Fukuhara, Shinji

Kawazumi, Nariya

Kuno, Yusuke

English

Osaka University Knowledge Archive

Title SELF-INTERSECTIONS OF CURVES ON A SURFACE ANDBERNOULLI NUMBERSAuthor(s) Fukuhara, Shinji; Kawazumi, Nariya; Kuno, YusukeCitation Osaka Journal of Mathematics. 55(4) P.761-P.768Issue Date 2018-10Text Version publisherURL https://doi.org/10.18910/70820DOI 10.18910/70820rightsNoteOsaka University Knowledge Archive : OUKAhttps://ir.library.osaka-u.ac.jp/repo/ouka/all/Osaka UniversityFukuhara, S., Kawazumi, N. and Kuno, Y.Osaka J. Math.55 (2018), 761–768SELF-INTERSECTIONS OF CURVES ON A SURFACE ANDBERNOULLI NUMBERSShinji FUKUHARA, Nariya KAWAZUMI and Yusuke KUNO(Received May 12, 2017, revised June 2, 2017)AbstractWe study an operation which measures self-intersections of curves on an oriented surface. Itturns out that a certain computation on this topological operation is related to the Bernoullinumbers Bm, and our study yields a family of explicit formulas for Bm. As a special case, thisfamily contains the celebrated formula for Bm due to Kronecker.1. Introduction1. IntroductionThe Bernoulli numbers Bm (m ≥ 0) are defined by the generating functionxex − 1 =∞∑m=0Bmm!xm.We have: B0 = 1, B1 = −1/2, B2 = 1/6, B4 = −1/30, . . ., and Bm = 0 for all odd m ≥ 3.The appearance of the Bernoulli numbers is ubiquitous in mathematics, and a large numberof identities involving the Bernoulli numbers has been known [3] [4] [9] [10].In this article, we show that the Bernoulli numbers arise naturally from the topology ofsurfaces, i.e., 2-manifolds. In more detail, by studying self-intersections of curves on anoriented surface, we obtain the following family of explicit formulas for Bm:Theorem 1. Let m ≥ 2. For any integers a and n satisfying 0 ≤ a ≤ m ≤ n, we have(1) Bm = (−1)an+1∑k=1(−1)k+1k(n + 1k) k−1∑i=1i a(k − i)m−a.Notice that the formula above has two parameters a and n. When a = 0 and n = m, theformula (1) reduces to the celebrated formula for Bm due to Kronecker ([7], see also [4] [5][9] [10]]): for m ≥ 2,(2) Bm =m+1∑k=1(−1)k+1k(m + 1k) k−1∑i=1im.In fact, using the classical formula for the sum of powers (known as Faulhaver’s formula)and a property of binomial coefficients (see Lemma 2), one can derive the formula (1) from2010 Mathematics Subject Classification. 11B68, 57N05, 57M99.The authors would like to thank Professor Noriko Yui for useful comments to an earlier version of thispaper. S.F. is supported by JSPS KAKENHI (No.26400098). N.K. is partially supported by JSPS KAKENHI(No.24224002), (No.24340010) and (No.15H03617). Y.K. is supported by JSPS KAKENHI (No.26800044).762 S. Fukuhara, N. Kawazumi and Y. Kunothe Kronecker formula (2). However, our derivation of the formula (1) is self-contained andmore direct.Our proof of Theorem 1 is motivated by a topological consideration on an oriented sur-face. In §2, we consider an operation μ to a curve on the surface. This operation wasintroduced in [6] inspired by a construction of Turaev [11], and, among other things, it com-putes self-intersections of curves. The key is to compute μ(log γ) for a simple loop γ andwe find that it involves the Bernoulli numbers (Theorem 2). Here, we work with a suitablecompletion to be able to consider log γ. In §3, we formalize the topological argument in §2and prove the main results. In §4, we give another self-contained proof of Theorem 1 byintroducing a certain generating function.The Bernoulli numbers have already appeared in the study of intersections of two curveson an oriented surface [8]. Our formula provides yet another evidence for a close con-nection between the topology of surfaces and the Bernoulli numbers. This connection hasbeen developed in [1] to an unexpected connection between the operation μ, or equivalently,the Turaev cobracket, and the Kashiwara-Vergne problem in the formulation by Alekseev-Torossian [2].2. Self-intersection map and Bernoulli numbers2. Self-intersection map and Bernoulli numbersLet S be a compact connected oriented surface with ∂S  ∅. Fix a basepoint ∗ ∈ ∂S andset π1(S ) := π1(S , ∗). We denote by π̂(S ) the set of free homotopy classes of oriented loopson S . For any p ∈ S , we denote by | | : π1(S , p)→ π̂(S ) the forgetful map of the basepoint.We recall the operation μ : Qπ1(S )→ Qπ1(S ) ⊗ (Qπ̂(S )/Q1), which has been introducedin [6] inspired by a construction of Turaev [11]. Here, 1 is the class of a constant loop. Letγ : [0, 1] → S be an immersed based loop. We arrange so that the pair of tangent vectors(γ̇(0), γ̇(1)) is a positive basis of the tangent space T∗S , and that the self-intersections of γ(except for the base point ∗) lie in the interior Int(S ) and consist of transverse double points.Let Γ be the set of such double points of γ. For p ∈ Γ we denote γ−1(p) = {tp1 , tp2 }, so that0 < tp1 < tp2 < 1. We defineμ(γ) := −∑p∈Γε(γ̇(tp1 ), γ̇(tp2 )) (γ0tp1γtp2 1) ⊗ |γtp1 tp2 | ∈ Qπ1(S ) ⊗ (Qπ̂(S )/Q1).Here,• the sign ε(γ̇(tp1 ), γ̇(tp2 )) is +1 if the pair (γ̇(tp1 ), γ̇(tp2 )) is a positive basis of TpS , and is−1 otherwise,• the based loop γ0tp1γtp2 1 is the conjunction of the paths γ|[0,tp1 ] and γ|[tp2 ,1],• the element γtp1 tp2 ∈ π1(S , p) is the restriction of γ to [tp1 , tp2 ] and we understand that|γtp1 tp2 | = 0 if the loop γtp1 tp2 is homotopic to a constant loop.Remark 1. The operation μ is essentially the same as Turaev’s operation μT : π1(S ) →Qπ1(S ) in [11]. In fact, we have μT (γ)γ = −(id ⊗ ε)μ(γ) for any γ ∈ π1(S ), where ε(α) = 1for any α ∈ π̂(S ) \ {1}. Conversely, one can express μ in terms of μT . The alternating part of(| | ⊗ 1)μ(γ) is exactly the Turaev cobracket [12] of the free loop |γ|.We observe that if γ is simple and the pair (γ̇(0), γ̇(1)) is a positive basis of T∗S , then forany integer k ∈ Z,Self-Intersections of Curves and Bernoulli Numbers 763∗Fig.1. computation of μ(γk) for a simple γ (k = 4).(3) μ(γk) =⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩−∑k−1i=1 γi ⊗ |γk−i| (k > 0)0 (k = 0)∑|k|−1i=0 γ−i ⊗ |γk+i| (k < 0).See Fig.1.In [6] §4, it was shown that the map μ extends to a map between completions μ : Qπ1(S )̂→ Qπ1(S )̂ ⊗̂Qπ̂(S )̂. Here Qπ1(S )̂ and Qπ̂(S )̂ are the completions of the group ring Qπ1(S )and the Goldman-Turaev Lie bialgebra Qπ̂(S )/Q1, respectively, with respect to the augmen-tation ideal of Qπ1(S ). Then we can consider log γ =∑∞i=1((−1)i+1/i)(γ − 1)i ∈ Qπ1(S )̂.As the following result shows, if γ is simple then one can compute μ(log γ) explicitly andthe formula involves the Bernoulli numbers.Theorem 2. Let γ ∈ π be represented by a simple loop, and assume that the pair(γ̇(0), γ̇(1)) is a positive basis of the tangent space T∗S . Then we have(4) μ(log γ) = −∑m=0Bmm!m∑p=0(−1)p(mp)(log γ)p ⊗̂ |(log γ)m−p|.3. Proof of Theorem 1 and Theorem 23. Proof of Theorem 1 and Theorem 2First of all, we describe a preliminary construction.Let Q[[Z]] (resp. Q[[X, Y]]) be the commutative ring of formal power series in an in-determinate Z (resp. in indeterminates X and Y). For a non-negative integer p, let FZp(resp. FX,Yp ) be the set of formal power series in Q[[Z]] (resp. Q[[X, Y]]) which has onlyterms of (total) degree ≥ p. We have natural isomorphisms Q[[Z]]  lim←− pQ[[Z]]/FZp andQ[[X, Y]]  lim←− pQ[[X, Y]]/FX,Yp .Set z := eZ =∑∞i=0(1/i!) Zi. Then the Laurent polynomial ring Q[z, z−1] is a subring ofQ[[Z]]. The augmentation ideal I is defined byI = Ker(Q[z, z−1]→ Q,∑ja jz j →∑ja j).Then I gives a filtration {I p}p of Q[z, z−1]. By the inclusion map Q[z, z−1] ↪→ Q[[Z]],the filtration {FZp }p restricts to {I p}p. Moreover, we have a natural isomorphism Q[[Z]] lim←− pQ[z, z−1]/I p.764 S. Fukuhara, N. Kawazumi and Y. KunoMotivated by the formula (3), we define a Q-linear map μ̂ : Q[z, z−1]→ Q[[X, Y]] by(5) μ̂(zk) =⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩−∑ki=1 eiXe(k−i)Y (k > 0)0 (k = 0)∑|k|−1i=0 e−iXe(k+i)Y (k < 0).From the definition of μ̂ it is easy to see that(e−XeY − 1)μ̂(zk) = ekX − ekY , k ∈ Z.Therefore, we have(6) (e−XeY − 1)μ̂( f (z)) = f (eX) − f (eY)for any Laurent polynomial f (z) ∈ Q[z, z−1]. ConsiderΦ(X, Y) :=∞∑i=0Bii!(−X + Y)i.Then we have (e−XeY − 1)Φ(X, Y) = −X + Y . Multiplying Φ(X, Y) to both sides of (6), wehave(7) (−X + Y)μ̂( f (z)) = ( f (eX) − f (eY))Φ(X, Y)for any f (z) ∈ Q[z, z−1].Lemma 1. There is a unique continuous extension μ̂ : Q[[Z]] → Q[[X, Y]] of the map μ̂in (5).Proof. It is sufficient to prove that μ̂(I p) ⊂ FX,Yp−1 for any p ≥ 1. Suppose f (z) ∈ I p. Thenf (eX) and f (eY) lie in FX,Yp . This means that the right hand side of (7) is an element of FX,Yp .Therefore, μ̂( f (z)) ∈ FX,Yp−1. Now for each k ≥ 1 we can put f (z) = (log z)k = Zk in (7), and we obtain(−X + Y)μ̂(Zk) = (Xk − Yk)Φ(X, Y).This shows that μ̂(Zk) ∈ FX,Yk−1. Setting k = 1, we have(8) μ̂(Z) = −Φ(X, Y) = −∞∑i=0Bii!i∑j=0(−1) j(ij)X jYi− j.This formula is essentially the same as the assertion of Theorem 2:Proof of Theorem 2. We identify the ring Q[[X, Y]] with the complete tensor productQ[[Z]]⊗̂Q[[Z]] by the map X → Z⊗̂1 and Y → 1⊗̂Z. Then the computation (8) impliesμ̂(log z) = −∞∑m=0Bmm!m∑p=0(−1)p(mp)(log z)p ⊗̂ (log z)m−p.(9)From (3) and (5) it follows that the substitution z → γ commutes with μ and μ̂. Thus weobtain (4). Self-Intersections of Curves and Bernoulli Numbers 765Further, by expanding the left hand side of (8) in terms of μ̂(zk)’s modulo higher degreeterms, we have the following:Proposition 1. Let m, n, a be integers satisfying 0 ≤ a ≤ m ≤ n. Then it holds thatBm = (−1)an+1∑k=1(−1)k+1k(n + 1k) ⎡⎢⎢⎢⎢⎢⎢⎣k−1∑i=1ia(k − i)m−a + δa,mkm⎤⎥⎥⎥⎥⎥⎥⎦ .Here δa,m is the Kronecker delta.Proof. In what follows, ≡means an equality in Q[[X, Y]] modulo FX,Yn+1. For k = 1, . . . , n+1, we have(10) μ̂(zk) = μ̂(ekZ) =∞∑i=1kii!μ̂(Zi) ≡n+1∑i=1kii!μ̂(Zi).Consider the square matrix D = (Dki)k,i of order n + 1, where Dki = ki/i!. Then D isinvertible since det D is a non-zero multiple of Vandermonde’s determinant det(ki−1)k,i. Theinverse matrix of D has the first row (a1, . . . , an+1), whereak =(−1)k+1k(n + 1k).(To see this, for instance, one can use Lemma 2 below to get (a1, . . . , an+1)D = (1, . . . , 0).)From (10) we have(11) μ̂(Z) ≡n+1∑k=1akμ̂(zk) =n+1∑k=1(−1)k+1k(n + 1k)μ̂(zk).Furthermore, for k = 1, . . . , n + 1, from (5) we have(12) μ̂(zk) = −k−1∑i=1∞∑a,b=0ia(k − i)ba!b!XaYb −∞∑a=0kaa!Xa.By (11) and (12), the coefficient of XaYm−a in μ̂(Z) isn+1∑k=1(−1)kk(n + 1k) ⎡⎢⎢⎢⎢⎢⎢⎣k−1∑i=1ia(k − i)m−aa!(m − a)! + δm,akmm!⎤⎥⎥⎥⎥⎥⎥⎦ .On the other hand, by (8), this coincides with(−1)a+1 Bmm!(ma)=(−1)a+1a!(m − a)! Bm.This completes the proof. Now, we can derive Theorem 1 from Proposition 1 by applying the following lemma.Although it might be well known, we give its proof for the sake of completeness.Lemma 2. Let m, n be integers satisfying 0 ≤ m ≤ n. Then it holds thatn+1∑k=1(−1)k(n + 1k)km =⎧⎪⎪⎨⎪⎪⎩0 if m ≥ 1,−1 if m = 0.766 S. Fukuhara, N. Kawazumi and Y. KunoProof. Set f (x) := (ex − 1)n+1. Since m ≤ n, the coefficient of xm in the series expansionof f (x) is zero.On the other hand, we computef (x) =n+1∑k=0(−1)n+1−k(n + 1k)ekx= (−1)n+1⎡⎢⎢⎢⎢⎢⎢⎣n+1∑k=1(−1)k(n + 1k)ekx + 1⎤⎥⎥⎥⎥⎥⎥⎦= (−1)n+1⎡⎢⎢⎢⎢⎢⎢⎣n+1∑k=1(−1)k(n + 1k) ∞∑a=0kaa!xa + 1⎤⎥⎥⎥⎥⎥⎥⎦ .Since the coefficient of xm in the last expression is equal to⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩(−1)n+1m!n+1∑k=1(−1)k(n + 1k)km if m ≥ 1,(−1)n+1⎡⎢⎢⎢⎢⎢⎢⎣n+1∑k=1(−1)k(n + 1k)+ 1⎤⎥⎥⎥⎥⎥⎥⎦ if m = 0,the assertion follows. 4. Another proof of Theorem 14. Another proof of Theorem 1Introducing a generating function of two variables, we give another self-contained proofof Theorem 1. Since we have Lemma 2, it is sufficient to prove Proposition 1.Let f (x, y) and g(x, y) be functions in variables x and y defined byf (x, y) :=∫ yx(et − 1)n+1dt, and g(x, y) := f (x, y)ey−x − 1 .We will examine the coefficient of xaym−a in the series expansion of g(x, y).First we compute f (x, y) as follows:f (x, y) =∫ yx(et − 1)n+1dt=∫ yxn+1∑k=0(−1)n+1−k(n + 1k)ektdt= (−1)n+1n+1∑k=1(−1)kk(n + 1k)(eky − ekx) + (−1)n+1(y − x).Sinceeky − ekxey−x − 1 =ekx(ek(y−x) − 1)ey−x − 1 =k−1∑i=1eixe(k−i)y + ekx,Self-Intersections of Curves and Bernoulli Numbers 767we can compute g(x, y) as follows:g(x, y) =f (x, y)ey−x − 1=(−1)n+1n+1∑k=1(−1)kk(n + 1k)(eky − ekx)ey−x − 1 + (−1)n+1 y − xey−x − 1=(−1)n+1n+1∑k=1(−1)kk(n + 1k) ⎡⎢⎢⎢⎢⎢⎢⎣k−1∑i=1eixe(k−i)y + ekx⎤⎥⎥⎥⎥⎥⎥⎦+ (−1)n+1∞∑b=0Bbb!(y − x)b.Then using the identities:eixe(k−i)y =∞∑b,c=0ib(k − i)cb!c!xbyc and ekx =∞∑b=0kbb!xb,we see that the coefficient of xaym−a in g(x, y) is given by(−1)n+1n+1∑k=1(−1)kk(n + 1k) ⎡⎢⎢⎢⎢⎢⎢⎣k−1∑i=1i aa!(k − i)m−a(m − a)! + δa,mkmm!⎤⎥⎥⎥⎥⎥⎥⎦+ (−1)n+1+a Bmm!(ma).This is equal to ((−1)n+1+a/m!)(ma)times(13) (−1)an+1∑k=1(−1)kk(n + 1k) ⎡⎢⎢⎢⎢⎢⎢⎣k−1∑i=1i a(k − i)m−a + δa,mkm⎤⎥⎥⎥⎥⎥⎥⎦ + Bm.Secondly, we expand g(x, y) in a different way. Put g1(x, y) = f (x, y)/(y − x). Then wehaveg(x, y) =f (x, y)y − xy − xey−x − 1 = g1(x, y)∞∑b=0Bbb!(y − x)b.Writing (et − 1)n+1 = ∑i≥n+1 aiti, we havef (x, y) =∫ yx(et − 1)n+1dt =∑i≥n+1aii + 1(yi+1 − xi+1).Thus the series expansion of g1(x, y) has all terms of degree ≥ n + 1, so does that of g(x, y).In particular, the coefficient of xaym−a in this expansion is zero. Therefore, the expression(13) is zero, and we obtain Proposition 1. 768 S. Fukuhara, N. Kawazumi and Y. KunoReferences[1] A. Alekseev, N. Kawazumi, Y. Kuno and F. Naef: Higher genus Kashiwara-Vergne problems and theGoldman-Turaev Lie bialgebra, C.R. Math. Acad. Sci. Paris, 355 (2017), 123–127.[2] A. Alekseev and C. Torossian: The Kashiwara-Vergne conjecture and Drinfeld’s associators, Ann. of Math.175 (2012), 415–463.[3] K. Dilcher, L. Skula and I. Slavutskii: Bernoulli Numbers Bibliography (1713–1990), Queen’s Papers inPure and Applied Mathematics, Kingston, Ontario, 1987.[4] H. Gould: Explicit formulas for Bernoulli numbers, Amer. Math. Monthly 79 (1972), 44–51.[5] J. Higgins: Double series for the Bernoulli and Euler numbers, J. London Math. Soc. (2) 2 (1970), 722–726.[6] N. Kawazumi and Y. Kuno: Intersections of curves on surfaces and their applications to mapping classgroups, Ann. Inst. Fourier (Grenoble) 65 (2015), 2711–2762.[7] L. Kronecker: Ueber die Bernouillischen Zahlen, J. Reine Angew. Math. 94 (1883), 268–269.[8] G. Massuyeau and V. Turaev: Fox pairings and generalized Dehn twists, Ann. Inst. Fourier (Grenoble) 63(2013), 2403–2456.[9] N. Nielsen: Traité élémentaire des nombres de Bernoulli, Gauthier-Villars et Cie, Paris, 1923.[10] L. Saalschütz: Vorlesungen über die Bernoullischen Zahlen, ihren Zusammenhang mit den Secanten-Coefficienten und ihre wichtigeren Anwendungen, Julius Springer, Berlin, 1893.[11] V. Turaev: Intersections of loops in two-dimensional manifolds, Math. USSR Sbornik 35 (1979), 229–250.[12] V. Turaev: Skein quantization of Poisson algebras of loops on surfaces, Ann. sci. École Norm. Sup. (4) 24(1991), 635–704.Shinji FukuharaDepartment of MathematicsTsuda University2–1–1 Tsuda–machi, Kodaira-shi Tokyo 187–8577Japane-mail: fukuhara@tsuda.ac.jpNariya KawazumiDepartment of Mathematical SciencesUniversity of Tokyo3–8–1 Komaba, Meguro-ku Tokyo 153–8914Japane-mail: kawazumi@ms.u-tokyo.ac.jpYusuke KunoDepartment of MathematicsTsuda University2–1–1 Tsuda-machi, Kodaira-shi Tokyo 187–8577Japane-mail: kunotti@tsuda.ac.jp

SELF-INTERSECTIONS OF CURVES ON A SURFACE AND BERNOULLI NUMBERS

https://ir.library.osaka-u.ac.jp/repo/ouka/all/70820/ojm55_04_761.pdf

SELF-INTERSECTIONS OF CURVES ON A SURFACE AND BERNOULLI NUMBERS

Abstract

Similar works

Full text

Available Versions

Osaka University Knowledge Archive