Axioms for a local Reidemeister trace in fixed point and coincidence theory on differentiable manifolds

Axioms for a lo cal Reidemeis ter trace in fixed p oi n t and coincidence theory on differen tiable manifolds P . Christopher Staec k er August 24, 2021 Abstract W e give axioms whic h chara ct erize t h e lo cal Reidemeister trace for orien t able differentiable manif olds. The local Reidemeister trace in fixed p oin t theory is al ready known, a nd w e p ro vide both uniquen ess a n d exis- tence results for the local Reidemeister trace in coincidence theory . 1 In tro duction The Reidemeis ter trace is a fundament al in v ar ia nt in top olo gical fixe d point theory , generalizing b oth the Lefschetz and Nielsen num b ers. It was or iginally defined by Reidemeister in [11]. A mor e mo dern treatment, under the name “genera lized Lefschetz num b er,” was given by Husseini in [9]. If X is a finite connected CW-complex with universal covering s pa ce e X and fundamental g r oup π , then the cellular chain complex C q ( e X ) is a free Z π - mo dule. If f : X → X is a cellular ma p and e f : e X → e X is a lift of f , then the induced map e f q : C q ( e X ) → C q ( e X ) can be vie wed as a matrix with entries in Z π (with resp ect to some chosen Z π basis for C q ( e X )). W e then define R T ( f , e f ) = ∞ X q =0 ( − 1) q ρ (tr( e f q )) , where tr is the sum o f the dia g onal entries of the matrix, a nd ρ is the pro jection int o the “Reidemeister classes” of π . The Reidemeister trace, then, is an element of Z R , where R is the s e t of Reidemeister cla sses. W eck en, in [13], pr oved what we will refer to as the We cken T r ac e The or em , that R T ( f , e f ) = X [ α ] ∈ R ind([ α ]) [ α ] , where ind ([ α ]) is the index of the Nielsen fixed p oint clas s asso ciated to [ α ] (see e.g. [10]). Thus the num b er of terms app ear ing in the Reidemeister trace with 1 nonzero co efficient is equa l to the Nielsen num b er of f , and b y the Lefschetz- Hopf Theorem, the sum o f the co efficients is equa l to the Lefschetz num b er of f . Recent work of F ur i, Pera, and Spadini in [6 ] has given a new pro of of the uniqueness of the fix ed p oint index on orientable manifolds with resp ect to three natural axioms. In [12] their appr o ach was ex tended to the coincidence index. The r esult is the following theor em: Theorem 1. L et X and Y b e oriente d differ entiable manifolds of the same dimension. The c oincidenc e index ind( f , g , U ) of two m appings f , g : X → Y over some op en set U ⊂ X is the un ique int e ger-value d function satisfyi ng the fol lowing ax ioms: • (A dditivity) If U 1 and U 2 ar e disjoint op en subsets of U whose union c on- tains al l c oincide nc e p oints of f and g on U , then ind( f , g , U ) = ind( f , g , U 1 ) + ind( f , g , U 2 ) . • (Homotopy) If f and g ar e “admissably homotopic” to f ′ and g ′ , t hen ind( f , g , U ) = ind ( f ′ , g ′ , U ) • (Normalization) If L ( f , g ) denotes the c oincidenc e L efschetz numb er of f and g , then ind( f , g , X ) = L ( f , g ) . In the spirit of the ab ov e theor em, we demonstrate the existence and unique- ness of a lo ca l Reidemeister tr a ce in coincidence theor y sub ject to five axioms. A lo cal Reidemeister trace for fixed p oint theory was given by F ares a nd Har t in [5], but no Reidemeister trace (lo cal or o therwise) has a ppe a red in the literature for coincidence theory . W e note that recent work b y Gon¸ calves and W eb er in [8] g ives axioms for the Reidemeister tra c e in fixed p oint theory using entirely different metho ds. Their work uses no lo cality prop erties, and is ba sed on ax ioms for the Le fschetz nu m be r by Ar ko witz and Brown in [1]. In Section 2 we present our axiom s et, and we prove the uniquenes s in coincidence theory in Sec tio n 3. In the sp ecial case of lo cal fixed p oint theory , we can obtain a slightly stro nger uniq ue ne s s r esult which we discuss in Section 4. Section 5 is a demonstr ation of the exis tence in the setting o f coincidence theory . This pap er co nt ains pieces of the author’s do ctora l dissertation. The author would like to thank his dissertation advisor Rob ert F. Brown for assistanc e with bo th the disser tation work and with this pap er. The author would a lso like to thank Peter W ong , who g uided the early disser tation work a nd interested him in the coincidence Reidemeister tra ce. 2 2 The Axioms Throughout the pap er, unless otherwise s tated, let X a nd Y denote connected orientable differentiable manifolds of the same dimensio n. All maps f , g : X → Y will b e a ssumed to b e co nt inu ous. The universal cov er ing spa ces of X and Y will b e denoted e X and e Y wit h pro jection maps p X : e X → X and p Y : e Y → Y . A lift of some map f : X → Y is a ma p e f : e X → e Y with p Y ◦ e f = f ◦ p X . Let f , g : X → Y b e maps, with induced homomorphisms φ, ψ : π 1 ( X ) → π 1 ( Y ) resp ectively . W e will view elemen ts o f π 1 ( X ) and π 1 ( Y ) as cov ering transformatio ns, s o that for any e x ∈ e X a nd σ ∈ π 1 ( X ), w e hav e e f ( σ e x ) = φ ( σ ) e f ( e x ) and e g ( σ e x ) = ψ ( σ ) e g ( e x ). W e will pa rtition the elements of π 1 ( Y ) into equiv alence class es defined b y the “doubly twisted conjugacy” rela tio n: α ∼ β ⇐ ⇒ α = ψ ( σ ) − 1 β φ ( σ ) . The eq uiv ale nce classe s with resp ect to this relation (denoted e.g . [ α ]) are called R eidemeister classes . The set of Reidemeister classe s is deno ted R [ f , g ]. F or any set S , let Z S denote the free ab elian gr oup genera ted by S , who se elements we write as sums of elemen ts of S with integer c o efficients. F or a ny such ab elian gro up, ther e is a homomor phism c : Z S → Z defined as the sum of the co efficients: c X i k i s i ! = X i k i , for s i ∈ S and k i ∈ Z , and i ranging ov e r a finite set. F or so me ma ps f , g : X → Y and a n op en subset U ⊂ X , let Coin( f , g , U ) = { x ∈ U | f ( x ) = g ( x ) } . W e say that the tr iple ( f , g , U ) is admissab le if Coin( f , g , U ) is compact. Two triples ( f , g , U ) and ( f ′ , g ′ , U ) are admissably homotopic if there is s ome pair o f homotopies F t , G t : X × [0 , 1] → X o f f , g to f ′ , g ′ with { ( x, t ) ∈ U × [0 , 1] | F t ( x ) = G t ( x ) } compact. Let C ( X , Y ) b e the set of admissable tuples , all tuples of the form ( f , e f , g , e g , U ) where f , g : X → Y a re maps , ( f , g , U ) is an admissable triple, and e f and e g a re lifts of f a nd g . Let ( f , e f , g , e g , U ) , ( f ′ , e f ′ , g ′ , e g ′ , U ) ∈ C ( X , Y ) with ( f , g , U ) admissably ho- motopic to ( f ′ , g ′ , U ) by homo topies F t , G t . By the homotopy lifting prop er t y , there ar e unique lifted homoto pies e F t , e G t : e X × [0 , 1] → e Y with e F 0 = e f and e G 0 = e g . If we additionaly hav e e F 1 = e f ′ and e G 1 = e g ′ , then we say that the tuples ( f , e f , g , e g, U ) and ( f ′ , e f ′ , g ′ , e g ′ , U ) are admisssably homo topic . Throughout the following, let R T b e any function whic h to an admiss a ble tuple ( f , e f , g , e g , U ) ∈ C ( X , Y ) ass o ciates a n element of Z R [ f , g ]. Our first three axioms for the lo cal Reidemeister trace are mo deled after the axioms of Theorem 1. 3 Axiom 1 (Additivit y) . Given ( f , e f , g , e g , U ) ∈ C ( X , Y ) , if U 1 and U 2 ar e disjoi nt op en subsets of U with Coin( f , g , U ) ⊂ U 1 ∪ U 2 , then R T ( f , e f , g , e g , U ) = R T ( f , e f , g , e g, U 1 ) + R T ( f , e f , g , e g, U 2 ) . Axiom 2 (Homo topy) . If ( f , e f , g , e g, U ) and ( f ′ , e f ′ , g ′ , e g ′ , U ) ar e admissably ho- motopic admissable tuples, then R T ( f , e f , g , e g , U ) = R T ( f ′ , e f ′ , g ′ , e g ′ , U ) . Axiom 3 (Norma liz ation) . If ( f , e f , g , e g , X ) ∈ C ( X , Y ) , then c ( R T ( f , e f , g , e g , X )) = L ( f , g ) , wher e L ( f , g ) is the L efschetz n u mb er of f and g . W e will req uire one a dditional a xiom to make s ome connections with Nielsen theory , based on a well-kno wn pro pe r ty of the Reidemeister trace: Axiom 4 (Lift in v ar iance) . F or any ( f , e f , g , e g , U ) ∈ C ( X , Y ) , and any α, β ∈ π 1 ( Y ) we have c ( R T ( f , e f , g , e g, U )) = c ( R T ( f , α e f , g , β e g , U )) . The four axioms ab ove are enoug h to demonstra te some relations hips b e- t ween R T and the coincidence index. Prop ositio n 1. If R T satisfies the homotopy, additivity, normalization, and lift invarianc e ax ioms, then c ( R T ( f , e f , g , e g , U )) = ind( f , g , U ) for any ( f , e f , g , e g, U ) ∈ C ( X, Y ) , wher e ind denotes t he c oincidenc e index (se e [7]). Pr o of. Let ω = c ◦ R T : C ( X , Y ) → Z . By the lift inv ariance a xiom, ω is independent of the choice of lifts. Thus ω can b e viewed as a function fro m the set of all admissable triples to Z . It is clear that ω satisfies the three axioms o f Theorem 1, since they are implied by our additivity , homoto p y , and normalizatio n ax ioms for R T (disregarding the lift parameters). Th us ω is the coincidence index. Prop ositio n 2. If R T satisfies t he additivity, homotopy, normalization, and lift invarianc e axioms and c ( R T ( f , e f , g , e g, U )) 6 = 0 , then t her e is s ome σ ∈ π 1 ( Y ) such that σ e f and e g have a c oincid en c e on p − 1 X ( U ) . Pr o of. By Prop osition 1 , if c ( R T ( f , e f , g , e g , U )) 6 = 0 then ind( f , g , U ) 6 = 0, and so f and g hav e a co incidence on U . Let x ∈ U b e this coincidence p oint, and choos e e x ∈ p − 1 X ( x ). Then since e f a nd e g are lifts, the points e f ( e x ) and e g ( e x ) will pr o ject to the sa me point of Y b y p Y . Th us there is some cov er ing transformatio n σ with σ e f ( e x ) = e g ( e x ). 4 The four ax ioms given ab ov e are not sufficient to uniquely characterize the Reidemeister trace in fixed p oint or c oincidence theory . F or insta nc e, the func- tion defined by T ( f , e f , g , e g, U ) = ind( f , g , U )[1] , where [1] is the Reidemeister class of the trivia l element 1 ∈ π 1 ( Y ), s atisfies a ll of the a xioms ab ove, but provides none of the exp ected data conce rning R [ f , g ], and s o that function cannot b e the Reidemeister trace. An a dditional a xiom is needed, one which somehow indica tes the elemen ts of R [ f , g ] whic h are to app ear in the Reidemeister tra ce. Our final ax iom is a s ort of strengthening of Pro p o sition 2, which sp ecifies the Reidemeister data asso ciated to the coincidence p oints. Axiom 5 (Coincidence of lifts) . If [ α ] app e ars with nonzer o c o efficient in R T ( f , e f , g , e g , U ) , then α e f and e g have a c oincid en c e on p − 1 X ( U ) . An y function R T which to a tuple ( f , e f , g , e g , U ) ∈ C ( X , Y ) asso ciates an element of Z R [ f , g ], and satisfies the additivity , homoto p y , normaliza tio n, lift inv ar iance, and coincidence of lifts axioms w e will call a lo c al Rei demeister tr ac e . Our main result (Theorem 3) states that there is a unique such function. 3 Uniqueness Let ( f , e f , g , e g , U ) ∈ C ( X , Y ), let e U = p − 1 X ( U ), and le t C ( e f , e g , e U , [ α ]) = p X (Coin( α e f , e g , e U )) . F or each α we have C ( e f , e g , e U , [ α ]) ⊂ Co in( f , g , U ), a nd such coincidence sets are called c oincidenc e classes . Tha t these classes a re well defined is a c o nsequence of the fo llowing lemma, which app ears in slightly different langua ge as Lemma 2.3 o f [4]. Lemma 1. L et α, β ∈ π 1 ( Y ) , maps f , g : X → Y , and an op en subset U ⊂ X b e given. Then: • [ α ] = [ β ] if and only if p X Coin( α e f , e g , e U ) = p X Coin( β e f , e g , e U ) for any lifts e f , e g . • If [ α ] 6 = [ β ] , then p X Coin( α e f , e g , e U ) and p X Coin( α e f , e g , e U ) ar e disjoint for any lifts e f , e g . Given the ab ov e notation, the coincidence of lifts axiom could b e r estated as follows: If [ α ] app ear s with nonzero co e fficie n t in R T ( f , e f , g , e g , U ), then C ( e f , e g , e U , [ α ]) is nonempt y . F o r each coincidence point x in U , define [ x e f , e g ] ∈ R [ f , g ] a s that class [ α ] for which x ∈ C ( e f , e g , e U , [ α ]). 5 Theorem 2. If R T is a lo c al R eidemeister tr ac e and Coin( f , g , U ) is a set of isolate d p oints, then R T ( f , e f , g , e g , U ) = X x ∈ Coin( f ,g ,U ) ind( f , g , U x )[ x e f , e g ] , wher e U x is an isolating neighb orho o d for the c oincidenc e p oint x . Pr o of. By the additivity pro pe r ty , we need only show that R T ( f , e f , g , e g , U x ) = ind( f , g , U x )[ x e f , e g ] . First, we observe that no elemen t of R [ f , g ] other than [ x e f , e g ] appear s as a term with nonzero co efficient in R T ( f , e f , g , e g, U x ): If some [ β ] do es app ear with nonzero co efficient, then we know b y the coincidence of lifts axiom that β e f and e g have a coincidenc e o n e U x = p − 1 X ( U x ). Pr o jection of this coincidence p oint gives a coincidence p oint in U x which neces sarily must b e x , since x is the only coincidence po in t in U x . Thus x ∈ p X Coin( β e f , e g , e U x ), which means that [ β ] = [ x e f , e g ]. Since [ x e f , e g ] is the only element of R [ f , g ] app ea ring in R T ( f , e f , g , e g , U ), we hav e R T ( f , e f , g , e g , U x ) = k [ x e f , e g ] for some k ∈ Z (p ossibly k = 0). Prop osition 1 says that the co efficient sum m ust eq ua l the index, and so k = ind( f , g , U x ) as des ir ed. The ab ove is a strong result for maps whose coincidence sets ar e isolated. In order to leverage this result for a rbitrary maps, we will make use of a technical lemma, a c o mbination of Lemmas 13 a nd 15 fro m [1 2]. Lemma 2. L et ( f , g , U ) b e an admissab le triple, and let V ⊂ U b e an op en subset c ontaining Coin( f , g , U ) with c omp act closur e ¯ V ⊂ U . Then ( f , g , V ) is admissably homotopic to an admissable triple ( f ′ , g ′ , V ) , wher e f ′ and g ′ have isolate d c oincidenc e p oints in V . The ab ov e lemma is used to a pproximate a ny ma ps by maps having isola ted coincidence p oints, and we obtain our uniqueness theorem: Theorem 3. Ther e is at most one lo c al R eidemeister tr ac e define d on C ( X, Y ) . Pr o of. Let R T be lo ca l Reidemeister trace, and take ( f , e f , g , e g , U ) ∈ C ( X , Y ). Then by Lemma 2 there is an op en subs et V ⊂ U with Coin( f , U ) ⊂ V and maps f ′ , g ′ with isolated co inc ide nce p o ints with ( f , g , V ) admissa bly homotopic to ( f ′ , g ′ , V ). Then by the homotopy axiom there are lifts e f ′ , e g ′ of f and g with R T ( f , e f , g , e g , U ) = R T ( f ′ , e f ′ , g ′ , e g ′ , V ) . The coincidence p oints o f f ′ and g ′ in V ar e isolated, so we hav e R T ( f , e f , g , e g, U ) = X x ∈ Coin( f ′ ,g ′ ,V ) ind( f ′ , g ′ , V x )[ x e f ′ , e g ′ ] , 6 where V x is an isolating neig hborho o d of the coincidence p oint x . This g ives an explicit form ula for the computation of R T ( f , e f , g , e g, U ). The only choice made in the computatio n is o f the a dmissable homo to py to ( f ′ , g ′ , V ), but any alternative choice m ust give the same lo ca l Reidemeister tr ace by the homotopy axiom. Th us all lo cal Reidemeister traces must b e computed in the same way , giving the same result, which means that there can b e only one. 4 Uniqueness in fi x ed p oin t theory In the sp ecia l ca se where Y = X and g is taken to b e the identit y map id : X → X , the ab ov e metho d can b e used w ith slig ht mo difications to prov e a uniqueness result for the lo cal Reidemeister trace in the fixed p o int theory of po ssibly nonorientable manifolds. W e have not in this pa pe r made e xplicit use of the orientabilit y hypothesis, but it is a nece ssary hypothesis for the unique nes s of the coincidence index in Theorem 1, which was used in Prop osition 1. An accounting of or ie n tations is needed in coincidence theory to dis tinguish b etw een p o ints of index +1 and index − 1 (though see [4] for an approa ch to an index for no norientable ma nifolds, which doe s no t alwa ys give an integer). Orie n tability is no t needed in lo cal fixed po int theory , since the notion of an orientation preserving selfmap is w e ll-defined lo cally , even on a manifold with no g lo bal or ie n tation. Th us the uniqueness of the fixed p oint index in [6] do es not require orientabilit y , a nd we will not require it here. Let C ( X ) b e the s e t of all tuples of the form ( f , e f , e ı, U ), where f : X → X is a selfmap, e f : e X → e X is a lift of f , the map e ı : e X → e X is a lift of the ide ntit y map, and U is an op en subset of X with compact fix e d p oint set Fix( f , U ) = Coin( f , id , U ). Let R [ f ] = R [ f , id]. Two tuples ( f , e f , e ı, U ) and ( f ′ , e f ′ , e ı, U ) are s aid to b e admissably homoto pic if there is some homotopy F t of f to f ′ with { ( x, t ) | F t ( x ) = x } compact, and F t lifts to a homotopy of e f to e f ′ . Our uniqueness theorem is then: Theorem 4. If X is a (p ossibly nonorientable) differ entiable manifold, t hen ther e is a unique function taking an admissable tuple ( f , e f , e ı, U ) to an element of Z R [ f ] satisfying t he fol lowing axioms: • (A dditivity) If U 1 and U 2 ar e disjoint op en subsets of U with Fix( f , U ) ⊂ U 1 ∪ U 2 , then R T ( f , e f , e ı, U ) = R T ( f , e f , e ı, U 1 ) + R T ( f , e f , e ı, U 2 ) • (Homotopy) If ( f , e f , e ı, U ) is admissably homotopic to ( f ′ , e f ′ , e ı, U ) , then R T ( f , e f , e ı, U ) = R T ( f ′ , e f ′ , e ı, U ) 7 • (We ak n ormalization) If f is a c onstant map, then c ( R T ( f , e f , e ı, U )) = 1 • (Lift invarianc e) F or any α, β ∈ π 1 ( X ) , we have c ( R T ( f , e f , e ı, U )) = c ( R T ( f , α e f , β e ı, U )) • (Coincid enc e of lifts) If [ α ] app e ars with nonzer o c o efficient in R T ( f , e f , e ı, U ) , then α e f and e ı have a c oincidenc e p oint on p − 1 X ( U ) . Pr o of. First we note that a r e sult analago us to Pr op osition 1 can b e obtained in the fixed p oint setting using only the weak normaliza tion axiom: Using the three axioms of [6], which make use of an appropr iately weakened normalizatio n axiom, we see that c ◦ R T is the fixed p o int index . Then letting g = id in the pro of of Theorem 2, w e hav e that, if f ha s is olated fixed p oints, R T ( f , e f , e ı, U ) = X x ∈ Fix( f ,U x ) ind( f , U x )[ x e f , e ı ] , where ind denotes the fixed point index, and U x is an isolating neigh b o r ho o d for the fixed p o int x . A fixed point v er sion of Lemma 2 can b e found in Lemmas 4.1 and 3.3 o f [6], and the pro of of Theo rem 3 can be mimic ked to obtain our uniqueness result. Note that the uniqueness in fixed p oint theory requir e s o nly a weakened version of the nor malization a xiom. A uniqueness result for co incidence theory using only the weak normalizatio n ax iom can b e obtained if we restrict ourselves to self-maps of a particular (not necessarily orientable) manifold. This would use a pr o of similar to the a bove, using r esults from Section 5 of [12]. 5 Existence The exis tence of a lo cal Reidemeister trace in fixed p oint theory fo r connected finite dimensional lo cally co mpa ct p olyhedr a is esta blished by F ares a nd Har t in [5]. Ther e , the slightly more genera l lo cal H -Reide meis ter trace is defined, called “the lo cal gener alized H -Lefschetz num b er” . An extension of this pap er to the mo d H theo r y would not be difficult. The fact that the mo d H Reidemeister classes a r e unions o f ordina ry Reidemeister classes a llows the same r esults to b e obtained without substantial mo difications . In [5], the additivity and homo to py axio ms ar e proved in Pro po sition 3.2 .9 and P r op osition 3.2.8, r esp ectively . A strong version o f the lift inv a riance ax - iom (see our T he o rem 6 ) is pr ov ed in P rop osition 3 .2 .4. The coincidence of lifts axiom is not stated explicitly by F a r es and Hart, but is a s tr aightforw a rd con- sequence of their trace-like definition (if some [ α ] has no nzero co efficient in the 8 Reidemeister trace, it neccesa rily comes from some simplex in the cov ering space containing a fixed p oint of α e f ). A result analogo us to the W ec ken T race Theo- rem (which trivia lly implies the normalizatio n and weak normaliza tion ax io ms) is g iven in Theorem 3.3 .1. No Reidemeister trace fo r co incidence theory , either lo cal or global, has app eared previo usly in the literature. The pro of of The o rem 3 furnishes the appropria te definition, a s follows: Given a n admissable tuple ( f , e f , g , e g , U ), we find (by Lemma 2 ) a n admissably ho motopic tuple ( f ′ , e f ′ , g ′ , e g ′ , V ) with iso lated coincidence p oints, and we define R T ( f , e f , g , e g, U ) = X x ∈ Coin( f ′ ,g ′ ,V ) ind( f ′ , g ′ , V x )[ x e f ′ , e g ′ ] , where V x is an isolating neig hborho o d for the coincidence p oint x . The ab ov e is well defined pro vided that it is indep endent o f the choice of the admissa bly homotopic tuple. This is ensured by the following lemma: Lemma 3. If ( f , e f , g , e g , U ) and ( f ′ , e f ′ , g ′ , e g ′ , U ) ar e admissably homotopic tu - ples with isolate d c oincidenc e p oints, then X x ∈ Coin( f ,g ,U ) ind( f , g , U x )[ x e f , e g ] = X x ′ ∈ Coin( f ′ ,g ′ ,U ) ind( f ′ , g ′ , U x ′ )[ x ′ e f ′ , e g ′ ] , wher e U x is an isolating neighb orho o d for the c oincidenc e p oint x ∈ Co in( f , g , U ) , and U x ′ is an isolating n eighb orho o d of the c oincidenc e p oint x ′ ∈ Coin( f ′ , g ′ , U ) . Pr o of. W e define the index of a coincidence class C of f and g as follows: ind C = X x ∈ C ind( f , g , U x ) . A class is called essential if its index is nonzero. Since f a nd g are homoto pic to f ′ and g ′ , we hav e R [ f , g ] = R [ f ′ , g ′ ]. Call this common set of Reidemeis ter classes R . Letting e U = p − 1 X ( U ), the statement of the Lemma is equiv alent to X [ α ] ∈ R ind C ( e f , e g , e U , [ α ])[ α ] = X [ α ] ∈ R ind C ( e f ′ , e g ′ , e U , [ α ])[ α ] , and we need only s how that ind C ( e f , e g , e U , [ α ]) = ind C ( e f ′ , e g ′ , e U , [ α ]) for any [ α ]. W e will prove this using Bro o ks’s notio n of homoto p y-relatedness of c oincidence classes, exp osited in detail in [2] a nd briefly in [3]. Let F t , e F t , G t , e G t be homotopies realizing the admissable homo topy of ( f , e f , g , e g , U ) and ( f ′ , e f ′ , g ′ , e g ′ , U ). Two coincidence p oints x ∈ Coin( f , g , U ) and x ′ ∈ Coin( f ′ , g ′ , U ) are ( F t , G t ) –r elate d if there is some path γ ( t ) in X connecting x to x ′ such that the paths F t ( γ ( t )) and G t ( γ ( t )) are ho motopic in Y as paths with fixed end- po int s. Two coincidence cla s ses are related if at least o ne p oint of o ne is rela ted 9 to at least one p oint of the other. Theor em I I.2 2 of [2] shows that the notion of ( F t , G t )-relatedness gives a bijective cor r esp ondence b etw e e n the ess e n tial coin- cidence classes of ( f , g ) and those of ( f ′ , g ′ ). Theorem IV.24 of [2] further shows that any t wo such rela ted classes will hav e the s ame index. What rema ins is an elementary ar g ument using co vering-space theory . Let C = C ( e f , e g , e U , [ α ]), and let C ′ be the unique coincidence class of ( f ′ , g ′ ) which is ( F t , G t )-related to C . W e need only show that C ′ = C ( e f ′ , e g ′ , e U , [ α ]), and th us (since homotopy-relatedness preser ves the index) tha t ind C ( e f , e g , e U , [ α ]) = ind C ( e f ′ , e g ′ , e U , [ α ]). Cho ose a po int x ∈ C , a nd let x ′ be a p oint in C ′ which is ( F t , G t ) related to x . Then there is some path γ in X from x to x ′ with F t ( γ ( t )) homoto pic to G t ( γ ( t )). Let e x b e some p oint with p X ( e x ) = x and α e f ( e x ) = e g ( e x ). W e can lift γ to a path e γ in e X s ta rting at e x . Since F t ( γ ( t )) is homotopic to G t ( γ ( t )), we will hav e e F t ( e γ ( t )) homotopic to e G t ( e γ ( t )), whic h in particular means that they will hav e the same e ndpo in t. This commo n endp oint is α e f ′ ( e γ (1)) = e g ′ ( e γ (1)), which must pro ject by p X to the p oint x ′ . Thus x ′ ∈ p X (Coin( α e f ′ , e g ′ , e U )), and so C ′ = C ( e f ′ , e g ′ , e U , [ α ]), as desir ed. W e hav e th us pr o duced a meaningful definition of a lo cal coincidence Reide- meister tra ce o n o rientable differentiable manifolds of the sa me dimension, and the pro of ab ov e suffices to give: Theorem 5 (W eck en Coincidence T ra ce Theorem) . L et R T b e the un ique lo- c al c oinci denc e R eidemeister tr ac e satisfying our five axioms. Then for any ( f , e f , g , e g, U ) ∈ C ( X, Y ) with e U = p − 1 X ( U ) , we have R T ( f , e f , g , e g , U ) = X [ α ] ∈R [ f ,g ] ind C ( e f , e g , e U , [ α ])[ α ] . In conclusion we pr ove a stro nger form of the lift inv a riance a xiom, a c o in- cidence version of a well-known prop er t y o f the Reidemeis ter trace. Theorem 6. L et R T b e the unique lo c al c oincid en c e R eidemeister tra c e s atis- fying our fi ve ax ioms. If R T ( f , e f , g , e g , U ) = X [ σ ] ∈ R [ f ,g ] k [ σ ] [ σ ] for k [ σ ] ∈ Z , then for any α, β ∈ π 1 ( Y ) , we have R T ( f , α e f , g , β e g , U ) = X [ σ ] ∈ R [ f ,g ] k [ σ ] [ β σ α − 1 ] . Pr o of. Letting e U = p − 1 X ( U ), by Theorem 5 we know that k [ σ ] = ind C ( e f , e g , e U , [ σ ]). Then we hav e C ( α e f , β e g , [ σ ]) = p X Coin( σ α e f , β e g , e U ) = p X Coin( β − 1 σ α e f , e g , e U ) = C ( e f , e g , [ β − 1 σ α ]) , 10 and thus ind C ( α e f , β e g , e U , [ σ ]) = k [ β − 1 σα ] . Now by Theo r em 5 a gain, we hav e R T ( f , α e f , g , β e g , e U ) = X [ σ ] ∈ R [ f ,g ] ind C ( α e f , β e g , [ σ ])[ σ ] = X [ σ ] ∈ R [ f ,g ] k [ β − 1 σα ] [ σ ] = X [ γ ] ∈R [ f ,g ] k [ γ ] [ β γ α − 1 ] , as desired. References [1] M. Arkowitz and R. Brown. The Lefschetz-Hopf theor em and a xioms fo r the Lefschetz num b er. Fixe d Point The ory and Applic ations , 1 :1–11, 2 004. [2] R. B ro oks. Coincidenc es, R o ots, and Fix e d Points . Do ctoral disserta tio n, Univ ersity of Califo r nia, Los Ange le s, 1 967. [3] R. Br o oks and R. Brown. A low er b ound on the ∆-Nielsen num b er. T r ans- actions of the Americ a n Mathematic al So ciety , 14 3:555– 564, 1 9 69. [4] R. Dobre ´ nko and J. J e zierski. The coincidence Nielse n num b er on nonori- ent able manifolds. The Ro ck y Mount ain Journ al of Mathematics , 2 3:67– 85, 1993. [5] J. F ares and E . Hart. A generalized Lefschetz num b er for lo cal Niels en fixed p oint theor y . 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