We give a formula for the coincidence Reidemeister trace of selfmaps on
bouquets of circles in terms of the Fox calculus. Our formula reduces the
problem of computing the coincidence Reidemeister trace to the problem of
distinguishing doubly twisted conjugacy classes in free groups.Comment: 17 pages, 4 figure

Staecker, P. Christopher

English

arXiv

Christopher P. Staecker

CiteSeerX

A formula for the coincidence Reidemeister trace of selfmaps on bouquets of circles

We give a formula for the coincidence Reidemeister trace of selfmaps on
bouquets of circles in terms of the Fox calculus. Our formula reduces
the problem of computing the coincidence Reidemeister trace to the
problem of distinguishing doubly twisted conjugacy classes in free
groups

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We give a formula for the coincidence Reidemeister trace of selfmaps on bouquets of circles in terms of the Fox calculus. Our formula reduces the problem of computing the coincidence Reidemeister trace to the problem of distinguishing doubly twisted conjugacy classes in free groups

Staecker, Christopher P.

Fairfield University: DigitalCommons@Fairfield

Fairfield University DigitalCommons@Fairfield Mathematics Faculty Publications Mathematics Department 2009 A formula for the coincidence Reidemeister trace of selfmaps on bouquets of circles Christopher P. Staecker Fairfield University, cstaecker@fairfield.edu Follow this and additional works at: https://digitalcommons.fairfield.edu/mathandcomputerscience-facultypubs Copyright 2009 Topological Methods in Nonlinear Analysis. Included with permission from the publisher. Peer Reviewed Repository Citation Staecker, Christopher P., "A formula for the coincidence Reidemeister trace of selfmaps on bouquets of circles" (2009). Mathematics Faculty Publications. 33. https://digitalcommons.fairfield.edu/mathandcomputerscience-facultypubs/33 Published Citation Staecker, P. Christopher, A formula for the coincidence Reidemeister trace of selfmaps on bouquets of circles. Topological Methods in Nonlinear Analysis 33, 2009, 41–50. This Article is brought to you for free and open access by the Mathematics Department at DigitalCommons@Fairfield. It has been accepted for inclusion in Mathematics Faculty Publications by an authorized administrator of DigitalCommons@Fairfield. For more information, please contact digitalcommons@fairfield.edu. Topological Methods in Nonlinear AnalysisJournal of the Juliusz Schauder CenterVolume 33, 2009, 41–50A FORMULAFOR THE COINCIDENCE REIDEMEISTER TRACEOF SELFMAPS ON BOUQUETS OF CIRCLESP. Christopher StaeckerAbstract. We give a formula for the coincidence Reidemeister trace ofselfmaps on bouquets of circles in terms of the Fox calculus. Our formulareduces the problem of computing the coincidence Reidemeister trace to theproblem of distinguishing doubly twisted conjugacy classes in free groups.1. IntroductionFadell and Husseini, in [3] proved the following:Theorem 1.1 (Fadell, Husseini, 1983). Let X be a bouquet of circles, G =pi1(X), and let G0 be the set of generators of G. If f :X → X induces the mapϕ:G→ G, then there is some lift f˜ : X˜ → X˜ to the universal covering space withRT(f, f˜) = ρ(1−∑a∈G0∂∂aϕ(a)),where ρ:ZG→ ZR(f) is the linearization of the projection into twisted conjugacyclasses, and ∂ denotes the Fox derivative.Theorem 1.1 reduces the calculation of the Reidemeister trace (and thusof the Nielsen number) in fixed point theory to the computation of twistedconjugacy classes. Our goal for this paper is to obtain a similar result in2000 Mathematics Subject Classification. 54H25, 55M20.Key words and phrases. Coincidence theory, Nielsen number, Redemeister trace.c©2009 Juliusz Schauder Center for Nonlinear Studies4142 P. Ch. Staeckercoincidence theory of selfmaps — a formula for the coincidence Reidemeistertrace RT(f, f˜ , g, g˜) in terms of Fox derivatives which reduces the computationto twisted conjugacy decisions.The proof of Theorem 1.1 given in [3] is brief, thanks to a natural trace-likeformula for the Reidemeister trace in fixed point theory. No such formula existsfor the coincidence Reidemeister trace, and this will complicate our derivation.Our argument is based on first specifying a particular regular form for maps inSection 3 and for pairs of maps in Section 4. In Section 5 we give our mainresult.The author would like to thank Philip Heath, Nirattaya Khamsemanan, andSeungwon Kim for helpful and encouraging conversations, and Robert Brownand the referee for helpful comments on the paper.2. PreliminariesThroughout the paper, let X be a bouquet of circles meeting at the basepoint x0. Let G = pi1(X), a free group, and let G0 be the set of generators of G.Let X˜ be the universal covering space of X with projection pX : X˜ → X, andchoose once and for all a base point x˜0 ∈ X˜ with pX(x˜0) = x0.Given maps f, g:X → X and their induced homomorphisms ϕ,ψ:G → G,we define an equivalence relation on G as follows: two elements α, β ∈ G are(doubly) twisted conjugate ifα = ϕ(γ)βψ(γ)−1.The equivalence classes with respect to this relation are the Reidemeister classes,and we denote the set of such classes as R(ϕ,ψ). Let ρ:G → R(ϕ,ψ) be theprojection into Reidemeister classes.For any pair of maps f, g:X → X, denote their coincidence set byCoin(f, g) = {x ∈ X | f(x) = g(x)}.The set of coincidence points are partitioned into coincidence classes of theform pX(Coin(α−1f˜ , g˜)), where α ∈ G and f˜ , g˜: X˜ → X˜ are specified lifts of fand g. Lemma 2.3 of [2] shows that Coin(α−1f˜ , g˜) = Coin(β−1f˜ , g˜) if and onlyif ρ(α) = ρ(β), and that Coin(α−1f˜ , g˜) and Coin(β−1f˜ , g˜) are disjoint if ρ(α) 6=ρ(β). Thus the coincidence classes are represented by Reidemeister classes in G,and so each particular coincidence point has an associated Reidemeister class.For x ∈ Coin(f, g), let [xef,eg] ∈ R(ϕ,ψ) denote the Reidemeister class ρ(α) forwhich x ∈ pX(Coin(α−1f˜ , g˜)).Let f, g:X → X be mappings with isolated coincidence points, and for eachcoincidence point x let Ux ⊂ X be a neighborhood of x containing no othercoincidence points. Then we wish to define the coincidence Reidemeister traceCoincidence Reidemeister Trace of Selfmaps 43as a sum of the classes [xef,eg] with each coefficient given by the coincidence indexat x.The coincidence index is in general not defined for nonmanifolds such as ourbouquet of circles X, but we can construct an appropriate index without toomuch difficulty. It will be shown in the next section that we may assume thatg is homotopic to the identity in a neighborhood of the base point x0, and thusthat the coincidence index of f and g on a neighborhood of x0 is simply the fixedpoint index of f at x0 (which is defined for any ANR). Near any point x otherthan x0, the space X is an orientable differentiable manifold, and we define thecoincidence index as usual for that setting (see [5]):ind(f, g, Ux) = sign(det(dgx − dfx)),where dfx and dgx denote the derivatives of f and g at x. (That this determinantcan be assumed to be nonzero is a consequence of our construction below.)Thus we define the coincidence Reidemeister trace as:RT(f, f˜ , g, g˜) = ι(f, Ux0)[x0 ef,eg] +∑x∈Coin(f,g)−x0ind(f, g, Ux)[x ef,eg],where ι denotes the fixed point index. Indeed we can define a local Reidemeistertrace: for any open set U not cointaining x0, defineRT(f, f˜ , g, g˜, U) =∑x∈Coin(f,g,U)ind(f, g, Ux)[x ef,eg],where Coin(f, g, U) = Coin(f, g) ∩ U . If U does contain x0, then we addι(f, Ux0)[x0 ef,eg] into the sum in the obvious way. Clearly this local Reidemeistertrace is equal to the nonlocal version if U is taken to be X, and has the followingadditivity property: if V and W are disjoint subsets of U with Coin(f, g, U) ⊂V ∪W , thenRT(f, f˜ , g, g˜, U) = RT(f, f˜ , g, g˜, V ) + RT(f, f˜ , g, g˜,W ).As for defining the Reidemeister trace for any pair of maps f, g:X → X(perhaps having nonisolated coincidence points) with lifts f˜ , g˜: X˜ → X˜, we firstchange f and g by a homotopy to f ′, g′ so that they have isolated coincidencepoints (that this is possible will be a consequence of our construction in Theo-rem 4.1). Now the homotopies of f , g to f ′, g′ can be lifted to a homotopy off˜ , g˜ to some lifts f˜ ′, g˜′. We then defineRT(f, f˜ , g, g˜) = RT(f ′, f˜ ′, g′, g˜′).That this is well defined will be a consequence of the homotopy invariance ofthe coincidence index and the homotopy-relatedness of coincidence classes (seeLemma 5.1 of [6]).44 P. Ch. StaeckerWe will now review the necessary properties of the Fox calculus (see e.g. [1]).If {xi} are the generators of a free group G, then the operators∂∂xi:G→ ZGare defined by:∂∂xi1 = 0,∂∂xixj = δij ,∂∂xi(uv) =∂∂xiu+ u∂∂xiv,where δij is the Kronecker delta, and u, v ∈ G are any words. Two importantformulas can be obtained from the above:∂∂xix−1i = −x−1i ,∂∂xi(h1 . . . hn) =n∑k=1h1 . . . hk−1∂∂xihk.3. A regular form for mappingsIn this section we will describe a standard form for selfmaps of X. Eachcircle of X is represented by some generator of the fundamental group. For eachgenerator a ∈ G0, let |a| ⊂ X be the circle represented by a (including the basepoint x0).For simplicity in our notation, we parameterize each circle by the interval[0, 1] with endpoints identified. The circles of X will be parameterized so thatthe base point x0 is identified with 0 (or equivalently, with 1). For any generatora ∈ G and any x ∈ [0, 1], let [x]a denote the point of |a| ⊂ X which hascoordinate x. Interval-like subsets of X will be denoted e.g. (x1, x2)a for thesubset of points in |a| parameterized by the interval (x1, x2).Homotopy classes of mappings of X are characterized by their induced map-pings on the fundamental groups. Consider the example where G = 〈a, b〉 andthe mapping f :X → X induces ϕ:G→ G withϕ(a) = ab−1a−1b2.Geometrically speaking, the above formula for ϕ indicates that f is homotopicto a map fixing the base point x0 and mapping some interval (0, x1)a bijectivelyonto |a|−x0, maps some interval (x1, x2)a bijectively onto |b|−x0 (in the “reversedirection”), and so on.We can represent the action of this map on |a| pictorially as in Figure 1,where in this case xi = [i/5]a, and the label on each interval indicates that theinterval is being mapped bijectively onto the corresponding circle. Note thatsliding the points xi (i 6= 0) around the circle will not change the homotopyclass of f , provided that no xi ever moves across another, and the ordering ofthe labels is preserved.Coincidence Reidemeister Trace of Selfmaps 45◦a)) b−1vva−1HHbbFigure 1. Diagram of the action on |a| of a map with ϕ(a) = ab−1a−1b2.The base point x0 is circled at leftThus any map f :X → X is homotopic to a map which is characterized asfollows: for each generator a ∈ G0, specify na intervals Ia1 , . . . Iana together withlabels {ha1 , . . . , hana}, where each of hai are letters of G (a letter of G is an elementwhich is either a generator or the inverse of a generator of G). These intervalswill be called intervals of f , and the labels will be called the labels of f .Since we are concerned only with homotopy classes of maps, we may assumethat f maps (xi, xi+1)a affine linearly onto (0, 1)b for some generator b ∈ G. Amap specified by intervals and labels which is linear on each interval in this waywill be called regular.Specifying a map f by intervals and labels gives precise information whichcan be used to compute the derivatives of f at any point. For any intervalI = (xi, xi+1)a, define w(I), the width of I, as the real number xi+1 − xi. It iseasy to verify that for any coincidence point x ∈ I, we havedfx = ± 1w(I),where the sign above is + when I is labeled by a generator, and − when I islabeled by the inverse of a generator.4. The Reidemeister trace of a pair of regular mapsLet f, g:X → X be a regular pair of maps, and choose lifts f˜ and g˜ of f andg respectively so that f˜(x˜0) = g˜(x˜0). In this section we will describe a methodfor calculating RT(f, f˜ , g, g˜).46 P. Ch. StaeckerFor each generator a of G, writef(a) = ha1 . . . hana , g(a) = la1 . . . lama ,where all hai and laj are letters of G. Without loss of generality, we will assumethe action of f and g on |a| is specified by intervals and labels as pictured inFigure 2.◦1VVha1??. . .  hanaa−1hh1VVahh1aa−1PPla188. . .lamaa−1nnaFigure 2. Diagram of the action of f and g on |a|. Outer labels give theaction of f , while inner labels give the action of g.Given this construction of f and g, we see that Coin(f, g) consists of thepoints x0 and the various [1/2]a, along with finitely many isolated coincidencepoints occuring away from the interval endpoints.In order to compute the Reidemeister trace, we partition X into severalintervals: let Ia1 , . . . , Iana be the intervals of f in (0, 1/2)a labeled by ha1 , . . . , hana ,and let Ja1 , . . . , Jama be the intervals of g in (1/2, 1)a labeled by la1 , . . . , lama . LetIa be the interval of f containing [1/2]a, and let I0 be the union of all intervalsof f having x0 as an endpoint.We define four more intervals to cover the remaining portions of X: let Ka1be the interval of f in (0, 1/2)a which follows Iana and is labeled by a−1. Let V abe the first interval of g in (1/2, 1)a (this interval is labeled by a−1) and we setKa2 to be the interior of Va− Ia. Let Ka3 be the interval of g following Jama , thisinterval is labeled a−1 by g. Finally, let Ua be the last interval of g, and let Ka4be the interior of Ua − I0. Since our regular maps are constructed to have noCoincidence Reidemeister Trace of Selfmaps 47coincidences at interval endpoints except for x0 and the [1/2]a, we haveCoin(f ′, g′) ⊂ I0 ∪⋃a∈G0(Ia ∪na⋃i=1Iai ∪ma⋃j=1Jaj ∪4⋃k=1Kak),and so we can compute RT(f ′, f˜ ′, g′, g˜′) as a sum of the Reidmeister traces overthe various intervals in the above union.The coincidence points at the various [1/2]a are removable, as g can be de-formed away from x0 in a neighborhood of [1/2]a. Thus we have RT(f, g, f˜ , g˜, Ia)= 0 for each a. We can also observe that in a neighbourhood of x0, the map gis homotopic to the identity, and thus that ind(f, g, I0) is simply the fixed pointindex of a constant map, which is 1. Since x˜0 is a coincidence point of f˜ and g˜,we have [x0 ef,eg] = ρ(1), and so RT(f, f˜ , g, g˜, I0) = ρ(1).The union above thus gives the Reidemeister trace as a sum:(4.1) RT(f, f˜ , g, g˜) = ρ(1) +∑a∈G0( na∑i=1RT(Iai ) +ma∑j=1RT(Jaj ) +4∑k=1RT(Kak )),where for brevity we write RT( · ) for RT(f, f˜ , g, g˜, · ).The RT(Iai ) terms can be computed as follows: given a coincidence pointx ∈ Iai , we have RT(Iai ) = ind(f, g, Iai )[x ef,eg]. The index is computed as ±1depending on the sign of dgx− dfx (which in turn depends on whether hai is a ora−1). A standard covering space argument shows that [xef,eg] is either ha1 . . . hai−1or ha1 . . . hai , again depending on whether hai is a or a−1. In particular, it isstraightforward to show thatRT(Iai ) = −ρ(ha1 . . . hai−1∂∂ahai).Summing over i givesna∑i=1RT(Iai ) =na∑i=1−ρ(∂∂a(h1 . . . hana))= −ρ(∂∂aϕ(a)).Completely analagous arguments will show thatRT(Jaj ) = ρ(ϕ(a)a−1(aa−1laa . . . laj−1∂∂alaj)−1),and thus thatma∑j=1RT(Jaj ) =ma∑j=1ρ(ϕ(a)a−1(aa−1laa . . . laj−1∂∂alaj)−1)= −ρ(ϕ(a)a−1i(∂∂aψ(a))),48 P. Ch. Staeckerwhere i:ZG→ ZG is the involution defined byi(∑kckak)=∑kcka−1k .Similar computations of Reidemeister classes and indices will show thatRT(Ka1 ) and RT(Ka2 ) will cancel in (4.1), and thatRT(Ka3 ) = −ρ(ϕ(a)ψ(a)−1).To complete the computation of (4.1) we note that the coincidence point inKa4 is removable by a homotopy, and thus we obtainTheorem 4.1. Let f, g:X → X be maps which induce the homomorphismsϕ,ψ:G→ G. Then there are lifts f˜ and g˜ such thatRT(f, f˜ , g, g˜) = ρ(1−∑a∈G0(∂∂aϕ(a) + ϕ(a)ψ(a)−1 − ϕ(a)a−1i(∂∂aψ(a)))).We end this section with an interpretation of the above formula with respectto a variation on the Fox calculus. For generators {xi} of G, define the operators∆∆xi:G→ ZG as follows:∆∆xi1 = 0,∆∆xixj = δij ,∆∆xi(uv) =(∆∆xiu)v +∆∆xiv.Analogous to the properties given for the Fox calculus we can derive:∆∆xix−1i = −xi,∆∆xi(hn . . . h1) =n∑k=1(∆∆xihk)hk−1 . . . h1We will use one further property, relating our new operator to the ordinaryFox calculus operator, whose proof is left as an exercise:∆∆xjw = x−1j i(∂∂xjw)w.The identity ρ(ϕ(w)z) = ρ(zψ(w) can be used in Theorem 4.2 to obtain:Corollary 4.2. With notation as in Theorem 4.1, we haveRT(f, f˜ , g, g˜) = ρ(1−∑a∈G0(∂∂aϕ(a)− ∆∆aψ(a) + ϕ(a)ψ(a)−1)).Coincidence Reidemeister Trace of Selfmaps 495. Some examplesFirst we will note that Theorem 1.1 is a simple consequence of our mainresult. Letting ψ be the identity map gives ρ(ϕ(a)a−1) = ρ(a−1ψ(a)) = ρ(1),and Theorem 1.1 immediately follows from Theorem 4.1.Our formula also gives the classical formula for the coincidence Nielsen num-ber on the circle:Example 5.1. Let G = 〈a〉, and let f and g be maps which induce thehomomorphismsϕ(a) = an, ψ(a) = am.Without loss of generality, we will assume that n ≥ m. Our formula then givesRT(f, f˜ , g, g˜) = ρ(1− ∂∂aan +∆∆aam − ana−m)= ρ(1− (1 + . . .+ an−1) + (1 + . . .+ am−1)− an−m)= ρ(1− am − . . .− an−1 − an−m).A simple calculation shows that ρ(ai) = ρ(aj) if and only if i ≡ j mod n −m.Thus ρ(1) = ρ(an−m), and all other terms in the above sum are in distinctReidemeister classes. Thus the Nielsen number is n−m, as desired.We conclude with one nontrivial computation of a coincidence Reidemeistertrace of two selfmaps of the bouquet of three circles.Example 5.2. Let X be a space with fundamental group G = 〈a, b, c〉, andlet f and g be maps which induce homomorphisms as follows:ϕ :a 7→ acb−1b 7→ abc 7→ bψ :a 7→ a−1cb−1b 7→ cc 7→ b−1aOur formula givesRT(f, f˜ , g, g˜) = ρ(1− (1 + a−1cb−1 + a2)− (a+ abc−1)− (ba−1b))= ρ(−a− a2 − abc−1 − ba−1b− a−1cb−1),and we must decide the twisted conjugacy of the above elements. We use thetechnique from [4] of abelian and nilpotent quotients.Checking in the abelianization suffices to show that a is not twisted conjugateto any of the other terms. We also see that a2 and ba−1b are twisted conjugatein the abelianization, and our computation reveals that ρ(a2) = ρ(ba−1b) withconjugating element γ = ac−1. Similarly, we find that ρ(a2) = ρ(abc−1) by theelement γ = ab−1.50 P. Ch. StaeckerIt remains to decide whether or not a2 and a−1cb−1 are twisted conjugate,and a check in the class 2 nilpotent quotient shows that they are not. ThusRT(f, f˜ , g, g˜) = −ρ(a)− 3ρ(a2)− ρ(a−1cb−1)is a fully reduced expression for the Reidemeister trace, and so in particular theNeilsen number is 3.References[1] R. Crowell and R. Fox, Introduction to Knot Theory, Springer, 1963.[2] R. Dobren´ko and J. Jezierski, The coincidence Nielsen number on nonorientablemanifolds, Rocky Mountain J. Math. 23 (1993), 67–85.[3] E. Fadell and S. Husseini, The Nielsen number on surfaces, Contemp. Math. 21(1983), 59–98.[4] P. C. Staecker, Computing twisted conjugacy classes in free groups using nilpotentquotients (2007), preprint.[5] , On the uniqueness of the coincidence index on orientable differentiable mani-folds, Topology Appl. 154 (2007), 1961–1970.[6] , Axioms for a local Reidemeister trace in fixed point and coincidence theory ondifferentiable manifolds, J. Fixed Point Theory Appl. (2008) (to appear).Manuscript received October 12, 2007P. Christopher StaeckerDepartment of Mathematical SciencesMessiah CollegeGrantham PA, 17027, USAE-mail address: cstaecker@messiah.eduTMNA : Volume 33 – 2009 – No 1

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A formula for the coincidence Reidemeister trace of selfmaps on bouquets of circles

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