Periodic homeomorphisms on surfaces and singular points of curves (Susumu Hirose) Department of Mathematics, Faculty of Science and Engineering Saga University 1. Introduction 8 1 2 3 4 5 6 7 Figure 1 Dehn [7] showed that mapping class groups of closed orientable surfaces are generated by Dehn twists illustrated in the left hand of Figure 1 (precisely saying, a twist illustrated is called a right handed Dehn twist), Lickorish [15] reduced the number of generators and Humphries [13] showed that Dehn twists about the circles shown in the right hand side of Figure 1 generate the mapping class group M 3 of Σ 3, a closed oriented surface of genus 3. By the symbol k, we also mean the right handed Dehn twist about the curve indicated by k. Let φ be an element of M 3 defined by: φ = 6 5 4 3 2 8, where i j means apply i first then apply j. By using Teruaki for Mathematica (T4M), implemented by K. Ahara (Meiji Univ.) and T. Sakasai (Univ. of Tokyo), we can check the action of φ on the simple closed curves shown in Figure 1 and see that φ 12 brings these simple closed curves back to their original position (off course, up to isotopy). This means that φ 12 commutes with the Dehn twists about these simple closed curves, so, by the above mentioned result of Humphries, φ 12 commutes with any elements of M 3. Powell [18] showed that, if g 3, M g is centerless, i.e. the element of M g which commutes with any elements of M g should be id Σg. Therefore, φ 12 = id Σ3. In this note, we will introduce a method to find a word of Dehn twists, which represent periodic map. 1
For the detail of fact shown in this note, please see a preprint [12], which is available from http://www.ms.saga-u.ac.jp/~hirose/work.html. From the next section, we write this note in Japanese. If you cannot read Japanese, please learn Japanese or check this preprint. This research was partially supported by Grant-in-Aid for Encouragement of Young Scientists (No. 16740038), Ministry of Education, Science, Sports and Culture, Japan. 2. Nielsen Σ g g Σ g Σ g f (periodic) 1 n f n = id Σg n f (period) n Σ g f Σ g f n Σ g p 0 < k < n k f k (p) = p p f multiple point M f f multiple point Σ g f Σ g /f f (orbit space) Σ g p Σ g /f p [p] π f n Σ g /f π f n [f multiple point ] Σ g /f [f multiple point ] f branch point B f (= π f (M f )) f branch point π f Σg\Mf : Σ g \ M f (Σ g /f) \ B f n n Ω f : π 1 ((Σ g /f) \ B f, x) Z n ( x (Σ g /f) \ B f π f ( x) = x Σ g x π 1 ((Σ g /f) \ B f, x) x (Σ g /f) \ B f loop 2
l : [0, 1] (Σ g /f)\b f l(0) = l(1) = x loop l Σ g lift l : [0, 1] Σ g l(0) = x π f ( l(1)) = l(1) = x l(1) l(0) = x f k f k ( x) = l(1) Ω f ([l]) = k Z n Z n π 1 ((Σ g /f) \ B f, x) H 1 ((Σ g /f) \ B f ) Z n ω f Ω f Σ g f, f g f = g f g 1 f f f f (conjugate) Nielsen topologically equavalent f branch point B f = {Q 1, Q 2,, Q b } Q i branch point S Qi Theorem 2.1. [17] Σ g f, f (1) f = f (2) B f = B f (3) B f = {Q 1, Q 2,, Q b } i ω f(s Qi ) = ω f (S Q i ) θ i = ω f (S Qi ) [g, n; θ 1,, θ b ] f total valency (n, θ 1 /n+ +θ b /n) [5] total valency Introduction (12, 1/12+2/3+1/4) Σ 3 Dehn twist 3
3. Introduction Dehn twist 1 f Σ g n f branch point b D 2 C. Σ g D 2 (x, t) (f(x), exp(2πi/n) t) Σ g D 2 Z n (Σ g D 2 )/Z n [f branch point, 0] b Hirzebruch-Jung resolution, Q f branch point valency θ/n L 1 L 0 L 1 n L 0 θ k 0 = θ k 0 n k 0 n L 0 L 1 L 1 n + k 1 k 0 k 1 L i k i k i 1 k i k i 1 L i L i+1 k i 1 + k i+1 k i k i+1 multiplicity L 1 B D 2 L ki S i D 2 S i D 2 Euler S i S i S i S i S i = L i multiplicity k i D 2 plumbing (Σ g D 2 )/Z n [Q, 0] [Q, 0] 4 4
b 1 Figure 2. 1 Σ 3 f = (12, 1/12 + 2/3 + 1/4) (Σ 3 /f) 0 (Σ 3 D 2 )/Z 12 Figure 2 multiplicty 12 Σ 3 /f f branch point Figure 2 blow-down 1 null-homotopic Dehn twist (0, 0) x 4 = y 3 x 4 = y 3 Ψ : C 2 C Ψ(x, y) = x 4 y 3 Ψ 1 (0) ϵ S 3 ϵ = {(x, y) C 2 x 2 + y 2 = ϵ 2 }, L ϵ = S 3 ϵ Ψ 1 (0) ψ : S 3 ϵ L ϵ S 1 ψ(x, y) = Ψ(x,y) Ψ(x,y) L ϵ Seifert 5
S 1 fibration ψ : Sϵ 3 L ϵ S 1 x 4 y 3 = 0 Milnor-fibration [16] x 4 y 3 = 0 fibration f isotopic Dehn twist f x 4 y 3 = 0 (4, 3)-torus knot Milnor [16] well-known f A Campo divide [2, 3, 4] real morsification Gusein-Zade [1] x 4 y 3 = 0 pertub [10] Figure 3 x 4 y 3 ) 1 90 90 perturb x 4 y 3 = 0 proper divide D proper generic 2 1 P divide Divide P L(P ) L(P ) = {(u, v) T D u P, v T u P, u 2 + v 2 = 1} S 3. 6
Figure 3 P Figure 3 divide, L(P ) x 4 y 3 = 0 L ϵ isotopic. Figure 3, P divide ordered Morse divide. Ordered Morse divide, O. Couture, B. Perron [6], [10], L(P ). divide Figure 4 8 2 3 4 5 6 Figure 4 Figure divide F (P ) L(P ), F (P ) L(P ). F (P ) Figure 4 7
. 3 Figure 1 S 3 R S 3 R 1, R 2 ; (1) R = R 1 R 2, = µ 1 ν 1 µ n ν n µ i (resp. ν i ) R 1 (resp. R 2 ) proper arc (2) S 3 3 B 1, B 2 : B 1 B 2 = S 3, B 1 B 2 = B 1 = B 2 = S 2, R 1 B 1, R 2 B 2, R 1 B 1 = R 2 B 2 =. Gabai [8, 9], (S 3, R i ) R i fiber φ i monodromy fibered link, R 1 R 2 R (S 3, R) R fiber φ 1 φ 2 monodromy fibered link. Figure 5 S 3 Figure 5 annulus (positive)hopf band (positive) Hopf link. (Psitive) Hopf link (positive) Hopf band fibered link monodromy core right handed Dehn twist Figure 4, B i i core Hopf band F (P ) B i Gabai L(P ) monodromy 6 5 4 3 2 8 f = (12, 1/12 + 2/3 + 1/4) right handed Dehn twist [12] 4 Dehn twist Hirzebruch-Jung resolution blow-down 8
Dehn twist Acknowledgments. The author would like to express his gratitude to Professors Tadashi Ashikaga, Ikuko Awata, Hisaaki Endo, Mikami Hirasawa, Masaharu Ishikawa, Mizuho Ishizaka, Yukio Matsumoto, Haruko Nishi, Yoshihisa Sato, Shigeru Takamura and Yuichi Yamada for fruitful discussions and comments, and to Professors Kazushi Ahara and Takuya Sakasai for their useful program Teruaki for Mathematica. References [1] V. I. Arnol d, S. M. Guseĭn-Zade and A. N. Varchenko, Singularities of differentiable maps. Vol. II. Monodromy and asymptotics of integrals, (Translated from the Russian by Hugh Porteous. Translation revised by the authors and James Montaldi.) Monographs in Mathematics, 83. Birkhauser, 1988 [2] N. A Campo, Real deformations and complex topology of plane curve singularities, Ann. Fac. Sci. Toulouse Math. (6) 8, (1999), 5 23. [3] N. A Campo, Generic immersions of curves, knots, monodromy and Gordian number, Inst. Hautes Etudes Sci. Publ. Math. 88, (1998), 151 169. [4] N. A Campo, Planar trees, slalom curves and hyperbolic knots, Inst. Hautes Etudes Sci. Publ. Math. 88, (1998), 171 180. [5] T. Ashikaga and M. Ishizaka, Classification of degenerations of curves of genus three via Matsumoto-Montesinos theorem, Tohoku Math. J. (2) 54, (2002), 195 226. [6] O. Couture, B. Perron, Representative braids for links associated to plane immersed curves, Journal of Knot Theory and Its Ramifications, 9, (2000), 1-3. [7] M. Dehn, Die Gruppe der Abbildungsklassen, Acta Math. 69, (1938), 135 206. [8] D. Gabai, The Murasugi sum is a natural geometric operation, Low-dimensional topology (San Francisco, Calif., 1981), 131 143, Contemp. Math., 20, Amer. Math. Soc., Providence, RI, 1983. [9] D. Gabai, The Murasugi sum is a natural geometric operation. II, Combinatorial methods in topology and algebraic geometry (Rochester, N.Y., 1982), 93 100, Contemp. Math. 44, Amer. Math. Soc., Providence, RI, 1985. [10] H. Goda, M. Hirasawa, and Y. Yamada, Lissajous curves as A Campo divides, torus knots and their fiber surfaces, Tokyo Jour. Math., 25, (2002), 485-491. [11] R. E. Gompf and A. I. Stipsicz, 4-Manifolds and Kirby Calculus, Graduate Studies in Math., 20, Amer. Math. Soc., 1999. [12] S. Hirose, Presentations of periodic maps on oriented closed surfaces of genera up to 4, preprint available from http://www.ms.saga-u.ac.jp/~hirose/work.html [13] S. P. Humphries Generators for the mapping class group, In: Topology of low-dimensional manifolds (Proc. Second Sussex Conf., Chelwood Gate, 1977), Lecture Notes in Math., 722, 44 47 Springer, 1979. 9
[14] M. Ishizaka, Presentation of hyperelliptic periodic monodromies and splitting families, preprint. [15] W.B.R. Lickorish, A finite set of generators for the homeotopy group of a 2-manifold, Proc. Cambridge Philos. Soc. 60(1964), 769 778, Corrigendum: Proc. Cambridge Philos. Soc. 62(1966), 679 681. [16] J. Milnor, Singular points of complex hypersurfaces, Annals of Mathematics Studies, No. 61 Princeton University Press, 1968 [17] J. Nielsen, Die Struktur periodischer Transformationen von Flächen, Math. -fys. Medd. Danske Vid. Selsk. 15, nr.1 (1937) (English transl. in Jakob Nielsen collected works, Vol.2, 65 102). [18] J. Powell, Two theorems on the mapping class group of a surface, Proc. Amer. Math. Soc. 68 (1978), 347 350. [19] P.A. Smith, Abelian actions on 2-manifolds, Michigan Math. J., 14 (1967), 257 275. [20] K. Yokoyama, Classification of periodic maps on compact surfaces. I, Tokyo J. Math. 6 (1983), 75 94. 10