Maps on graphs can be deformed to be coincidence-free

Maps on graphs can b e deformed to b e coincidence free ∗ P . Christopher Staec k er † ‡§ No v em b er 16, 2018 Abstract W e giv e a construction to remo ve coincidence p oin ts of con tinuous maps on graphs (1-complexes) by changing th e maps b y homotopies. When the co domain is not homeomorphic to the circle, we sho w th at an y pair of maps can b e c hanged b y homotopies to be coincidence free. This means that there can b e no n on trivial coincidence index, N ie lsen coinci- dence num b er, or coincidence Reidemeister trace in this setting, and the results of our previous paper “A form ula for the coincidence Reidemeister trace of selfmaps o n b ouquets of circles” are in v alid. 1 In tro duction Let X and Y be gra phs, whic h are alwa ys assumed to b e nontrivial. Throughout, we will consider contin uous maps f , g : X → Y (contin uo us maps of X and Y as dimension 1 CW-complexes) and examine the coincidence set Coin( f , g ) = { x | f ( x ) = g ( x ) } . The pap er [3] attempts, in the specia l case o f b ouquets of circles, to study coincidence p oints o f f a nd g b y computing the Reidemeister tra ce, which would then allow the computation of the Nielsen num b er of the pair ( f , g ). This Nielsen nu mber would b e a low er b ound on the minimal num ber o f coincidence po in ts achiev able b y deforming f and g . A serious e r ror in [3] render s the a pproach fundamentally misguided. The approach mak es heavy use o f the co incidence index, which is not well-behav ed for b ouquets of cir cles. Our main result (Theorem 3) is that maps f , g : X → Y of graphs with Y not homeomorphic to the cir cle c a n alwa ys be changed by homotopy to be coincidence f ree. Thu s an y coincidence index in this setting ∗ MSC2000: 54H25, 55M20 † Address: Departmen t of Mathematics and Computer Science, F airfield Universit y , F air - field CT, USA ‡ Email: cstaec ke r@f ai rfield.edu § Keyw ords: Nielsen theo ry , coinciden ce theory 1 m ust alwa ys b e zero , and so any Nielsen num b er or Reidemeister tra c e which were b eing computed in [3] must ha ve the v a lue zero. In Section 2 we give our main result. W e conclude in Sectio n 3 with a note on the spe c ific errors in [3]. W e w ould like to thank Rob ert F. Br o wn for many helpful suggestions on the organiza tion of the pap er, and the refere e f or s uggestions which substantially simplified the pap er. 2 Remo ving coincidences b y homotop y Our str ategy for r emo ving coincidences can b e intuitiv ely described using a road traffic analogy . C o nsider a co incidence p oin t which o ccurs on the in terior of a n edge of the domain spa ce. Then w e pa rameterize this edge (1 -cell) as the time int erv al [0 , 1], and w e c an view the ma ps f and g as being represent ed by a pair of points which travel a round the spa ce Y . Let us imagine that these p oin ts represent ca rs trav eling on a ne twork of single-lane roads (so that the cars ma y not pass one another), and a coincidence po in t of the maps will represent a collisio n of the cars. A remov a l o f a coincidence po in t b y a homotopy will consist of a strategy for letting the t w o cars pass one another without colliding. Avoiding a collisio n is p o ssible provided that there is a fork in the netw ork of roads where at least three roa ds meet: W hen tw o cars are ab out to collide , one of them reverses direction until the fork is reached. A t this p oin t, the car which reversed direction mov es onto the third road and a llo ws the other to pass. The cars can now pro ceed ba c k to their or iginal meeting point, this time with their p ositions reversed. Rep eating this pro cess befo r e each imminen t c ollision allows the ca rs to complete their trips without colliding. This strategy is forma liz ed as follows: Theorem 1 . L et f , g : X → Y b e maps of c onne cte d gr aphs with Y n ot a manifold, and let x ∈ Coin( f , g ) b e a c oincidenc e p oint in the interior of some e dge. Then ther e is an arbitr arily smal l n eig hb orho o d U of x on which f and g c an b e change d by homotopy to b e c oincidenc e fr e e. Pr o of. Let σ b e the edge (1- c ell) containing x , which we identif y with its attach- ing map σ : [0 , 1] → X . Let x = σ ( t 0 ), and le t U = σ ([ t 0 − ǫ, t 0 + ǫ ]) for s ome small ǫ > 0. W e may assume that f ( x ) = g ( x ) is a point on the in terior of so me 1-cell ρ : [0 , 1] → Y . W e can parameterize σ and ρ so that f ( x ) = g ( x ) = ρ (1 / 2), and that f and g b eha ve acco rding to the graph in Figure 1. (W e ma y perhaps hav e to in terchange the r oles o f f and g .) The assumption that Y is not a manifold m eans that we may cho o se the CW-complex structure on Y so that the v ertex σ (0) meets tw o other 1-cells γ , λ : [0 , 1] → Y with γ (0) = λ (0) = σ (0). This vertex is the “fork in the roa d”. Now we change f and g by ho motop y on U to maps f ′ and g ′ according to Figure 2. Informa lly , the t w o maps retreat to the fork p oint, use the fo r k to pass o ne another without colliding, and return to their or iginal po sitions at time 2 σ ( t 0 − ǫ ) σ ( t 0 ) σ ( t 0 + ǫ ) ρ (0) ρ ( 1 4 ) ρ ( 1 2 ) ρ ( 3 4 ) ρ (1) σ ( t ) ρ ( s ) ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? f                                  g ◦ Figure 1: Behavior o f f and g on U . t 0 + ǫ . The maps f ′ and g ′ are fre e of co incidences on U , a nd the theor em is prov ed. The ab ov e theorem implies our main result, with the help of a lemma which is true for muc h mor e general space s , though we only r equire it for complexes. Its pro of is an exercise. Lemma 2. L et f , g : X → Y wher e X and Y ar e c onne cte d c omplexes, and let x ∈ Coin( f , g ) . Then for any neighb orho o d U of x , we may change f and g by homotopy on U so that x is no longer a c oincidenc e p oint. The lemma above means that w e ma y assume that every coincidence of o ur maps occurs on the interior of an edge, and then T heo rem 1 can be applied rep eatedly to remo ve them. Thus w e obtain: Theorem 3. If f , g : X → Y ar e maps on c onne cte d gr aphs with Y not home o- morphic t o the cir cle, then f and g c an b e change d by homotopy to b e c oincidenc e fr e e. Pr o of. If Y is homeomorphic to the interv al [0 , 1], then f and g are trivially nu llhomotopic and th us can be made to be coincidence free b y deforming them int o different constant maps. Thus we may a s sume that Y is not a manifold, and w e may fr eely use Theorem 1. First, w e ma y change our maps by homotopy to be “linear” as in [3] so that Coin( f , g ) is a finite s et. F ur thermore by the lemma we may assume that all coincidences o ccur at interior p oin ts o f edges. Then rep eated applica tio n of Theorem 1 will remov e all coincidences. 3 σ ( t 0 − ǫ ) σ ( t 0 ) σ ( t 0 + ǫ ) 0 1 4 1 2 3 4 1 σ ( t ) s                               g ′ . . . . . . . .         2 2 2 2 2 2 2        f ′ ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( Figure 2: Behavior of f ′ and g ′ on U . Solid line indicates v alues in ρ ( s ), dotted line indicates v alues in γ ( s ), and dashed line indicates v alues in λ ( s ). See the end of Section 3 for a note on the case where Y is the cir c le . The ab o ve theorem highlights the fact that coincidence theor y on graphs is not a generalizatio n of fixed p oin t theor y . It is cer tainly poss ible for a selfma p f o n e.g. a bo uquet of 2 circ les to hav e fixed points which cannot be removed b y homotopy , even though (by Theorem 3) any coincidence s of f with the identit y map c an b e r emo ved. This o ccurs bec a use our r emo v al c onstruction c hanges the second map b y homotopy as w ell as the first. This distinction do es not o ccur b et ween fixed p oint and coincidence theory on manifolds and some other spaces, as demonstrated by Bro oks in [1], but Bro oks’ result do es not apply to complexes in general. 3 The error of [3 ], and the coincidenc e index The form ula g iv en for the Reidemeister trace in [3] uses ess e ntially tw o ingredi- ent s: the computatio n of the Reidemeister cla ss for each coincide nc e point, a nd the computation of the coincidence index for e a c h coincidence p oin t. The ma- terial concer ning the Reidemeister class is es sen tially corr ect, and the ma terial concerning the index is incorrect. The erro r sp ecifically arises on p age 4 3 o f [3]: “ Near any p oin t x other than x 0 , the space X is an o r ien table differentiable manifold, and w e define the coincidence index as usual for that setting.” This formulation of the coincidence index is not well-behaved under homotopy . If, ov er the course o f the homo top y , the coincidence v alue (the common v a lue o f f ( x ) and g ( x )) tra vels through the 4 wedge p oin t y 0 , this “index” will change unpredictably . In fact, tw o fundamental pro perties of the coincidence index ar e that it is inv ar ian t under homoto pie s of f and g , and that the index is zer o when f and g are coincidence-free on U . Since (b y Theorem 3) the coincidence se t for maps of gra phs can alwa ys be made empt y by homotopies, any “coincidence index” in this setting mu st always give the v a lue zero. Some nontrivial indices can be de fined b y restricting the structure of either the domain or the co domain space s. Gon¸ calves in [2] gives a co incidence index for maps from a complex in to a manifo ld of the s ame dimension, w hich suffices to address the exceptional cas e from Section 2, the case where Y is the cir c le . In this ca se Gon¸ calves’s index do e s provide a nontrivial coincidence index which generalizes the fixed po in t index. Thu s there are many exa mples of maps f , g : X → S 1 for which MC ( f , g ), the minimal num ber of coincidence p o in ts when f a nd g ar e c hanged b y ho - motopy , is nonzero (this will o ccur whenever Gon¸ calves’s index is nonzero ). In particular when X is also S 1 , it is known that MC ( f , g ) is the Nielsen num ber N ( f , g ) = | deg ( f ) − deg( g ) | , which is easily made nonzero. References [1] R. B r ooks . On removing co inc ide nc e s of t w o maps when o nly one, rather than both, o f them ma y be deformed b y a homotopy . Pacific Journal of Mathematics , 139:45 –52, 1971 . [2] D. L. Gon¸ ca lv es. Coincidenc e theor y for maps from a complex into a mani- fold. T op olo gy and Its Applic ations , 92:63–7 7, 1999 . [3] P . C. Staeck er. A formula for the coincidence Reidemeis ter tra ce o f selfmaps on bouquets of circles. T op olo gic al Metho ds in N online ar Analysis , 33:41– 50, 2009. 5