FIXED POINTS OF n-valued MULTIMAPS OF THE CIRCLE

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1 FIXED POINTS OF -VALUED MULTIMAPS OF THE CIRCLE Robert F. Brow Departmet of Mathematics Uiversity of Califoria Los Ageles, CA November 15, 2005 Abstract A multifuctio φ: X Y is -valued if φ(x) is a uordered subset of poits of Y for each x X. The (cotiuous) -valued multimaps φ: S 1 S 1 are classified up to homotopy by a iteger-valued degree. I the Nielse fixed poit theory of such multimaps, due to Schirmer, the Nielse umber N(φ) of a -valued φ: S 1 S 1 of degree d equals d ad φ is homotopic to a -valued power map that has exactly d fixed poits. Thus the Wecke property, that Schirmer established for maifolds of dimesio at least three, also holds for the circle. A -valued multimap φ: S 1 S 1 of degree d splits ito selfmaps of S 1 if ad oly if d is a multiple of. Subject Classificato 55M20; 54C20, 55M25 1 Itroductio A multifuctio φ: X Y is a fuctio such that φ(x) is a subset of Y for each x X. For S a subset of Y, the set φ 1 (S) cosists of the poits x X such that φ(x) S ad the set φ 1 + (S) cosists of the poits x X such that φ(x) S. A multifuctio φ is said to be upper semicotiuous (usc) if U ope i Y implies φ 1 (U) is ope i X. It is lower semicotiuous (lsc) if U ope i Y implies φ 1 + (U) is ope i X. A multifuctio that is both upper semicotiuous ad lower semicotiuous is said to be cotiuous. Although the term multimap is sometimes used for a more geeral cocept, i this paper it will mea a cotiuous multifuctio. A -valued multifuctio φ: X Y is a fuctio that assigs to each x X a uordered subset of exactly poits of Y. Thus a -valued multimap is a cotiuous -valued multifuctio. 1

2 O Neill [5] proved a versio of the Lefschetz fixed poit theorem for a class of multimaps φ: X X of fiite polyhedra that icludes the -valued multimaps. He itroduced a iteger-valued Lefschetz umber Λ(φ) such that Λ(φ) 0 implies that φ has a fixed poit, that is, x φ(x) for some x X. The Nielse fixed poit theory of -valued multimaps was developed by Schirmer i a series of papers [6], [7], [8]. For φ: X X a -valued multimap of a fiite polyhedro, the Nielse umber N(φ) has the property that for ay -valued cotiuous homotopy : X I X with (x, 0) = φ(x), the multimap ψ : X X defied by ψ(x) = (x, 1) has at least N(φ) fixed poits. The mai result of [8] exteded a celebrated theorem of Wecke [9] i the followig way. If φ: X X is a -valued multimap where X is a compact triagulable maifold, with or without boudary, of dimesio at least three, the there is a -valued multimap ψ : X X homotopic to φ such that ψ has exactly N(φ) fixed poits. As i the sigle-valued theory, we will refer to this property as the Wecke property for -valued multimaps. If f 0, f 1,..., f 1 : X X are maps such that j k implies f j (x) f k (x) for all x X the φ(x) = {f 0 (x), f 1 (x),..., f 1 (x)} defies a -valued multimap φ: X X that is called split i [7]. Oly two examples of osplit -valued multimaps are icluded i Schirmer s papers; see page 75 of [6] ad page 219 of [7]. The examples are of -valued multimaps o the uit circle S 1 ad thus the Wecke theorem of [8] does ot apply to them. I both cases, the umber of fixed poits of the map φ that Schirmer defies is precisely N(φ), but there is o geeral such result about -valued multimaps of the circle. We recall that, i the sigle-valued case, amog the maifolds oly surfaces ca fail to have the Wecke property that a selfmap f : X X is homotopic to a map with exactly N(f) fixed poits. With regard to the 1-dimesioal maifolds, the Wecke property holds for maps of the iterval because they are all homotopic to a costat map. For X = S 1, there is the followig well-kow argumet that establishes the Wecke property for sigle-valued maps. By the classificatio theorem ([4], page 39), if f : S 1 S 1 is of degree d, the f is homotopic to the power map φ d defied by viewig S 1 as the uit circle i the complex plae ad settig φ d (z) = z d. Thus N(f) = N(φ d ). It has log bee kow that N(φ d ) = 1 d ad clearly φ d has 1 d fixed poits except i the case d = 1. Sice φ 1, the idetity map, is homotopic to a fixed poit free map, 2

3 every selfmap f o the circle is homotopic to a map with N(f) fixed poits. The Wecke property is easily see to hold for -valued multimaps of the iterval I, as follows. Let φ: I I be a multimap. Defie : I I I by (s, t) = φ(st), the is cotiuous by Theorems 1 ad 1 o page 113 of [2]. Thus φ is homotopic to the costat -valued multimap κ: I I defied by κ(t) = φ(0), which has fixed poits, whereas N(κ) = by Corollary 7.3 of [7]. The purpose of this paper is to prove that the circle also has the Wecke property for -valued multimaps. I outlie, the argumet follows that of the sigle-valued settig, but there are several sigificat issues that must be addressed i the -valued case. I Sectio 2, we exted the defiitio of the degree of a selfmap of the circle to defie the degree of a -valued multimap of the circle ad we discuss its properties. Sectio 3 itroduces a collectio of -valued multimaps we call -valued power maps φ,d : S 1 S 1 ad we exted the classificatio theorem by provig that a -valued multimap φ: S 1 S 1 of degree d is homotopic to φ,d. We prove i Sectio 4 that φ,d has d fixed poits if d ad the that N(φ,d ) = d for all ad d. I Sectio 5, the Wecke property for -valued multimaps of the circle is easily see to follow from the previous results. Moreover, we characterize the split -valued multimaps of the circle: a -valued multimap is split if ad oly if its degree is a multiple of. 2 The degree of a -valued multimap of the circle We begi with some geeral properties of -valued multimaps. The followig result is a special case of a theorem of O Neill [5] but, accordig to [8], it was essetially kow much earlier [1]. Lemma 2.1. (Splittig Lemma) Let φ: X Y be a -valued multimap ad let Γ φ = {(x, y) X Y : y φ(x)} be the graph of φ. The map p 1 : Γ φ X defied by p 1 (x, y) = x is a coverig space. It follows that if X is simply-coected, the ay -valued multimap φ: X Y is split. Theorem 2.1. Let : X I Y be a -valued homotopy; write = {δ t : X Y }. If δ 0 is split, so also is. Thus a -valued multimap homotopic to a split -valued multimap is also split. 3

4 Proof. Write δ 0 = {f 0 0, f 0 1,..., f 0 1 } where f 0 j : X Y. Defie ˆf 0 0 : X {0} Γ (X I) Y by ˆf 0 0(x, 0) = ((x, 0), f 0 0(x)). Sice p 1 : Γ X I is a coverig space by Lemma 2.1 the, by the coverig homotopy property, there is a map ˆf 0 : X I Γ such that p 1 ˆf0 is the idetity map of X I. Let p 2 : Γ Y be projectio, the p 2 ˆf0 (x, t) δ t (x) so p 2 ˆf0 is a selectio for ad we ca write = {p 2 ˆf0, } where : X Y is a ( 1)-valued homotopy = {δ t } with δ 0 = {f1 0,..., f 1 0 }. Repeated applicatio of the coverig homotopy property produces a splittig = {p 2 ˆf0, p 2 ˆf1,..., p 2 ˆf 1 }. If a -valued multimap ψ : X Y is homotopic to a split -valued multimap φ = {f 0,..., f 1 } by a homotopy with δ 0 = φ ad δ 1 = ψ, the ψ = {f0 1,..., f 1 1 } where f j 1(x) = p ˆf 2 j (x, 1). Now we tur our attetio to the circle ad let p: R S 1 be the uiversal coverig space where p(t) = e i2πt. We will deote poits of the circle by p(t) for 0 t < 1. Let φ: S 1 S 1 be a -valued multimap, the the -valued fuctio φp: I S 1 is cotiuous by Theorems 1 ad 1 o page 133 of [2]. Therefore φp is split ad, usig the orderig o S 1 imposed by p from the orderig of R, we write φp = {f 0, f 1,..., f 1 } where the maps f j : I S 1 have the property f j (0) = p(t j ) for 0 t 0 < t 1 < < t 1 < 1. Let f j : I R be the lift of f j such that f j (0) = t j. We ote that if 0 j < k 1, the f j (t) < f k (t) for all t I because f j (p(t)) f k (p(t)). Sice φ is well-defied, it must be that the sets φp(0) ad φp(1) are idetical. Cosequetly, f0 (1) = v + t J for some itegers v, J where 0 J 1. We defie Deg(φ), the degree of the -valued multimap φ: S 1 S 1, by Deg(φ) = v + J. The degree ca be defied just i terms of f0 (1) because that value determies f j (1) for all j, as the ext result demostrates. Lemma 2.2. Let φ: S 1 S 1 be a -valued multimap of degree Deg(φ) = v + J. For φp = {f 0, f 1,..., f 1 } where the maps f j : I S 1 have the property f j (0) = p(t j ) with 0 t 0 < t 1 < < t 1 < 1 ad f j the lift of f j such that f j (0) = t j, we have f 1 (1) f 0 (1) < 1. Therefore, fj (1) = v + t J+j for j = 0,..., ( 1) J ad, if J 1, the f j (1) = v t j ( J) for j = J,..., 1. 4

5 Proof. Defie F : I R by F (t) = f 1 (t) f 0 (t) the F (0) = t 1 t 0 < 1. If F (1) > 1, the F (t ) = 1 for some t (0, 1) ad thus f 1 (t ) = f 0 (t ) + 1. But f j is a lift of f j so we would have p f 1 (t ) = f 1 (p(t )) = p( f 0 (t ) + 1) = p( f 0 (t )) = f 0 (p(t )) cotrary to the defiitio of a splittig. The formulas for the f j (1) the follow because f 0 (t) < f 1 (t) < < f 1 (t) for all t I. The fact that this defiitio of degree agrees with the classical defiitio whe = 1 is a special case of the followig result. Theorem 2.2. If φ: S 1 S 1 is a split -valued multimap, the Deg(φ) equals times the classical degree of the maps i the splittig. Proof. Write φ = {f 0, f 1,..., f 1 } where f j (p(0)) = p(t j ) ad 0 t 0 < t 1 < < t 1 < 1. Let f j : I R be the lift of f j p: I S 1 such that f j (0) = t j. Sice f 0 : S 1 S 1, the f 0 (1) = v+ f 0 (0) = v + t 0 for some iteger v ad thus Deg(φ) = v. Moreover, Lemma 2.2 implies that f j (1) = v + t j for j = 0,..., 1. O the other had, by the argumet o page 39 of [4], each map f j is homotopic to the power map φ v : S 1 S 1 ad therefore it is of classical degree deg(f j ) = v, so Deg(φ) = deg(f j ). Theorem 2.3. If -valued multimaps φ, ψ : S 1 S 1 are homotopic, the Deg(φ) = Deg(ψ). Proof. Let = {δ t }: S 1 S 1 be a -valued homotopy with φ = δ 0 ad ψ = δ 1. We will show that there exists ɛ > 0 such that if t t < ɛ, the Deg(δ t ) = Deg(δ t ), that is, the degree is locally costat. Sice the degree is iteger-valued, that will imply that it is costat ad therefore Deg(φ) = Deg(ψ). Write δ t p = {f0 t, f 1 t,..., f 1 t } where f j t(0) = p(t j) for 0 t 0 < t 1 < < t 1 < 1. Let f j t : I R be the lift of fj t such that f j(0) t = t j. We use the correspodig otatio for δ t. If f j t(1) = v+t J where t J > 0 the, by the cotiuity of, if ɛ > 0 is small eough, t t < ɛ t implies that f j (1) = v + t J where t J > 0 ad therefore Deg(δ t ) = Deg(δ t ) = v + J. If f t 0 (1) = v = v + 0, that meas t 0 = 0 so f t j(1) = v + t j = v + f t j(0) for all j by Lemma 2.2. Therefore, the f t j : S1 S 1 defied by f t j(p(s)) = p f t j(s) splits δ t ad thus Deg(δ t ) = deg(f t 0) by Theorem 5

6 2.2. Sice δ t is homotopic to δ t, by Theorem 2.1 δ t is also split ad f0 t is homotopic to f0 t so, for the classical degrees, deg(f0) t = deg(f0 t ) ad thus Deg(δ t ) = Deg(δ t ). 3 The classificatio theorem For itegers d ad 1, we defie the -valued multimap we call the -valued power map φ,d : S 1 S 1 by Sice φ,d (p(t)) = {p( d t), p( d t + 1 ),..., p( d t + 1 )}. φ 1,d (p(t)) = p(dt) = e i2πdt = (e i2πt ) d = (p(t)) d, we see that φ 1,d = φ d. The example o page 75 of [6] is φ 2,1 ad the example o page 219 of [7] is φ 2, 1. Lemma 3.1. The degree of φ,d is d. Proof. We see that φ,d p = (p f 0,..., p f 1 ) where f j (t) = d t + j so f j (0) = j = t j. Write d = v + J where 0 J 1 the f 0 (1) = d = v + J = v + f J (0) = v + t J so, from the defiitio, Deg(φ,d ) = v + J = d. Theorem 3.1. (Classificatio Theorem) If φ: S 1 S 1 is a - valued multimap of degree d, the φ is homotopic to φ,d. Proof. We agai write φp = {f 0, f 1,..., f 1 } : I S 1 ad lift f j to f j : I R such that f j (0) = t j where f j (0) = p(t j ) ad 0 t 0 < t 1 < t 1 < 1. Defie maps h s j : I I R by h s j(t) = s( d t + j) + (1 s) f j (t) the it is clear that j < k implies h s j (t) < hs k (t) for all s, t I. Write Deg(φ) = d = v + J where 0 J 1. Suppose 0 j ( 1) J the, by Lemma 2.2, we have h s j (1) h s J+j (0) = v. For J 1 ad J j 1, Lemma 2.2 implies that h s j (1) h s j ( J) (0) = v + 1. Thus, for all s I, the sets {p h s j (0)} ad {p h s j (1)} are idetical. Therefore, settig (p(t), s) = {p h s 0 (t), p h s 1 (t),..., p h s 1 (t)} we obtai a homotopy : S 1 I S 1 betwee φ ad φ,d. 6

7 4 Properties of the -valued power maps Theorem 4.1. If d, the the -valued power map φ,d has d fixed poits, each of ozero idex, ad o two fixed poits are i the same fixed poit class, therefore N(φ,d ) = d. Proof. If p(t) φ,d (p(t)) for some t such that 0 t < 1 the, for some j = 0, 1,... 1, we have p( d t + j ) = p(t) ad therefore d t + j (d )t t = + j = r for some iteger r. Sice d, the possible solutios are of the form t = r j d where r ad j are itegers ad 0 j 1. We require that 0 t < 1 so if d > 0, the 0 r j < d whereas if d < 0, the 0 r j > d. I either case, there are d such itegers ad we coclude that φ,d has d fixed poits. Each of the d fixed poits of φ,d is trasversal ad therefore of idex ±1 (see page 210 of [7]). It remais to prove that o two of the fixed poits of φ,d are equivalet i the sese of [7]. Notig that the fixed poits are of the form p( r j ), we will make use of the fact that d ( ) d r j + j d = r + r j d. For k = 0, 1, let x k = p( r k j k d ) = p( x k) be two fixed poits of φ,d ad let a: I S 1 be a path such that a(k) = x k. Let ã: I R be the lift of a such that a(0) = x 0. Sice a = pã, we ca write φ,d a(t) = φ,d p(ã(t)) = {p( d ã(t)), p( d ã(t) + 1 d 1 ),..., p( + ã(t) )} = {g 0 (t), g 1 (t),..., g 1 (t)}, a split multimap. The fixed poits x 0 ad x 1 are i the same fixed poit class if there exists a path a coectig them ad some j 7

8 with 0 j 1 such that g j (x k ) = x k for k = 0, 1 ad the paths a, g j : I S 1 are homotopic relative to the edpoits (see [7], page 214). We claim that the coditio g j (x 0 ) = x 0 implies that j = j 0. To prove it, we ote that sice a(0) = x 0, the ad therefore p ( d d ( r0 j 0 d for some iteger m, which implies ) + j ) = p ( r0 j 0 d ( ) r0 j 0 + j d = r 0 j 0 d + m, r 0 + r 0 j 0 d + j j 0 = r 0 j 0 d ) + m so j j 0 = m r 0, a iteger. But 0 j, j 0 1 ad therefore j = j 0. This establishes the claim ad we write g = g j = g j0 : I S 1 as the path from x 0 to x 1 that is homotopic to a relative to the edpoits. Let g : I R be the lift of g defied by g(t) = d ã(t) + j 0 r 0 the g(0) = x 0 = ã(0). Sice ag 1 is a cotractible loop, the its lift ã g 1 is also a loop ad thus g(1) = ã(1) = x 1 + q, for some iteger q. Now g(1) = d ( r1 j 1 d ) + q + j 0 r 0 = r 1 + r 1 j 1 d + j 0 j 1 + d q r 0 which implies that ad thus that q = r 1 r 0 + j 0 j 1 + d q q = r 1 j 1 d r 0 j 0 d = x 1 x 0. The 0 x 0, x 1 < 1 implies that q = 0 so x 0 = x 1 ad therefore x 0 = x 1. We coclude that o two distict fixed poits of φ,d are i the same fixed poit class. 8

9 5 The Wecke property ad split multimaps Theorem 5.1. (The Wecke Property) The circle has the Wecke property for -valued multimaps because, if φ: S 1 S 1 is a - valued multimap of degree d, the N(φ) = d ad there is a -valued multimap homotopic to φ that has exactly d fixed poits. Proof. By Theorem 3.1, φ is homotopic to φ,d so N(φ) = N(φ,d ) by Theorem 6.5 of [7]. If d =, the φ is homotopic to φ,. Choose 0 < ɛ < 1 ad defie : S1 I S 1 by (p(t), s) = {p(t + sɛ), p(t + sɛ ),..., p(t + sɛ + )}. The φ, is homotopic by to a fixed poit free multimap. Furthermore, N(φ) = N(φ, ) = 0. If d, the Theorem 4.1 completes the proof because N(φ) = N(φ,d ) = d ad φ,d has d fixed poits. Theorem 5.2. The power map φ,d is split if ad oly if d is a multiple of. Proof. The graph of φ,d is Γ φ,d = {(p(t), p( d t + j )): t R, j = 0, 1,..., 1}. For j {0, 1,..., 1} defie γ j : I Γ φ,d by γ j (t) = (p(t), p( d t + j )). Let Γ j Γ φ,d be the compoet of the graph cotaiig (p(0), p( j )), the p 1j : Γ j S 1, the restrictio of p 1 to Γ j, is a coverig space ad γ j is a path i Γ j from (p(0), p( j )) to (p(0), p( d + j )). Write d = v + J where 0 J 1, the p( d + j ) = p(r + J + j ) = p( J + j ) tells us that p( j ) = p( d + j ) ad thus γ j(0) = γ j (1) if, ad oly if, J = 0, that is, if ad oly if d is a multiple of. If d is ot a multiple of, the we have show that the fiber of every coverig space p 1j : Γ j S 1 obtaied by restrictig p 1 to a compoet of Γ φ,d cotais at least two poits. If φ,d were split, it would have a selectio, that is, there would be a map f : S 1 S 1 such that f(p(t)) φ,d (p(t)) for each t I. I particular, (p(0), f(p(0))) Γ j 9

10 for some j ad thus σ : S 1 Γ j defied by σ(p(t)) = (p(t), f(p(t)) is a cross-sectio of the coverig space p 1j : Γ j S 1, that is, p 1j σ is the idetity map of S 1. Thus p 1j σ would iduce the idetity isomorphism o the fudametal group of S 1. But that is impossible because the idex of the image of the homomorphism iduced by p 1j i that fudametal group equals the cardiality of the fiber of the coverig space, which is greater tha oe. O the other had, if d is a multiple of, the φ,d splits as φ,d = {f 0, f 1,... f 1 } where the map f j : S 1 S 1 is defied by f j (p(t)) = p( d t + j ). Corollary 5.1. If φ: S 1 S 1 is a -valued multimap of degree d, the φ is split if ad oly if d is a multiple of. Proof. By Theorem 3.1, φ is homotopic to φ,d. Therefore, by Theorem 2.1, φ is split if ad oly if φ,d is split which, by Theorem 5.2, occurs if ad oly if d is a multiple of. Refereces [1] Baach, B. ad Mazur, S. Über mehrdeutige stetige Abbilduge, Studia Math. 5, (1934). [2] Berge, C. Topological Spaces, Oliver & Boyd, [3] Góriewicz, L., Topological Fixed Poit Theory of Multivalued Mappigs, Kluwer Academic Publishers, [4] Hu, S., Homotopy Theory, Academic Press, [5] O Neill, B., Iduced homology homomorphisms for set-valued maps, Pacific J. Math. 7, (1957). [6] Schirmer, H., Fix-fiite approximatios of -valued multifuctios, Fud. Math. 121, (1984). [7] Schirmer, H., A idex ad Nielse umber for -valued multifuctios, Fud. Math. 124, (1984). [8] Schirmer, H., A miimum theorem for -valued multifuctios, Fud. Math. 126, (1985). [9] Wecke, F., Fixpuktklasse, III, Math. A. 118, (1942). 10