ¯ f i k ( α ˜ x )) } = { p ( φ i ( α ) ¯ f σ − α ( i ) (˜ x )) , . . . , p ( φ i k ( α ) ¯ f σ − α ( i k ) (˜ x )) } = { p ( ¯ f σ − α ( i ) (˜ x )) , . . . , p ( ¯ f σ − α ( i k ) (˜ x )) } = { p ( ¯ f i (˜ x )) , . . . , p ( ¯ f i k (˜ x )) } = p k (¯ g (˜ x )) , as desired. Theorem 4.3.
Let g : X → D k ( X ) be a submap of f : X → D n ( X ) , andlet ¯ f = ( ¯ f , . . . , ¯ f n ) be the basic lifting of f . Then g has a lifting of the form ¯ g = ( ¯ f i , . . . , ¯ f i k ) for some indices i j ∈ { , . . . , n } .Furthermore, if g is irreducible, then { ¯ f i , . . . , ¯ f i k } is a σ -class of ¯ f .Proof. For the first statement, let x ∈ X be the basepoint, and we have g ( x ) ⊂ f ( x ). Let ˜ x ∈ p − ( x ) be the basepoint, and let i j be chosen sothat p k ( ¯ f i (˜ x ) , . . . , ¯ f i k (˜ x )) = g ( x ). Then clearly ¯ g = ( ¯ f i , . . . , ¯ f i k ) is a liftingof g .For the second statement, we will prove the contrapositive. Assume that { ¯ f i , . . . , ¯ f i k } is not a σ -class, and we will show that g is reducible. Since { ¯ f i , . . . , ¯ f i k } is not a single σ -class, it has a nontrivial partition into σ -classes.Without loss of generality assume that { ¯ f i , . . . , ¯ f i l } is a σ -class for some l < k .Now let h = p l { ¯ f i , . . . , ¯ f i l } , and we have h ( x ) ⊂ g ( x ) and h ( x ) has cardinal-ity l < k for each x . Thus g is reducible, and we have proved the contrapositiveof the second statement. 6ombining the two theorems above shows that the partitions of f are exactlydetermined by the σ -classes: Theorem 4.4.
There is a bijective correspondence between the set of irreduciblesubmaps of f and the set of σ -classes of the ¯ f i . The next theorem shows that fixed point classes of f naturally respect par-titions of f . Theorem 4.5.
Let f be an n -valued map with a partition f = { g, h } . Thenevery fixed point class of g is disjoint from every fixed point class of h .Proof. Since g and h are submaps of f , each fixed point class of g has the form p Fix( α ¯ f i ) for some α ∈ π ( X ), and each fixed point class of h has the form p Fix( β ¯ f j ) for some β ∈ π ( X ). Furthermore, the lifts ¯ f i and ¯ f j belong todifferent σ -classes, since g and h do not contain a common irreducible submap.Thus there is no γ with σ γ ( j ) = i , and thus [( α, i )] = [( β, j )] and so p Fix( α ¯ f i ) ∩ p Fix( β ¯ f j ) = ∅ .Since the Nielsen number is the number of essential fixed point classes, weimmediately obtain: Corollary 4.6. If f has a partition f = { g, h } , then N ( f ) = N ( g ) + N ( h ) . Corollary 4.7. If f = { f , . . . , f k } is the partition into irreducibles, then N ( f ) = N ( f ) + · · · + N ( f k ) . (1) The n -valued Nielsen theory on tori has been studied in detail for the class of linear n -valued maps , introduced in [6]. This class of maps includes all n -valuedmaps on the circle (up to homotopy).Denote the universal covering space of the torus T q by p q : R q → T q where p ( t ) = t mod 1. A q × q integer matrix A induces a map f A : R q / Z q = T q → T q by f A ( p q ( v )) = p q ( A v ) = ( p ( A · v ) , . . . , p ( A q · v )) , where A j is the j -th row of A .Given x = ( x , . . . , x q ) , y = ( y , . . . , y q ) ∈ R q , we say that x = y mod n when x j = y j mod n for each j . Let c be the vector whose coordinates allequal 1. Define f kn,A : R q → R q by f kn,A ( t ) = 1 n ( At + k c ) . Theorem 5.1. ([6], Theorem 3.1, Theorem 4.2) A q × q integer matrix A induces an n -valued map f n,A : T q → D n ( T q ) defined by f n,A ( p q ( t )) = p q { f n,A ( t ) , . . . , f nn,A ( t ) } f and only if A i = A j mod n for all i, j ∈ { , . . . , q } .For this map, we have N ( f n,A ) = n | det( I − n A ) | , where I is the identitymatrix. In the case q = 1, our space is T = S , the circle. The theory of n -valuedmaps on the circle was extensively studied by Brown in [1]. Brown showed thatany n -valued map f : S → D n ( S ) is homotopic to some linear map. The q × q -matrix A for this linear map is a single integer, called the degree of f , andthe formula for the Nielsen number is simply N ( f ) = | n − d | .Brown also showed that f n,d : S → D n ( S ), the linear circle map of degree d , is split if and only if n | d . In that case, f splits into n maps, each homotopicto the map of degree d/n . We generalize this result to partitions of linear torusmaps as follows: Theorem 5.2.
Let f n,A : T q → D n ( T q ) be a linear torus map induced by some q × q -matrix A , such that, for each row A j , we have A j = [ l , . . . , l q ] mod n .Then f n,A has a nontrivial partition if and only if there is some m ∈ Z with m > which is a common factor of n and every l j .In this case, f n,A partitions into a a set of m maps, each homotopic to thelinear nm -valued map f nm , n A .Proof. We will compute the σ -orbits of f n,A . Let { e , . . . , e q } be the standardbasis vectors for R q . Since σ : π ( T q ) → Σ n is a homomorphism, it suffices tocompute σ e j for each j . Since every row A j of A has the form A j = [ l , . . . , l q ]mod n , every column A e j has every entry equal to l j mod n , that is, A e j = l j c mod n .We have: f kn,A ( v + e j ) = 1 n ( A ( v + e j ) + k c ) = 1 n ( A v + A e j + k c )= 1 n ( A v + ( k + l j ) c ) mod 1 = f k + l j n,A ( v ) mod 1and we see that σ e j ( k ) = k + l j . Thus the σ -class of k is all of { , . . . , n } if andonly if some l j has no nontrivial common factor with n , and we have proved thefirst statement of the theorem.For the second statement, assume that m > n and every l j . Then the σ -orbits in { , . . . , n } are simply the conjugacyclasses modulo n/m . Thus there are n/m different σ -orbits, each consistingof m elements. Then f n,A partitions into m maps, each being n/m -valued.Given some k ∈ { , . . . , n } , divide k by m to obtain a quotient and remainder k = sm + r with s ≤ k and 0 ≤ r < m . Then we have f kn,A ( v ) = 1 n ( A v + k c ) = 1 n ( A v + sm c ) + 1 n ( r c ) = mn ( 1 m A v + 1 m ( sm ) c ) + rn c = mn ( 1 m A v + s c ) + rn c = f s mn , m A ( v ) + rn c . k , . . . , k n/m each have the same remainder, i.e. k i = ( i − m + r ,then the set { f k n,A , . . . , f k n/m n,A } differs from the linear n/m -valued map f mn , m A by the constant rn c , and thus they are homotopic, proving the second statement.Letting m = n in Theorem 5.2 gives the following result, which was proveddirectly by Brown & Lin in [6]: Corollary 5.3.
With notation as in Theorem 5.2, the linear map f n,A : T q → D n ( T q ) splits if and only if n divides every l j , and in this case f n,A splits into n maps, each homotopic to f , n A . For the special case of circle maps (where q = 1), Theorem 5.2 gives: Corollary 5.4.
Let f : S → D n ( S ) be a circle map of degree d . Then f hasa nontrivial partition if and only if n and d have a common factor m > . Inthis case, f partitions into m maps, each an n/m -valued map of degree d/m . No specific example of n -valued maps on tori has been presented in theliterature other than linear maps. The following example is a torus map whichhas linear partitions, but is not itself linear. The example is specific, but notvery remarkable- it is easy to construct similar examples. Example 5.5.
Let: A = (cid:20) (cid:21) , B = (cid:20) − (cid:21) , and let g ( t ) be the linear 2-valued map g ( t ) = f ,A ( t ) and h ( t ) be the followingtranslation of a linear 2-valued map: h ( t ) = f ,B ( t ) + (cid:20) / (cid:21) where the entries of h ( t ) are read mod 1.Then g and h are each 2-valued maps g, h : T → D ( T ), and we claim that f = { g, h } is a 4-valued map f : T → D ( T ).We must show that f ( x ) is always a set of 4 elements. Clearly f lifts to¯ f = (¯ g , ¯ g , ¯ h , ¯ h ), where¯ g ( t ) = f ,A ( t ) , ¯ g ( t ) = f ,A ( t ) , ¯ h ( t ) = f ,B ( t ) + (cid:20) / (cid:21) , ¯ h ( t ) = f ,B ( t ) + (cid:20) / (cid:21) and so it suffices to show that p ( ¯ f ( t )) is always a set of 4 distinct elements, thatis, that the 4 elements: ¯ g ( t ) , ¯ g ( t ) , ¯ h ( t ) , ¯ h ( t ) never differ by integer vectors.9e already know that this is true for ¯ g and ¯ g , and for ¯ h and ¯ h . We mustcheck the others. For t = ( t , t ), we have:¯ g ( t ) − ¯ h ( t ) = 12 (cid:18) At + (cid:20) (cid:21)(cid:19) − (cid:18) (cid:18) Bt + (cid:20) (cid:21)(cid:19) + (cid:20) / (cid:21)(cid:19) = 12 ( A − B ) t − (cid:20) / (cid:21) = (cid:20) (cid:21) t − (cid:20) / (cid:21) = (cid:20) t − / t (cid:21) , and this vector cannot be integer valued. Similar computations show¯ g ( t ) − ¯ h ( t ) = (cid:20) (cid:21) t − (cid:20) − / − / (cid:21) = (cid:20) t + 1 / t + 1 / (cid:21) ¯ g ( t ) − ¯ h ( t ) = (cid:20) (cid:21) t − (cid:20) / / (cid:21) = (cid:20) t − / t − / (cid:21) ¯ g ( t ) − ¯ h ( t ) = (cid:20) (cid:21) t − (cid:20) / (cid:21) = (cid:20) t − / t (cid:21) and none of these vectors can be integer valued.Thus f = { g, h } is a 4-valued map. In [6] it is shown that N ( f n,A ) = n | det( I − n A ) | , where I is the identity matrix. Then by Corollary 4.6 we have N ( f ) = N ( g ) + N ( h ), and since h is homotopic to the linear map f ,B , we have: N ( f ) = N ( f ,A ) + N ( f ,B ) = 2 | det( I − A ) | + 2 | det( I − B ) | = 1 + 3 = 4 . In the example above, f is nonsplit and nonlinear, but it does partition intosubmaps which are homotopic to linear maps. It is not clear if all n -valued torusmaps can be partitioned in this way. Question 5.6.
Given any f : T q → D n ( T q ), does f have a partition intosubmaps, each of which is homotopic to a linear map? The Jiang subgroup for n -valued maps was defined in [4] as follows. Let f : X → D n ( X ) be an n -valued map with a chosen reference lift ¯ f : ˜ X → F n ( ˜ X, π ). Ahomotopy h : X × I → D n ( X ) is a cyclic homotopy of f if h ( x,
0) = h ( x,
1) = f ( x ) for all x ∈ X . A cyclic homotopy of f will lift to a homotopy ¯ h : ˜ X × I → F n ( ˜ X, π ) with ¯ h (˜ x,
0) = ¯ f (˜ x ) and ¯ h (˜ x,
1) = ( α ; η ) ¯ f (˜ x ) for some ( α ; η ) ∈ π ( X ) n ⋊ Σ n . The Jiang subgroup for n -valued maps J n ( ¯ f ) ⊆ π ( X ) n ⋊ Σ n isthe set of all elements ( α ; η ) ∈ π ( X ) n ⋊ Σ n obtained in this way from cyclichomotopies.Recall from the work of [4] that f determines a homomorphism ψ f : π ( X ) → π ( X ) n ⋊ Σ n .The following clarifies the work in [4], showing that the traditional Jiang-results concerning the equality of the fixed point index are satisfied when themap is irreducible. 10 heorem 6.1. Let ψ f ( π ( X )) ⊂ J n ( ¯ f ) , and let f be irreducible. Then all fixedpoint classes of f have the same index.Proof. Proposition 6.7 of [4] says that if ψ f ( π ( X )) ⊂ J n ( ¯ f ) and if there is some γ with σ γ ( j ) = i , then p Fix( α ¯ f i ) and p Fix( β ¯ f j ) have the same index for any α, β ∈ π ( X ). Since f is irreducible, such a γ does exist for any i and j , andthus all fixed point classes have the same index.We immediately obtain: Corollary 6.2.
Let ψ f ( π ( X )) ⊂ J n ( ¯ f ) , and let f = { f , . . . , f k } be a partitioninto irreducibles. Then for each i , all fixed point classes of f i have the sameindex. Parallel to the work of Schirmer & Brown on n -valued Nielsen theory, M. C.Crabb has developed a Nielsen theory in the context of multivalued maps f : X ⊸ X which satisfy f ( x ) ≤ n for each x . The theory of n -valued maps is aspecial case of Crabb’s theory, and in many cases Crabb’s point of view providesopportunities for simpler computation. Crabb’s survey paper [7] details theconnections between the two theories, which we will briefly review.In the most generality, Crabb discusses local maps on ENRs, but for ourpurposes we will focus on global maps on compact polyhedra. Let X be acompact polyhedron, let q : ˆ X → X be a finite covering map, and F : ˆ X → X be any map. When ˆ X is an n -fold cover, the map F : ˆ X → X resembles amap X → X , but with the points of the domain each being replicated n -timesaccording to the cover q . Specifically, f ( x ) = F ( q − ( x )) defines a multivaluedmap f : X ⊸ X with f ( x ) ≤ n for each x . Crabb writes the pair ( F, q )as a fraction
F/q , which he calls an “ n -valued map.” To avoid confusion interminology, we will call F/q a finite-valued map with cardinality at most n .Typically we will simply call it a finite-valued map , as the cardinality boundwill usually be clear (and will usually be n ).Any n -valued map f : X → D n ( X ) can naturally be expressed as a finite-valued map, as follows. The following appears in more generality as Proposition5.3 of [7]. Theorem 7.1.
Let f : X → D n ( X ) be an n -valued map. Let ˆ X = Γ f be thegraph of f , and let q : ˆ X → X be given by q ( x, y ) = x and F : ˆ X → X be givenby F ( x, y ) = y . Then F/q is a finite-valued map with cardinality at most n , and f = F ◦ q − for all x .Proof. It is clear that
F/q is finite-valued, we need only show that f = F ◦ q − .Take some x ∈ X , and we will show that f ( x ) and F ( q − ( x )) are equal as sets.First take y ∈ f ( x ), which means that ( x, y ) ∈ Γ f , which implies ( x, y ) ∈ q − ( x ). Then applying F gives y ∈ F ( q − ( x )).11onversely, take y ∈ F ( q − ( x )), so there is some pair z ∈ q − ( x ) ⊂ Γ f with F ( z ) = y . Since z ∈ q − ( x ), the first coordinate of z must be x . Since F ( z ) = y ,the second coordinate of z must be y . Thus z = ( x, y ), and so ( x, y ) ∈ Γ f andthus y ∈ f ( x ).When f : X → D n ( X ) and we use the coordinate projections q : Γ f → X and F : Γ f → X as above, the finite-valued map F/q will be called the finite-valued map associated to f .The fixed point set of a finite-valued map F/q is defined as a coincidenceset: Fix(
F/q ) = Coin(
F, q ) = { ˆ x ∈ ˆ X | F (ˆ x ) = q (ˆ x ) } . Note that, when
F/q is the finite-valued map associated to an n -valued map f ,we have Fix( F/q ) ⊂ ˆ X while Fix( f ) ⊂ X . In this case we will have q Fix(
F/q ) =Fix( f ). Generally for a finite-valued map F/q , the sets Fix(
F/q ) and q Fix(
F/q )may have a different number of elements.Crabb defines a fixed point index, Lefschetz number, and Nielsen numberfor finite-valued maps, and proves that his definitions match the correspondingdefinitions by Schirmer and Brown when
F/q corresponds to an n -valued map.For any finite-valued map F/q , we may consider the set of components ofˆ X , which we write ˆ X , . . . , ˆ X k . Then F : ˆ X → X naturally decomposes into k component maps F i : ˆ X i → X .From Theorem 2.1 we immediately obtain: Theorem 7.2.
Let f : X → D n ( X ) be an n -valued map, and let F/q bethe associated finite-valued map. Then there is an irreducible partition f = { f , . . . , f k } if and only if ˆ X has k components, and in this case we have f i = F i ◦ q − for each i . Thus if an n -valued map is presented as a finite-valued map F/q (in partic-ular if we are given the structure of the covering q : ˆ X → X ), then determiningthe irreducible partition of f becomes obvious: it is simply given by the com-ponent maps F i /q . Crabb discusses this decomposition of F/q into components F i /q , and obtains additivity formulas analogous to (1).Furthermore, the set of all finite-valued maps on a space X is often easy toclassify from this point of view. For example if X is a torus, then any connectedfinite cover of X is also a torus of the same dimension. Thus if F/q is a finite-valued map on X , then the cover ˆ X naturally decomposes into components ˆ X i ,each of which are tori, and so F/q has an irreducible partition into finite-valuedtorus maps F i /q . Each F i : ˆ X i → X is a single-valued self-map of a torus, andso can be linearized into a matrix A i , and thus N ( F i /q ) = | det( I − A i ) | , andso by (1) we have N ( F/q ) = X i | det( I − A i ) | . Thus we have a solution to the problem of computing N ( f ) when f : T q → D n ( T q ) is an n -valued torus map, provided that f is specified as a finite-valued12ap F/q and we are given enough information to deduce the matrices A i of thecomponent maps F i /q .The paragraph above would appear to answer Question 5.6 in the affirmative,but the situation is more subtle. Crabb’s work shows that every finite-valuedtorus map F/q is homotopic through finite-valued maps to one with linear par-titions. Even if
F/q corresponds to an n -valued map, there is no guarantee thatthe intermediate maps of the homotopy linearizing the components of F/q willbe n -valued.We note that although the Nielsen number defined by Crabb’s theory agreeswith that defined by Schirmer, the minimal number of fixed points may not.Specifically, if f : X → D n ( X ) is an n -valued map, let MF( f ) be the minimalnumber of fixed points of any n -valued map homotopic (by n -valued homotopy)to f , and if F/q is a finite-valued map, let MF(
F/q ) be the minimal number offixed points of any finite-valued map
G/q with G homotopic (as a single-valuedmap) to F .If F/q is the finite-valued map associated to f , then both MF( f ) and MF( F/q )will be bounded below by the Nielsen number N ( f ) = N ( F/q ). Any n -valuedhomotopy of f corresponds naturally to a finite-valued homotopy of F/q , butthe converse may not be true. Thus we will have N ( f ) ≤ MF(
F/q ) ≤ MF( f ).We suspect that the second inequality can be strict in some cases, but we donot have an example. Question 7.3.
Let f : X → D n ( X ) be an n -valued map, and let F/q be thecorresponding finite-valued map. Is MF(
F/q ) = MF( f )?The equality in question is related to the Wecken problem for n -valued maps:for which spaces and selfmaps f : X → D n ( X ) will we have N ( f ) = MF( f )?This can be a difficult question even for simple spaces, see [3] which proves theWecken property for any n -valued map of the sphere S . Whenever N ( f ) =MF( f ), we will automatically have MF( F/q ) = MF( f ). References [1] Robert F. Brown. Fixed points of n -valued multimaps of the circle. Bulletinof the Polish Academy of Sciences. Mathematics , 54(2):153–162, 2006.[2] Robert F. Brown. Nielsen numbers of n -valued fiber maps. J. Fixed PointTheory Appl. , 4(2):183–201, 2008.[3] Robert F. Brown, Michael Crabb, Adam Ericksen, and Matthew Stimpson.The 2-sphere is Wecken for n -valued maps. J. Fixed Point Theory Appl. ,21(2):Paper No. 55, 6, 2019.[4] Robert F. Brown, Charlotte Deconinck, Karel Dekimpe, and P. Christo-pher Staecker. Lifting classes for the fixed point theory of n -valued maps. Topology Appl. , 274:107125, 26, 2020.135] Robert F. Brown and Daciberg Lima Gon¸calves. On the topology of n -valued maps. Advances in Fixed Point Theory , 8(2):205–220, 2018.[6] Robert F. Brown and Jon T. Lo Kim Lin. Coincidences of projections andlinear n -valued maps of tori. Topology Appl. , 157(12):1990–1998, 2010.[7] Michael Crabb. On the definition of an n -valued map. 2018. preprint.[8] Daciberg Lima Gon¸calves and John Guaschi. Fixed points of n -valuedmaps, the fixed point property and the case of surfaces–A braid approach. Indagationes Mathematicae , 29(1):91 – 124, 2018. L.E.J. Brouwer, fiftyyears later.[9] Daciberg Lima Gon¸calves and John Guaschi. The roots of the full twist forsurface braid groups.
Math. Proc. Cambridge Philos. Soc. , 137(2):307–320,2004.[10] Allen Hatcher.
Algebraic topology . Cambridge University Press, Cambridge,2002.[11] Sandro Manfredini. Some subgroups of Artin’s braid group.