Computation of Nielsen and Reidemeister coincidence numbers for multiple maps
aa r X i v : . [ m a t h . A T ] J a n COMPUTATION OF NIELSEN AND REIDEMEISTER COINCIDENCENUMBERS FOR MULTIPLE MAPS
THA´IS F. M. MONIS AND PETER WONG
Abstract.
Let f , ..., f k : M → N be maps between closed manifolds, N ( f , ..., f k ) and R ( f , ..., f k )be the Nielsen and the Reideimeister coincidence numbers respectively. In this note, we re-late R ( f , ..., f k ) with R ( f , f ) , ..., R ( f , f k ). When N is a torus or a nilmanifold, we compute R ( f , ..., f k ) which, in these cases, is equal to N ( f , ..., f k ). Introduction
A central problem in classical Nielsen coincidence theory is the computation of the Nielsen coin-cidence number N ( f, g ) for two maps f, g : M → N between two closed orientable manifolds ofthe same dimension. Moreover, the classical Reidemeister number R ( f, g ) is an upper bound for N ( f, g ).In [16], P. C. Staecker stablished a theory for coincidences of multiple maps called Nielsen equalizertheory : given k maps f , . . . , f k : M ( k − n → N n between compact manifolds of dimension ( k − n and n , respectively, a Nielsen number N ( f , . . . , f k ) is defined such that it is a homotopy invariantand a lower bound for the cardinality of the setsCoin( f ′ , . . . , f ′ k ) = { x ∈ M | f ′ ( x ) = · · · = f ′ k ( x ) } , where f ′ i is homotopic to f i . The approach of Staecker to define the essentiality of a coincidence classis via a coincidence index or semi-index. Independently, Biasi, Libardi & Monis ([3]) introduceda Lefschetz coincidence class for k maps f , . . . , f k : X → N n , L ( f , . . . , f k ) ∈ H ( k − n ( X ; Z ),where N is a closed connected orientable n -manifold, with the property that if L ( f , . . . , f k ) is anon-trivial element of H ( k − n ( X ; Z ) then there is x ∈ X such that f ( x ) = · · · = f k ( x ). In [13],Monis & Spie˙z generalized this Lefschetz class to the case where N is not necessarily orientable by Date : January 22, 2020.2010
Mathematics Subject Classification.
Primary: 55M20; Secondary: 22E25.
Key words and phrases.
Topological coincidence theory, Nielsen coincidence number, nilmanifolds.This work is supported by FAPESP Grant 2018/03550-5. using twisted coefficients and in [14], Monis & Wong determined the obstruction class to deformthe maps f , . . . , f k : M ( k − n → N n to be coincidence free.In this paper, we focus on the computation of R ( f , ..., f k ) in terms of the Reidemeister coinci-dence numbers R ( f , f ) , R ( f , f ) , ..., R ( f , f k ). In particular, we study the cases when the targetmanifold is either a torus or a nilmanifold.We should point out that in the case of positive codimension, other Nielsen type invariants vianormal bordism have been introduced and Wecken type theorems have been proven. However,these invariants in general are not readily computable. In this work, the Nielsen coincidencenumber N ( F, G ) (and N ( f , ..., f k ) for multiple maps) we use is the geometric invariant that isindex free. Thus, the calculation is carried out at the fundamental group level.We thank the anonymous referee for his/her comments, the question related to the divisibilityresult in section 5, and the suggestion for extending Theorem 5.2. Nielsen coincidence number for multiple maps
In this section, we will define a geometric Nielsen coincidence number for multiple maps followingthe approach of [4]. Throughout this section, X is a connected finite CW-complex and Y is amanifold without boundary. Definition 1.
Let f , f , . . . , f k : X → Y be maps. Two points x , x ∈ Coin( f , . . . , f k ) are calledNielsen equivalent as coincidences with respect to f , . . . , f k if there is a path γ : [0 , → X suchthat γ (0) = x , γ (1) = x and f ( γ ) is homotopic to f j ( γ ) relative to the endpoints, j = 2 , . . . , k .Such relation defines an equivalence relation on Coin( f , . . . , f k ) and each equivalence class is calleda Nielsen coincidence class of ( f , . . . , f k ). Definition 2.
Let f , . . . , f k : X → Y be maps and { f t } , { f t } , . . . , { f tk } homotopies of f = f , f = f , . . . , f k = f k , respectively. Let F be a Nielsen coincidence class of ( f , . . . , f k ) and F aNielsen coincidence class of ( f , . . . , f k ). We say that F and F are { f t } , { f t } , . . . , { f tk } -related ifthere are x ∈ F , x ′ ∈ F and a path γ : [0 , → X with γ (0) = x , γ (1) = x ′ such that the paths t f t ( γ ( t )) and t f ti ( γ ( t )) are homotopic relative to the endpoints, i = 2 , . . . , k . Definition 3.
A Nielsen coincidence class F of ( f , . . . , f k ) is said to be (geometric) essential if forany homotopies { f t } , { f t } , . . . , { f tk } of f = f , f = f , . . . , f k = f k , there is a Nielsen coincidenceclass of ( f , . . . , f k ) to which it is { f t } , { f t } , . . . , { f tk } -related. Otherwise, it is called (geometric)inessential. OMPUTATION OF NIELSEN AND REIDEMEISTER COINCIDENCE NUMBERS FOR MULTIPLE MAPS 3
The (geometric) Nielsen coincidence number of ( f , . . . , f k ), N ( f , . . . , f k ), is the number of (geo-metric) essential Nielsen coincidence classes of ( f , . . . , f k ).A clear property with the above definition is that, for any permutation σ ∈ S k on { , ..., k } , N ( f , ..., f k ) = N ( f σ (1) , ..., f σ ( k ) ). Remark . Let F be an inessential Nielsen coincidence class of ( f , . . . , f k ). Roughly speak-ing, F disappears under some homotopy. More precisely, it means that there are homotopies { f t } , { f t } , . . . , { f tk } of f = f , f = f , . . . , f k = f k , respectively, such that every x ∈ F is not { f t } , { f t } , . . . , { f tk } -related to any coincidence x ′ ∈ Coin( f , . . . , f k ). On the other hand, F isessential if and only if for any homotopies { f t } , { f t } , . . . , { f tk } of f = f , f = f , . . . , f k = f k ,respectively, there exists x ′ ∈ Coin( f , . . . , f k ) and a path γ : [0 , → X with γ (0) = x ∈ F , γ (1) = x ′ such that t f t ( γ ( t )) and t f ti ( γ ( t )) are homotopic paths relative to the endpoints, i = 2 , . . . , k . Remark . Let
F, G : X → Y k − be defined by F = ( f , f , . . . , f ) and G = ( f , f , . . . , f k ). Itis not difficult to see that Coin( F, G ) = Coin( f , . . . , f k ) and that the Nielsen coincidence classesof ( f , . . . , f k ) coincide with the Nielsen coincidence classes of ( F, G ). In the next result, we provethat F is essential as a coincidence class of ( f , . . . , f k ) if and only if it is essential as a coincidenceclass of ( F, G ). Therefore, N ( f , . . . , f k ) = N ( F, G ). Proposition 1. N ( f , . . . , f k ) = N ( F, G ) .Proof. Let F be a Nielsen coincidence class of ( f , . . . , f k ) and, therefore, a Nielsen coincidenceclass of ( F, G ). We will show that F is essential as a coincidence class of ( f , . . . , f k ) if and only ifit is essential as a coincidence class of ( F, G ).Suppose F is essential as a coincidence class of ( F, G ). Let { f t } , { f t } , . . . , { f tk } be homotopies of f = f , f = f , . . . , f k = f k , respectively. Then, { ( f t , . . . , f t ) } , { ( f t , . . . , f tk ) } are homotopies of F and G , respectively. Since F is essential as a coincidence class of ( F, G ), there exists x ′ ∈ Coin(( f , . . . , f ) , ( f , . . . , f k )) = Coin( f , . . . , f k )and a path γ : [0 , → X with γ (0) = x ∈ F , γ (1) = x ′ such that t ( f t ( γ ( t )) , . . . , f t ( γ ( t ))) and t ( f t ( γ ( t )) , . . . , f tk ( γ ( t ))) are homotopic paths relative to the endpoints. Therefore, t f t ( γ ( t ))and t f ti ( γ ( t )) are homotopic paths relative to the endpoints, i = 2 , . . . , k . Hence, F is essentialas a coincidence class of ( f , . . . , f k ). THA´IS F. M. MONIS AND PETER WONG
Now, suppose F is essential as a coincidence class of ( f , . . . , f k ). Since Y is a manifold, it followsfrom [6, Corollary 2] that F is essential as a coincidence class of ( F, G ) iff it is essential with respectto homotopies of the form (
F, G t ), in other words, the homotopy of F can be kept constant. Since F is essential as a coincidence class of ( f , . . . , f k ), under the homotopies { f } , { f t } , . . . , { f tk } , where¯ G t = ( f t , . . . , f tk ), there exists a path γ : [0 , → X with γ (0) = x and γ (1) ∈ Coin( f , f , ..., f k ) =Coin( F, ¯ G ) such that the path { f ( γ )( t ) } and the path { f ti ( γ )( t ) } are homotopic relative to theendpoints for i = 2 , ..., k . Thus, by [6, Corollary 2], we conclude that F is essential as a coincidenceclass of ( F, G ). (cid:3) Remark . Since X is compact and the Nielsen coincidence classes are both open and closed inCoin( F, G ), this index-free geometric Nielsen number N ( F, G ) (and hence N ( f , ..., f k )) is finite, isa lower bound for the number of connected components in Coin( F, G ) (and hence Coin( f , ..., f k )),and is a homotopy invariant.3. Reidemeister coincidence classes for multiple maps
In [16], P. C. Staecker also developed a Reidemeister-type theory for coincidences of multiplemaps. Let X and Y be connected, locally path-connected, and semilocally simply connected spaces,and e X, e Y their universal covering spaces with projection maps p X : e X → X and p Y : Y → e Y ,respectivelly. Let f , . . . , f k : X → Y be maps, k >
2, and denoteCoin( f , . . . , f k ) = { x ∈ X | f ( x ) = · · · = f k ( x ) } the set of coincidences of the maps f , . . . , f k . Theorem 1 ([16], Theorem 2.1) . Let f , . . . , f k : X → Y be maps with lifts ˜ f i : e X → e Y and inducedhomomorphisms φ i : π ( X ) → π ( Y ) . Then: (1) Coin( f , . . . , f k ) = [ α ,...,α k ∈ π ( Y ) p X (Coin( ˜ f , α ˜ f , . . . , α k ˜ f k )) . (2) For α i , β i ∈ π ( Y ) , the sets p X (Coin( ˜ f , α ˜ f , . . . , α k ˜ f k )) and p X (Coin( ˜ f , β ˜ f , . . . , β k ˜ f k )) are disjoint or equal. (3) The above sets are equal if and only if there is some z ∈ π ( X ) with β i = φ ( z ) α i φ i ( z ) − for all i . OMPUTATION OF NIELSEN AND REIDEMEISTER COINCIDENCE NUMBERS FOR MULTIPLE MAPS 5
The above theorem gives rise to the definition of the set of Reidemeister classes and the Reidemeisternumber for φ , . . . , φ k : given ( α , . . . , α k ) and ( β , . . . , β k ) in π ( Y ) k − , ( α , . . . , α k ) ∼ ( β , . . . , β k )if and only if β i = φ ( z ) α i φ i ( z ) − , i = 2 , . . . , k , for some z ∈ π ( X ). The quotient of π ( Y ) k − by such equivalence relation is denoted by R ( φ , . . . , φ k ) = π ( Y ) k − / ∼ and each class is called aReidemeister class for φ , . . . , φ k . The cardinality of R ( φ , . . . , φ k ) is the Reidemeister number for φ , . . . , φ k and it is denoted by R ( φ , . . . , φ k ). Remark . Note that R ( φ , . . . , φ k ) = R (( φ , . . . , φ ) , ( φ , . . . , φ k )) the usual Reidemeister setfor the two homomorphisms ( φ , . . . , φ ) , ( φ , . . . , φ k ) : π ( X ) → π ( Y ) k − induced by the maps( f , . . . , f ) , ( f , . . . , f k ) : X → Y k − , respectivelly. Proposition 2.
For any permutation σ ∈ S k on { , ..., k } , R ( f , ..., f k ) = R ( f σ (1) , ..., f σ ( k ) ) . Proof.
Since a permutation is a product of transpositions, it is sufficient to prove the result for thetranspositions.Case 1: Suppose σ = ( ij ) with { i, j } ⊂ { , , . . . k } . Then, it is not difficult to see that R ( f , . . . , f k ) → R ( f σ (1) , ..., f σ ( k ) )[( α , . . . , α k )] → [( α σ (2) , . . . , α σ ( k ) )]is a well-defined bijection between the Reidemeister classes sets R ( f , . . . , f k ) and R ( f σ (1) , ..., f σ ( k ) ).Therefore, R ( f , ..., f k ) = R ( f σ (1) , ..., f σ ( k ) ).Case 2: Suppose σ = (1 i ), for some i ∈ { , . . . , k } . Then, R ( f , . . . , f k ) → R ( f σ (1) , ..., f σ ( k ) )[( α , . . . , α k )] → [( α − i α , . . . , α − i α i − , α − i , α − i α i +1 , . . . , α − i α k )]is a well-defined bijection between the Reidemeister classes sets R ( f , . . . , f k ) and R ( f σ (1) , ..., f σ ( k ) ).Therefore, R ( f , ..., f k ) = R ( f σ (1) , ..., f σ ( k ) ). (cid:3) Jiang type results for coincidences of multiple maps
In Nielsen coincidence theory, for a large class of spaces known as Jiang-type spaces, either N ( f, g ) =0 or N ( f, g ) = R ( f, g ). In this section, we also obtain similar results for multiple maps. THA´IS F. M. MONIS AND PETER WONG
Theorem 2.
Let X and N be compact nilmanifolds with dim X ≥ ( k −
1) dim N , k ≥ . For anymaps f , ..., f k , either N ( f , ..., f k ) = 0 or N ( f , ..., f k ) = R ( f , ..., f k ) .Proof. By Proposition 1, N ( f , ..., f k ) = N ( F, G ), where
F, G : X → N k − , F = ( f , . . . , f ) and G = ( f , . . . , f k ). Moreover, R ( f , ..., f k ) = R ( F, G ) (see Remark 4). From [8, Theorem 4.2],either N ( F, G ) = 0 or N ( F, G ) = R ( F, G ). Therefore, either N ( f , ..., f k ) = 0 or N ( f , ..., f k ) = R ( f , ..., f k ). (cid:3) When one of the maps is the constant map, we obtain similar results as in the classical root case.
Theorem 3.
Let X be a topological space, Y a topological manifold, c ∈ N and denote by ¯ c the constant map at c . For any maps f , ..., f k : X → N , k ≥ , either N (¯ c, f , ..., f k ) = 0 or N (¯ c, f , ..., f k ) = R (¯ c, f , ..., f k ) .Proof. It follows from [5, Corollary 2], since N (¯ c, f , ..., f k ) = N ((¯ c, . . . , ¯ c ) , ( f , . . . , f k ))and R (¯ c, f , ..., f k ) = R ((¯ c, . . . , ¯ c ) , ( f , . . . , f k )) . (cid:3) Analogous to [14, Theorem 6.3], we have the following
Theorem 4.
Let f , . . . , f k : M → N from a closed connected ( k − n -manifold M to a closedconnected orientable n -manifold N . Suppose N is a Jiang space; a nilmanifold, an orientable cosetspace G/K of a compact connected Lie group G by a closed subgroup K ; or a C -nilpotent spacewhose fundamental group has a finite index center where C is the class of finite groups. Then L ( f , . . . , f k ) = 0 ⇒ N ( f , ..., f k ) = 0 and L ( f , . . . , f k ) = 0 ⇒ N ( f , ..., f k ) = R ( f , ..., f k ) .Remark . Every compact nilmanifold is the total space of a principal S -bundle over anothernilmanifold. Using the upper central series, there is a sequence of S fibrations such that an 2 n -dimensional compact nilmanifold M can fiber over an n -dimensional nilmanifold N (see e.g. [18]).Consider such a fibration p : M → N and a point c ∈ N . If M is not symplectic then for anymap f : M → N , it follows from [14, Example 7.3] that the Lefschetz class L ( p, ¯ c, f ) = 0. Itfollows that N ( p, ¯ c, f ) = 0 so, by [7], R ( p, ¯ c, f ) = ∞ while R ( p, ¯ c ) < ∞ because p and ¯ c cannot bemade coincidence free. The existence of non-symplectic nilmanifolds is equivalent to the existenceof symplectic structure on nilpotent Lie algebras (see e.g., [2, 15, 1]). OMPUTATION OF NIELSEN AND REIDEMEISTER COINCIDENCE NUMBERS FOR MULTIPLE MAPS 7
It was shown in [13] that the Lefschetz class L ( f , ..., f k ) is related to the classes L ( f , f ), L ( f , f ), . . . , L ( f , f k ). Indeed, the obstruction o ( f , ..., f k ) to deforming the k -maps to be coincidence freeis the product o ( f , f ) ∪ · · · ∪ o ( f , f k ) ([14, Theorem 5.4]). Next, we investigate the relationshipbetween R ( f , ..., f k ) and the product R ( f , f ) · R ( f , f ) · · · R ( f , f k ) when these quantities arefinite.In general, there is a well-defined natural surjection(4.1) Ψ : R ( φ , . . . , φ k ) → R ( φ , φ ) × · · · × R ( φ , φ k )given by(4.2) Ψ ([( α , . . . , α k )]) = ([ α ] , . . . , [ α k ]) . Therefore, if R ( φ , . . . , φ k ) < ∞ then R ( φ , φ j ) < ∞ , j = 2 , . . . , k and(4.3) R ( φ , . . . , φ k ) ≥ R ( φ , φ ) · · · R ( φ , φ k ) . Coincidences of maps into a torus
Let ϕ, ψ : G → A be group homomorphisms, where A is an abelian group. In this case, therelation that determines the Reidemeister classes with respect to the pair ( ϕ, ψ ) can be writtenas follows: given α, β ∈ A , α ∼ β if and only if β − α = ϕ ( g ) − ψ ( g ) for some g ∈ G , that is, R ( ϕ, ψ ) = coker( ϕ − ψ ). Therefore, R ( ϕ, ψ ) = ϕ − ψ ).Let ϕ, ψ : Z m → Z n be homomorphisms, { α , . . . , α m } generators of Z m and { β , . . . , β n } gener-ators of Z n . If ϕ ( α j ) = P ni =1 a ij β i and ψ ( α j ) = P ni =1 b ij β i , the Reidemeister number R ( ϕ, ψ ) isdetermined by the integral matrices A = ( a ij ) and B = ( b ij ): if all n × n minor of A − B is zerothen R ( ϕ, ψ ) = ∞ . If some n × n minor of A − B is nonzero, then R ( ϕ, ψ ) is finite and its effectivecomputation can be done by using Smith normal form. Indeed, let C = A − B , by Smith normalform, there are matrices S ∈ M n ( Z ) and T ∈ M m ( Z ) with | det( S ) | = | det( T ) | = 1 and SCT is of
THA´IS F. M. MONIS AND PETER WONG the form l . . . l . . .
00 0 . . . ... ... ...... · · · l r . . . ...0 . . . ... . . .0 . . . It follows that R ( ϕ, ψ ) = C ) = SCT ) since coker( C ) and coker( SCT ) are isomor-phic. If we assume R ( ϕ, ψ ) < ∞ then r = n and SCT ) = l · l · · · · l n . The numbers l k aredefined by the following: l k = d k ( C ) d k − ( C ) , where d k ( C ) is the greatest common divisor of all k × k minors of C and d ( C ) = 1. For example, for the matrix C = ! ,l ( C ) = 1 and l ( C ) = 2. Therefore, C ) = 1 · m = n , R ( ϕ, ψ ) = ∞ if det( A − B ) = 0 and R ( ϕ, ψ ) = | det( A − B ) | ifdet( A − B ) = 0. Proposition 3.
Let f , f , f : T → S be maps and φ , φ , φ : π ( T ) → π ( S ) the inducedhomomorphisms. If R ( φ , φ , φ ) < ∞ then R ( φ , φ ) < ∞ , R ( φ , φ ) < ∞ and the product R ( φ , φ ) · R ( φ , φ ) divides R ( φ , φ , φ ) .Proof. Let φ i : π ( T ) → π ( S ) be the homomorphism induced by f i and denote φ i ! = a i and φ i ! = b i . For maps g, h : T → S , R ( g , h ) = h − g ). Thus,(5.1) R ( φ , φ j ) = φ j − φ ) = Z h a j − a , b j − b i , j = 2 , . Also, note that R ( φ , φ , φ ) = R (( φ , φ ) , ( φ , φ )), where ( φ , φ ) , ( φ , φ ) : π ( T ) → π ( T ) arethe homomorphisms induced by the maps ( f , f ) , ( f , f ) : T → T given by( f i , f j )( z, w ) = ( f i ( z ) , f j ( w )) . OMPUTATION OF NIELSEN AND REIDEMEISTER COINCIDENCE NUMBERS FOR MULTIPLE MAPS 9
Moreover, for maps h, g : T → T , R ( g , h ) = h − g ). Thus,(5.2) R ( φ , φ , φ ) = φ , φ ) − ( φ , φ )) = Z ⊕ Z * a − a a − a ! , b − b b − b !+ , If R ( φ , φ , φ ) < ∞ then det a − a b − b a − a b − b ! = 0 . Thus, for each j ∈ { , } , a j − a = 0 or b j − b = 0. Therefore,(5.3) R ( φ , φ j ) = φ j − φ ) = Z h a j − a , b j − b i = gcd( a j − a , b j − b ) < ∞ ,j = 2 , Z ⊕ Z * a − a a − a ! , b − b b − b !+ = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) det a − a b − b a − a b − b !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Therefore, R ( φ , φ , φ ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) det a − a b − b a − a b − b !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Let d = gcd( a − a , b − b ) = R ( φ , φ ), d = gcd( a − a , b − b ) = R ( φ , φ ), and A, B, C, D ∈ Z such that a − a = Ad , b − b = Bd , a − a = Cd and b − b = Dd . Thus,det a − a b − b a − a b − b ! = d d det A BC D ! = R ( φ , φ ) R ( φ , φ ) det A BC D ! . Therefore, R ( φ , φ , φ ) = R ( φ , φ ) · R ( φ , φ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) det A BC D !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (cid:3) The result above can easily be generalized to k >
3, as we show in Corollary 1. Note also that theReidemeister number R ( φ , φ , φ ) is divisible by the product R ( φ , φ ) · R ( φ , φ ). In fact, suchdivisibility is valid in general for abelian groups (not necessarily torsion-free).Let f, g : X → Y be maps and suppose π ( Y ) abelian. Then, R ( f , g ) is a group (abelian). Infact, R ( f , g ) = coker( g − f ) = π ( Y )im( g − f ) . Consequently, if π ( Y ) is an abelian group then R ( φ , . . . , φ k ), R ( φ , φ ), . . . , R ( φ , φ k ) are all(abelian) groups. Also, one can check that, in this case, Ψ is a group homomorphism. Hence,(5.5) R ( φ , . . . , φ k )ker(Ψ) ≃ R ( φ , φ ) × · · · × R ( φ , φ k ) . Therefore, if R ( φ , . . . , φ k ) < ∞ then(5.6) R ( φ , . . . , φ k ) = R ( φ , φ ) · · · · · R ( φ , φ k ) · | ker(Ψ) | . Corollary 1.
Suppose π ( Y ) an abelian group. If R ( φ , . . . , φ k ) < ∞ then R ( φ , φ j ) < ∞ , j =2 , . . . , k , and the product R ( φ , φ ) · · · · · R ( φ , φ k ) divides R ( φ , . . . , φ k ) .Remark . We should point out that if π ( Y ) is not abelian then the product R ( φ , φ ) · R ( φ , φ )may not divide R ( φ , φ , φ ) as we show in the next example. Example . Let Y be the Poincar´e 3-dimensional sphere, X = Y × Y , f = f = p : Y × Y → Y the projection onto the first coordinate and f = ¯ c : Y × Y → Y the constant map. The relationon π ( Y ) × π ( Y ) that defines R ( p, p, ¯ c ) is given by: ( α , α ) ∼ ( β , β ) if and only if ( β , β ) =( zα z − , zα ) for some z ∈ π ( Y ). So, it is not difficult to see that the number of elements in theclass of ( α , α ) is in 1-1 correspondence with | π ( Y ) | . Therefore, R ( φ , φ , φ ) = 120 /
120 = 120.On the other hand, the relation on π ( Y ) that defines R ( p, p ) is given by α ∼ β if and only if β = zαz − for some z ∈ π ( Y ), that is, R ( p, p ) is the set of conjugacy classes of π ( Y ), and it isknown that there are 9 conjugacy classes. Hence, R ( p, p ) = 9. For R ( p, ¯ c ), the relation on π ( Y ) isgiven by α ∼ β if and only if β = zα for some z ∈ π ( Y ). Hence, R ( p, ¯ c ) = 1. Now, 9 ∤ Remark . As suggested by the anonymous referee, one may ask if the divisibility in Proposition3 can be generalized for k > Q ki =1 R ( φ , . . . , φ i − , φ i +1 , . . . , φ k ) divides R ( φ , . . . , φ k ). The following example shows that such divisibility does not hold in general. OMPUTATION OF NIELSEN AND REIDEMEISTER COINCIDENCE NUMBERS FOR MULTIPLE MAPS 11
Example . Let f , f , f , f : T → S be maps with φ = f , φ = f , φ = f , φ = f : Z = π ( T ) → π ( S ) = Z given by φ = (cid:16) (cid:17) , φ = (cid:16) (cid:17) , φ = (cid:16) (cid:17) and φ = (cid:16) (cid:17) . We know that R ( φ , φ , φ , φ ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) det a − a b − b c − c a − a b − b c − c a − a b − b c − c (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , where φ = (cid:16) a b c (cid:17) , φ = (cid:16) a b c (cid:17) , φ = (cid:16) a b c (cid:17) and φ = (cid:16) a b c (cid:17) .Therefore, R ( φ , φ , φ , φ ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) det (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 10 . On the other hand, R ( φ , φ , φ ) = φ , φ ) − ( φ , φ )) = ! , R ( φ , φ , φ ) = φ , φ ) − ( φ , φ )) = ! and R ( φ , φ , φ ) = φ , φ ) − ( φ , φ )) = ! . By using Smith normal form, one can show that R ( φ , φ , φ ) = 2, R ( φ , φ , φ ) =1 and R ( φ , φ , φ ) = 2. Therefore, R ( φ , φ , φ ) · R ( φ , φ , φ ) · R ( φ , φ , φ ) = 4 does not divide R ( φ , φ , φ , φ ) = 10. 6. Coincidences of maps into a nilmanifold
As pointed out in Remark 4 that R ( f , ..., f k ) = R ( F, G ) where F = ( f , ..., f ) and G = ( f , ..., f k ),the computation of R ( f , ..., f k ) amounts to the computation of the Reidemeister number of twomaps. In this section, we show how the Reidemeister coincidence number R ( f, g ) can be computedfor two maps f, g : X → N when N is a nilmanifold.From [9], we know that for any ˜ f , ˜ g : X → N , there is a nilmanifold ˆ N with nilpotency class c ( π ( ˆ N )) ≤ c ( π ( N )) and two maps f, g : ˆ N → N such that, up to homotopy, ˜ f = f ◦ q and˜ g = g ◦ q where q : X → ˆ N induces a surjection on the fundamental groups. It follows that R ( f, g ) = R ( ˜ f , ˜ g ). This reduces to the case where the domain is also a compact nilmanifold.From now on, we let G = π ( ˆ N ), G = π ( N ), and c ( G ) ≤ c ( G ). Suppose ϕ, ψ are thecorresponding induced homomorphisms of f, g respectively. Central extension.
Let c = c ( G ) be the nilpotency class of G and { γ i ( G j ) } be the lowercentral series of G j , j = 1 ,
2. The subgroup γ c − ( G j ) is central in G j since { } = γ c ( G j ) =[ γ c − ( G j ) , G j ] for j = 1 , −−−−→ A i −−−−→ G p −−−−→ B −−−−→ ϕ ′ y ψ ′ ϕ y ψ ϕ y ψ −−−−→ A i −−−−→ G p −−−−→ B −−−−→ ϕ ′ = ϕ | A , ψ ′ = ψ | A , A j = γ c − ( G j ) so that B j = G j /γ c − ( G j ) for j = 1 , R ( ϕ, ψ ) = b i ( R ( ϕ ′ , ψ ′ )) · R ( ϕ, ψ ) . Here, the function b i is induced by the inclusion i as in (6.1). Lemma 1.
Given the commutative diagram (6.1) , if rk(
Coin ( ϕ ′ , ψ ′ )) = rk( A ) − rk( A ) then R ( ϕ, ψ ) · | Im δ | = R ( ϕ ′ , ψ ′ ) · R ( ϕ, ψ ) where δ : Coin ( ϕ, ψ ) → R ( ϕ ′ , ψ ′ ) is given by δ ( θ ) = [ ψ ( θ ) ϕ ( θ ) − ] where p ( θ ) = θ .Proof. It follows from [12] that there is an 8-term exact sequence1 → Coin ( ϕ ′ , ψ ′ ) → Coin ( ϕ, ψ ) p → Coin ( ϕ, ψ ) δ → R ( ϕ ′ , ψ ′ ) b i → R ( ϕ, ψ ) b p → R ( ϕ, ψ ) → A is central, it follows that R ( ϕ ′ , ψ ′ ) is an abelian group and δ is a group homomorphism.Since rk( Coin ( ϕ ′ , ψ ′ )) = rk( A ) − rk( A ), it follows from [17] that R ( ϕ ′ , ψ ′ ) is finite. It is straight-forward to verify that the map R ( ϕ ′ , ψ ′ ) / Im δ → b i ( R ( ϕ ′ , ψ ′ ))given by [ σ ]Im δ b i ([ σ ]) is a bijection of finite sets. The result follows from (6.2). (cid:3) Remark . The finiteness of R ( ϕ ′ , ψ ′ ) can be determined in another way, namely, R ( ϕ ′ , ψ ′ ) < ∞ iffrk(Im( ϕ ′ − ψ ′ )) = rk( A ). OMPUTATION OF NIELSEN AND REIDEMEISTER COINCIDENCE NUMBERS FOR MULTIPLE MAPS 13
Abelianization.
In the case where R ( ϕ ′ , ψ ′ ) = ∞ in the situation above, one can make useof the abelianization of the G j ’s as follows. Again, we have the following commutative diagram(6.3) 1 −−−−→ H i −−−−→ G p −−−−→ Q −−−−→ ϕ ′ y ψ ′ ϕ y ψ ϕ y ψ −−−−→ H i −−−−→ G p −−−−→ Q −−−−→ H i = [ G i , G i ] for i = 1 , ϕ ′ = ϕ | H , ψ ′ = ψ | H , Q = G ab , Q = G ab .At the space level, the abelianization G abj is the fundamental group of some torus T j , i = 1 ,
2. Since R ( ϕ, ψ ) < ∞ , R ( ϕ, ψ ) < ∞ . This forces (see e.g. [17] or [7]) dim T ≥ dim T , i.e., rk( Q ) ≥ rk( Q ).Following the argument from [17, Lemma 3], we can further assume that ϕ and ψ factor throughanother quotient Q of Q such that rk( Q ) = rk( Q ). Thus (6.3) becomes(6.4) 1 −−−−→ K i −−−−→ G p −−−−→ Q −−−−→ ϕ ′ y ψ ′ ϕ y ψ ϕ y ψ −−−−→ H i −−−−→ G p −−−−→ Q −−−−→ Q ) = rk( Q ) and R ( ϕ , ψ ) < ∞ , it follows that Coin ( ϕ , ψ ) = 1. Thus the Reidemeisterclasses R ( ϕ ′ , ψ ′ ) inject into R ( ϕ, ψ ). Thus, it follows from [17, Corollary 1] or [11, Theorem 2.1]that(6.5) R ( ϕ, ψ ) = X [ α ] ∈R ( ϕ ,ψ ) R ( τ α ϕ ′ , ψ ′ ) . Now, ϕ ′ , ψ ′ are homomorphisms induced by maps between compact nilmanifolds where the di-mension of the codomain is smaller than that of N . After a finite number of steps, we arrive atcomputing R ( τ α ϕ ′ , ψ ′ ) where the target group is the fundamental group of a torus, hence suchReidemeister numbers are simply the cokernels of the difference of the two homomorphisms. Inthis sense, the Reidemeister number of a pair of maps from X to a nilmanifold N can be computedusing the formula (6.5) together with the arguments described above.Using the arguments as in the previous subsections 6.1 and 6.2, one can compute R ( f, g ) when thecodomain is a compact nilmanifold.We end this section with an example illustrating the computation discussed here. Example . Consider the following finitely generated torsion-free nilpotent groups G = h a, b, c, d, e, t | [ a, b ] = c, [ a, d ] = e, [ a, c ] = [ a, e ] = [ a, t ] = [ b, c ]=[ b, d ] = [ b, t ] = [ c, d ] = [ c, t ]=[ d, e ] = [ d, t ] = [ e, t ] = 1 i and G = h α, β, γ | [ α, β ] = γ, [ α, γ ] = [ β, γ ] = 1 i . Note that, using Hall’s identity, [ b, e ] = [ c, e ] = 1 in G .Now, [ G , G ] = h c, e | [ c, e ] = 1 i is central and [ G , G ] = h γ i is the center of G . Moreover, G ab = h ¯ a, ¯ b, ¯ d, ¯ t i ∼ = Z and G ab = h ¯ α, ¯ β i ∼ = Z .For i = 1 , ,
3, we define homomorphisms ϕ i : G → G by ϕ : a α ; b β ; c γ ; d e t ϕ : a α ; b c d e t α ; ϕ : a β ; b α ; c γ − ; d α ; e γ − ; t . Consider the maps F = ( ϕ , ϕ ) and G = ( ϕ , ϕ ) and the commutative diagram(6.6) 1 −−−−→ A i −−−−→ G p −−−−→ B −−−−→ F ′ y G ′ F y G F y G −−−−→ A × A i × i −−−−→ G × G p × p −−−−→ B × B −−−−→ A i = [ G i , G i ] and B i = G abi , i = 1 ,
2. Now, F = ( ϕ , ϕ ) : a ( α , α ); b ( β, β ); c ( γ , γ ); d (1 , e (1 , t (1 , G = ( ϕ , ϕ ) : a ( α, β ); b (1 , α ); c (1 , γ − ); d (1 , α ); e (1 , γ − ); t ( α, . Similarly, we have F = ( ϕ , ϕ ) :¯ a ( ¯ α , ¯ α ); ¯ b ( ¯ β, ¯ β ); ¯ d (¯1 , ¯1); ¯ t (¯1 , ¯1) G = ( ϕ , ϕ ) :¯ a ( ¯ α, ¯ β ); ¯ b (¯1 , ¯ α ); ¯ d (¯1 , ¯ α ); ¯ t ( ¯ α, ¯1) . OMPUTATION OF NIELSEN AND REIDEMEISTER COINCIDENCE NUMBERS FOR MULTIPLE MAPS 15
It follows that F = and G = so that F − G = −
10 1 0 02 − − − and R ( F , G ) = | det( F − G ) | = 1 . Now, we have F ′ : c ( γ , γ ); e (1 ,
1) and G ′ : c (1 , γ − ); e (1 , γ − ). Thus, F ′ = ! and G ′ = − − ! so that F ′ − G ′ = ! and R ( F ′ , G ′ ) = | det( F ′ − G ′ ) | = 2 . To complete the calculation, we need to find Im δ where δ : Coin ( F , G ) → R ( F ′ , G ′ ). Note that F − G is invertible so the kernel, which is the same as Coin ( F , G ), must be trivial. Hence | Im δ | = 1.It follows from Lemma 1 that R ( F, G ) = 2 and hence R ( ϕ , ϕ , ϕ ) = 2.7. Concluding Remarks
Let f, g : M → N be maps between closed orientable manifolds where dim M ≥ dim N . When N isa Jiang type space as in Theorem 4, one can use the coincidence theory for multiple maps to givean upper bound for N ( f, g ) as follows.Suppose m = dim M, n = dim N and m = ( k − n for some positive integer k ≥
2. Let
F, G : M → N ( k − where F = ( f, f, ..., f ) and G = ( g, g, ..., g ). Now F, G are maps between two closedorientable manifolds of the same dimension so that L ( F, G ) is the usual homological Lefschetzcoincidence trace. Then we have the following result.
Theorem 5.
Let f, g : M → N be maps between two closed orientable manifolds where N is aJiang space; a nilmanifold; an orientable coset space G/K of a compact connected Lie group G by a closed subgroup K ; or a C -nilpotent space whose fundamental group has a finite index centerwhere C is the class of finite groups. Suppose dim M = ( k − n, dim N = n and k ≥ . Let F = ( f, f, ..., f | {z } ( k − ) and G = ( g, g, ..., g | {z } ( k − ) . If L ( F, G ) = 0 then k − p | L ( F, G ) | ≥ R ( f, g ) ≥ N ( f, g ) .Proof. It follows from Theorem 4 that L ( F, G ) = 0 ⇒ N ( F, G ) = R ( F, G ). By the inequality (4.3)we have R ( F, G ) = R ( f, g, g, ..., g ) ≥ ( R ( f, g )) ( k − . Since the Nielsen coincidence classes havecoincidence index of the same sign, it follows that | L ( F, G ) | ≥ N ( F, G ) = R ( F, G ). The inequalityfollows. (cid:3)
Next, we show that Theorem 5 can be applied to give an upper bound for N ( f, g ) in general. Lemma 2.
Let f, g : M → N be maps between closed connected orientable manifolds with m =dim M ≥ dim N = n . Let r ∈ N ∪ { } such that r + m = ( k − n for some integer k ≥ . Denoteby p : ˜ S r × M → M be the canonical projection where ˜ S r is the r -sphere if r > and ˜ S is a point.Let ˜ f = f ◦ p, ˜ g = g ◦ p . Then R ( f, g ) = R ( ˜ f , ˜ g ) .Proof. Suppose ϕ, ψ and ˜ ϕ, ˜ ψ denote the homomorphisms induced by f, g and by ˜ f , ˜ g , respectivelyon the fundmental groups. Consider the commutative diagram π ( ˜ S r ) −−−−→ π ( ˜ S r ) × π ( M ) p −−−−→ π ( M ) y ˜ ϕ y ˜ ψ ϕ y ψ { } −−−−→ π ( N ) = −−−−→ π ( N )Since ˜ ϕ ( a, σ ) = ϕ ( σ ) and ˜ ψ ( a, σ ) = ψ ( σ ), it follows that the ˜ ϕ - ˜ ψ twisted conjugacy classes in π ( N )coincide with the ϕ - ψ twisted conjugacy classes in π ( N ). Hence R ( f, g ) = R ( ˜ f , ˜ g ). (cid:3) Theorem 6.
Let f, g : M → N be the same as in Theorem 5 with no restrictions on dim M, dim N .Let r ∈ N ∪ { } such that r + m = ( k − n for some integer k ≥ . Denote by p : ˜ S r × M → M bethe canonical projection where ˜ S r is the r -sphere if r > and ˜ S is a point. Let ˜ f = f ◦ p, ˜ g = g ◦ p .Let F = ( ˜ f , ˜ f , ..., ˜ f | {z } ( k − ) and G = (˜ g, ˜ g, ..., ˜ g | {z } ( k − ) be maps from ˜ S r × M → N ( k − . If L ( F, G ) = 0 then k − p | L ( F, G ) | ≥ R ( f, g ) ≥ N ( f, g ) .Proof. The proof is a straightforward application of Lemma 2 and of Theorem 5. (cid:3)
OMPUTATION OF NIELSEN AND REIDEMEISTER COINCIDENCE NUMBERS FOR MULTIPLE MAPS 17
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Universidade Estadual Paulista (UNESP), Instituto de Geociˆencias e Ciˆencias Exatas (IGCE), Av 24A,1515, Bela Vista, CEP 13506-900, Rio Claro-SP, Brazil.
E-mail address : [email protected] Department of Mathematics, Bates College, Lewiston, ME 04240, U.S.A.
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