Firfield University DigitlCommons@Firfield Mth & Computer Science Fculty Publictions Mth & Computer Science Deprtment 1-1-2009 A formul for the coincidence Reidemeister trce of selfmps on bouquets of circles Christopher P. Stecker Firfield University, cstecker@firfield.edu Copyright 2009 Topologicl Methods in Nonliner Anlysis. Included with permission from the publisher. Peer Reviewed Repository Cittion Stecker, Christopher P., "A formul for the coincidence Reidemeister trce of selfmps on bouquets of circles" 2009). Mth & Computer Science Fculty Publictions. Pper 33. http://digitlcommons.firfield.edu/mthndcomputerscience-fcultypubs/33 Published Cittion Stecker, P. Christopher, A formul for the coincidence Reidemeister trce of selfmps on bouquets of circles. Topologicl Methods in Nonliner Anlysis 33, 2009, 41 50. This Article is brought to you for free nd open ccess by the Mth & Computer Science Deprtment t DigitlCommons@Firfield. It hs been ccepted for inclusion in Mth & Computer Science Fculty Publictions by n uthorized dministrtor of DigitlCommons@Firfield. For more informtion, plese contct digitlcommons@firfield.edu.
Topologicl Methods in Nonliner Anlysis Journl of the Juliusz Schuder Center Volume 33, 2009, 41 50 A FORMULA FOR THE COINCIDENCE REIDEMEISTER TRACE OF SELFMAPS ON BOUQUETS OF CIRCLES P. Christopher Stecker Abstrct. We give formul for the coincidence Reidemeister trce of selfmps on bouquets of circles in terms of the Fox clculus. Our formul reduces the problem of computing the coincidence Reidemeister trce to the problem of distinguishing doubly twisted conjugcy clsses in free groups. 1. Introduction Fdell nd Husseini, in [3] proved the following: Theorem 1.1 Fdell, Husseini, 1983). Let X be bouquet of circles, G = π 1 X), nd let G 0 be the set of genertors of G. If f: X X induces the mp ϕ: G G, then there is some lift f: X X to the universl covering spce with RTf, f) = ρ 1 G 0 ϕ) where ρ: ZG ZRf) is the lineriztion of the projection into twisted conjugcy clsses, nd denotes the Fox derivtive. Theorem 1.1 reduces the clcultion of the Reidemeister trce nd thus of the Nielsen number) in fixed point theory to the computtion of twisted conjugcy clsses. Our gol for this pper is to obtin similr result in 2000 Mthemtics Subject Clssifiction. 54H25, 55M20. Key words nd phrses. Coincidence theory, Nielsen number, Redemeister trce. ), c 2009 Juliusz Schuder Center for Nonliner Studies 41
42 P. Ch. Stecker coincidence theory of selfmps formul for the coincidence Reidemeister trce RTf, f, g, g) in terms of Fox derivtives which reduces the computtion to twisted conjugcy decisions. The proof of Theorem 1.1 given in [3] is brief, thnks to nturl trce-like formul for the Reidemeister trce in fixed point theory. No such formul exists for the coincidence Reidemeister trce, nd this will complicte our derivtion. Our rgument is bsed on first specifying prticulr regulr form for mps in Section 3 nd for pirs of mps in Section 4. In Section 5 we give our min result. The uthor would like to thnk Philip Heth, Nirtty Khmsemnn, nd Seungwon Kim for helpful nd encourging converstions, nd Robert Brown nd the referee for helpful comments on the pper. 2. Preliminries Throughout the pper, let X be bouquet of circles meeting t the bse point x 0. Let G = π 1 X), free group, nd let G 0 be the set of genertors of G. Let X be the universl covering spce of X with projection p X : X X, nd choose once nd for ll bse point x 0 X with p X x 0 ) = x 0. Given mps f, g: X X nd their induced homomorphisms ϕ, ψ: G G, we define n equivlence reltion on G s follows: two elements α, β G re doubly) twisted conjugte if α = ϕγ)βψγ) 1. The equivlence clsses with respect to this reltion re the Reidemeister clsses, nd we denote the set of such clsses s Rϕ, ψ). Let ρ: G Rϕ, ψ) be the projection into Reidemeister clsses. For ny pir of mps f, g: X X, denote their coincidence set by Coinf, g) = {x X fx) = gx)}. The set of coincidence points re prtitioned into coincidence clsses of the form p X Coinα 1 f, g)), where α G nd f, g: X X re specified lifts of f nd g. Lemm 2.3 of [2] shows tht Coinα 1 f, g) = Coinβ 1 f, g) if nd only if ρα) = ρβ), nd tht Coinα 1 f, g) nd Coinβ 1 f, g) re disjoint if ρα) ρβ). Thus the coincidence clsses re represented by Reidemeister clsses in G, nd so ech prticulr coincidence point hs n ssocited Reidemeister clss. For x Coinf, g), let [x ef,eg ] Rϕ, ψ) denote the Reidemeister clss ρα) for which x p X Coinα 1 f, g)). Let f, g: X X be mppings with isolted coincidence points, nd for ech coincidence point x let U x X be neighborhood of x contining no other coincidence points. Then we wish to define the coincidence Reidemeister trce
Coincidence Reidemeister Trce of Selfmps 43 s sum of the clsses [x ef,eg ] with ech coefficient given by the coincidence index t x. The coincidence index is in generl not defined for nonmnifolds such s our bouquet of circles X, but we cn construct n pproprite index without too much difficulty. It will be shown in the next section tht we my ssume tht g is homotopic to the identity in neighborhood of the bse point x 0, nd thus tht the coincidence index of f nd g on neighborhood of x 0 is simply the fixed point index of f t x 0 which is defined for ny ANR). Ner ny point x other thn x 0, the spce X is n orientble differentible mnifold, nd we define the coincidence index s usul for tht setting see [5]): indf, g, U x ) = signdetdg x df x )), where df x nd dg x denote the derivtives of f nd g t x. Tht this determinnt cn be ssumed to be nonzero is consequence of our construction below.) Thus we define the coincidence Reidemeister trce s: RTf, f, g, g) = ιf, U x0 )[x 0 ef,eg ] + indf, g, U x )[x ef,eg ], x Coinf,g) x 0 where ι denotes the fixed point index. Indeed we cn define locl Reidemeister trce: for ny open set U not cointining x 0, define RTf, f, g, g, U) = indf, g, U x )[x ef,eg ], x Coinf,g,U) where Coinf, g, U) = Coinf, g) U. If U does contin x 0, then we dd ιf, U x0 )[x 0 ef,eg ] into the sum in the obvious wy. Clerly this locl Reidemeister trce is equl to the nonlocl version if U is tken to be X, nd hs the following dditivity property: if V nd W re disjoint subsets of U with Coinf, g, U) V W, then RTf, f, g, g, U) = RTf, f, g, g, V ) + RTf, f, g, g, W ). As for defining the Reidemeister trce for ny pir of mps f, g: X X perhps hving nonisolted coincidence points) with lifts f, g: X X, we first chnge f nd g by homotopy to f, g so tht they hve isolted coincidence points tht this is possible will be consequence of our construction in Theorem 4.1). Now the homotopies of f, g to f, g cn be lifted to homotopy of f, g to some lifts f, g. We then define RTf, f, g, g) = RTf, f, g, g ). Tht this is well defined will be consequence of the homotopy invrince of the coincidence index nd the homotopy-reltedness of coincidence clsses see Lemm 5.1 of [6]).
44 P. Ch. Stecker We will now review the necessry properties of the Fox clculus see e.g. [1]). If {x i } re the genertors of free group G, then the opertors re defined by: x i 1 = 0, x i x j = δ ij, x i : G ZG uv) = u + u v, x i x i x i where δ ij is the Kronecker delt, nd u, v G re ny words. Two importnt formuls cn be obtined from the bove: x 1 i = x 1 n i, h 1... h n ) = h 1... h k 1 h k. x i x i x i k=1 3. A regulr form for mppings In this section we will describe stndrd form for selfmps of X. Ech circle of X is represented by some genertor of the fundmentl group. For ech genertor G 0, let X be the circle represented by including the bse point x 0 ). For simplicity in our nottion, we prmeterize ech circle by the intervl [0, 1] with endpoints identified. The circles of X will be prmeterized so tht the bse point x 0 is identified with 0 or equivlently, with 1). For ny genertor G nd ny x [0, 1], let [x] denote the point of X which hs coordinte x. Intervl-like subsets of X will be denoted e.g. x 1, x 2 ) for the subset of points in prmeterized by the intervl x 1, x 2 ). Homotopy clsses of mppings of X re chrcterized by their induced mppings on the fundmentl groups. Consider the exmple where G =, b nd the mpping f: X X induces ϕ: G G with ϕ) = b 1 1 b 2. Geometriclly speking, the bove formul for ϕ indictes tht f is homotopic to mp fixing the bse point x 0 nd mpping some intervl 0, x 1 ) bijectively onto x 0, mps some intervl x 1, x 2 ) bijectively onto b x 0 in the reverse direction ), nd so on. We cn represent the ction of this mp on pictorilly s in Figure 1, where in this cse x i = [i/5], nd the lbel on ech intervl indictes tht the intervl is being mpped bijectively onto the corresponding circle. Note tht sliding the points x i i 0) round the circle will not chnge the homotopy clss of f, provided tht no x i ever moves cross nother, nd the ordering of the lbels is preserved.
Coincidence Reidemeister Trce of Selfmps 45 b 1 1 b b Figure 1. Digrm of the ction on of mp with ϕ) = b 1 1 b 2. The bse point x 0 is circled t left Thus ny mp f: X X is homotopic to mp which is chrcterized s follows: for ech genertor G 0, specify n intervls I1,... In together with lbels {h 1,..., h n }, where ech of h i re letters of G letter of G is n element which is either genertor or the inverse of genertor of G). These intervls will be clled intervls of f, nd the lbels will be clled the lbels of f. Since we re concerned only with homotopy clsses of mps, we my ssume tht f mps x i, x i+1 ) ffine linerly onto 0, 1) b for some genertor b G. A mp specified by intervls nd lbels which is liner on ech intervl in this wy will be clled regulr. Specifying mp f by intervls nd lbels gives precise informtion which cn be used to compute the derivtives of f t ny point. For ny intervl I = x i, x i+1 ), define wi), the width of I, s the rel number x i+1 x i. It is esy to verify tht for ny coincidence point x I, we hve df x = ± 1 wi), where the sign bove is + when I is lbeled by genertor, nd when I is lbeled by the inverse of genertor. 4. The Reidemeister trce of pir of regulr mps Let f, g: X X be regulr pir of mps, nd choose lifts f nd g of f nd g respectively so tht f x 0 ) = g x 0 ). In this section we will describe method for clculting RTf, f, g, g).
46 P. Ch. Stecker For ech genertor of G, write f) = h 1... h n, g) = l 1... l m, where ll h i nd l j re letters of G. Without loss of generlity, we will ssume the ction of f nd g on is specified by intervls nd lbels s pictured in Figure 2.... h n h 1 1 1 1 l m... l 1 1 1 1 Figure 2. Digrm of the ction of f nd g on. Outer lbels give the ction of f, while inner lbels give the ction of g. Given this construction of f nd g, we see tht Coinf, g) consists of the points x 0 nd the vrious [1/2], long with finitely mny isolted coincidence points occuring wy from the intervl endpoints. In order to compute the Reidemeister trce, we prtition X into severl intervls: let I1,..., In be the intervls of f in 0, 1/2) lbeled by h 1,..., h n, nd let J1,..., Jm be the intervls of g in 1/2, 1) lbeled by l1,..., lm. Let I be the intervl of f contining [1/2], nd let I 0 be the union of ll intervls of f hving x 0 s n endpoint. We define four more intervls to cover the remining portions of X: let K1 be the intervl of f in 0, 1/2) which follows In nd is lbeled by 1. Let V be the first intervl of g in 1/2, 1) this intervl is lbeled by 1 ) nd we set K2 to be the interior of V I. Let K3 be the intervl of g following Jm, this intervl is lbeled 1 by g. Finlly, let U be the lst intervl of g, nd let K4 be the interior of U I 0. Since our regulr mps re constructed to hve no
Coincidence Reidemeister Trce of Selfmps 47 coincidences t intervl endpoints except for x 0 nd the [1/2], we hve Coinf, g ) I 0 G 0 n I i=1 I i m j=1 J j nd so we cn compute RTf, f, g, g ) s sum of the Reidmeister trces over the vrious intervls in the bove union. The coincidence points t the vrious [1/2] re removble, s g cn be deformed wy from x 0 in neighborhood of [1/2]. Thus we hve RTf, g, f, g, I ) = 0 for ech. We cn lso observe tht in neighbourhood of x 0, the mp g is homotopic to the identity, nd thus tht indf, g, I 0 ) is simply the fixed point index of constnt mp, which is 1. Since x 0 is coincidence point of f nd g, we hve [x 0 ef,eg ] = ρ1), nd so RTf, f, g, g, I 0 ) = ρ1). The union bove thus gives the Reidemeister trce s sum: 4.1) RTf, f, g, g) = ρ1) + G 0 n 4 k=1 m RTIi ) + RTJj ) + i=1 j=1 K k ), 4 k=1 ) RTKk ), where for brevity we write RT ) for RTf, f, g, g, ). The RTIi ) terms cn be computed s follows: given coincidence point x Ii, we hve RTI i ) = indf, g, I i )[x f,eg e ]. The index is computed s ±1 depending on the sign of dg x df x which in turn depends on whether h i is or 1 ). A stndrd covering spce rgument shows tht [x ef,eg ] is either h 1... h i 1 or h 1... h i, gin depending on whether h i is or 1. In prticulr, it is strightforwrd to show tht ) RTIi ) = ρ h 1... h i 1. Summing over i gives n i=1 n RTIi ) = i=1 h i ) ) ρ h 1... h n ) = ρ ϕ). Completely nlgous rguments will show tht RTJj ) = ρ ϕ) 1 1 l... lj 1 nd thus tht m m RTJj ) = ρ ϕ) 1 1 l... lj 1 j=1 j=1 l j l j ) 1 ) ) 1 ), )) = ρ ϕ) 1 i ψ),
48 P. Ch. Stecker where i: ZG ZG is the involution defined by ) i c k k k = k c k 1 k. Similr computtions of Reidemeister clsses nd indices will show tht RTK 1 ) nd RTK 2 ) will cncel in 4.1), nd tht RTK 3 ) = ρϕ)ψ) 1 ). To complete the computtion of 4.1) we note tht the coincidence point in K 4 is removble by homotopy, nd thus we obtin Theorem 4.1. Let f, g: X X be mps which induce the homomorphisms ϕ, ψ: G G. Then there re lifts f nd g such tht RTf, f, g, g) = ρ 1 G 0 ))) ϕ) + ϕ)ψ) 1 ϕ) 1 i ψ). We end this section with n interprettion of the bove formul with respect to vrition on the Fox clculus. For genertors {x i } of G, define the opertors x i : G ZG s follows: x i 1 = 0, x i x j = δ ij, ) uv) = u v + v. x i x i x i Anlogous to the properties given for the Fox clculus we cn derive: x i x 1 i = x i, x i h n... h 1 ) = n k=1 ) h k h k 1... h 1 x i We will use one further property, relting our new opertor to the ordinry Fox clculus opertor, whose proof is left s n exercise: ) w = x 1 j i w w. x j x j The identity ρϕw)z) = ρzψw) cn be used in Theorem 4.2 to obtin: Corollry 4.2. With nottion s in Theorem 4.1, we hve RTf, f, g, g) = ρ 1 G 0 ϕ) )) ψ) + ϕ)ψ) 1.
Coincidence Reidemeister Trce of Selfmps 49 5. Some exmples First we will note tht Theorem 1.1 is simple consequence of our min result. Letting ψ be the identity mp gives ρϕ) 1 ) = ρ 1 ψ)) = ρ1), nd Theorem 1.1 immeditely follows from Theorem 4.1. Our formul lso gives the clssicl formul for the coincidence Nielsen number on the circle: Exmple 5.1. Let G =, nd let f nd g be mps which induce the homomorphisms ϕ) = n, ψ) = m. Without loss of generlity, we will ssume tht n m. Our formul then gives RTf, f, g, g) = ρ 1 n + ) m n m = ρ1 1 +... + n 1 ) + 1 +... + m 1 ) n m ) = ρ1 m... n 1 n m ). A simple clcultion shows tht ρ i ) = ρ j ) if nd only if i j mod n m. Thus ρ1) = ρ n m ), nd ll other terms in the bove sum re in distinct Reidemeister clsses. Thus the Nielsen number is n m, s desired. We conclude with one nontrivil computtion of coincidence Reidemeister trce of two selfmps of the bouquet of three circles. Exmple 5.2. Let X be spce with fundmentl group G =, b, c, nd let f nd g be mps which induce homomorphisms s follows: ϕ : cb 1 b b c b ψ : 1 cb 1 b c c b 1 Our formul gives RTf, f, g, g) = ρ1 1 + 1 cb 1 + 2 ) + bc 1 ) b 1 b)) = ρ 2 bc 1 b 1 b 1 cb 1 ), nd we must decide the twisted conjugcy of the bove elements. We use the technique from [4] of belin nd nilpotent quotients. Checking in the beliniztion suffices to show tht is not twisted conjugte to ny of the other terms. We lso see tht 2 nd b 1 b re twisted conjugte in the beliniztion, nd our computtion revels tht ρ 2 ) = ρb 1 b) with conjugting element γ = c 1. Similrly, we find tht ρ 2 ) = ρbc 1 ) by the element γ = b 1.
50 P. Ch. Stecker It remins to decide whether or not 2 nd 1 cb 1 re twisted conjugte, nd check in the clss 2 nilpotent quotient shows tht they re not. Thus RTf, f, g, g) = ρ) 3ρ 2 ) ρ 1 cb 1 ) is fully reduced expression for the Reidemeister trce, nd so in prticulr the Neilsen number is 3. References [1] R. Crowell nd R. Fox, Introduction to Knot Theory, Springer, 1963. [2] R. Dobreńko nd J. Jezierski, The coincidence Nielsen number on nonorientble mnifolds, Rocky Mountin J. Mth. 23 1993), 67 85. [3] E. Fdell nd S. Husseini, The Nielsen number on surfces, Contemp. Mth. 21 1983), 59 98. [4] P. C. Stecker, Computing twisted conjugcy clsses in free groups using nilpotent quotients 2007), preprint. [5], On the uniqueness of the coincidence index on orientble differentible mnifolds, Topology Appl. 154 2007), 1961 1970. [6], Axioms for locl Reidemeister trce in fixed point nd coincidence theory on differentible mnifolds, J. Fixed Point Theory Appl. 2008) to pper). Mnuscript received October 12, 2007 P. Christopher Stecker Deprtment of Mthemticl Sciences Messih College Grnthm PA, 17027, USA E-mil ddress: cstecker@messih.edu TMNA : Volume 33 2009 N o 1