## Osaka Journal of Mathematics

### Presentations of periodic maps on oriented closed surfaces of genera up to 4

Susumu Hirose

#### Abstract

For oriented closed surfaces of genera up to 4, we list presentations of periodic maps by Dehn twists. As an application of these presentations, we provide examples of non-holomorphic Lefschetz fibrations.

#### Article information

Source
Osaka J. Math., Volume 47, Number 2 (2010), 385-421.

Dates
First available in Project Euclid: 23 June 2010

https://projecteuclid.org/euclid.ojm/1277298910

Mathematical Reviews number (MathSciNet)
MR2722366

Zentralblatt MATH identifier
1202.57022

#### Citation

Hirose, Susumu. Presentations of periodic maps on oriented closed surfaces of genera up to 4. Osaka J. Math. 47 (2010), no. 2, 385--421. https://projecteuclid.org/euclid.ojm/1277298910

#### References

• V.I. Arnol'd, S.M. Guseĭ n-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, Birkhäuser Boston, Boston, MA, 1988.
• N. A'Campo: Real deformations and complex topology of plane curve singularities, Ann. Fac. Sci. Toulouse Math. (6) 8 (1999), 5--23.
• N. A'Campo: Generic immersions of curves, knots, monodromy and Gordian number, Inst. Hautes Études Sci. Publ. Math. 88 (1998), 151--169.
• N. A'Campo: Planar trees, slalom curves and hyperbolic knots, Inst. Hautes Études Sci. Publ. Math. 88 (1998), 171--180.
• 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.
• J.S. Birman and H.M. Hilden: On the mapping class groups of closed surfaces as covering spaces; in Advances in the Theory of Riemann Surfaces (Proc. Conf., Stony Brook, N.Y., 1969), Ann. of Math. Studies 66, Princeton Univ. Press, Princeton, N.J., 81--115, 1971.
• Z.J. Chen: On the lower bound of the slope of a nonhyperelliptic fibration of genus 4, Internat. J. Math. 4 (1993), 367--378.
• O. Couture and B. Perron: Representative braids for links associated to plane immersed curves, J. Knot Theory Ramifications 9 (2000), 1--30.
• M. Dehn: Die Gruppe der Abbildungsklassen, Acta Math. 69 (1938), 135--206.
• S.K. Donaldson: Lefschetz pencils on symplectic manifolds, J. Differential Geom. 53 (1999), 205--236.
• H. Endo: Meyer's signature cocycle and hyperelliptic fibrations, Math. Ann. 316 (2000), 237--257.
• H. Endo and S. Nagami: Signature of relations in mapping class groups and non-holomorphic Lefschetz fibrations, Trans. Amer. Math. Soc. 357 (2005), 3179--3199.
• D. Gabai: The Murasugi sum is a natural geometric operation; in Low-Dimensional Topology (San Francisco, Calif., 1981), Contemp. Math. 20, Amer. Math. Soc., Providence, RI., 131--143, 1983.
• D. Gabai: The Murasugi sum is a natural geometric operation. II; in Combinatorial Methods in Topology and Algebraic Geometry (Rochester, N.Y., 1982), Contemp. Math. 44, Amer. Math. Soc., Providence, RI., 93--100, 1985.
• H. Goda, M. Hirasawa and Y. Yamada: Lissajous curves as A'Campo divides, torus knots and their fiber surfaces, Tokyo J. Math. 25 (2002), 485--491.
• R.E. Gompf and A.I. Stipsicz: $4$-Manifolds and Kirby Calculus, Graduate Studies in Mathematics 20, Amer. Math. Soc., Providence, RI, 1999.
• Y. Gurtas: Positive Dehn twist expressions for some new involutions in mapping class group, arXiv:math.GT/0404310.
• Y. Gurtas: Positive Dehn twist expressions for some new involutions in the mapping class group II, arXiv:math.GT/0404311.
• Y. Gurtas: Positive Dehn twist expressions for some elements of finite order in the mapping class group, arXiv:math.GT/0501385.
• M. Ishizaka: Monodromies of hyperelliptic families of genus three curves, Tohoku Math. J. (2) 56 (2004), 1--26.
• M. Ishizaka: Presentation of hyperelliptic periodic monodromies and splitting families, Rev. Mat. Complut. 20 (2007), 483--495.
• T. Ito: Splitting of singular fibers in certain holomorphic fibrations, J. Math. Sci. Univ. Tokyo 9 (2002), 425--480.
• M. Korkmaz: Noncomplex smooth 4-manifolds with Lefschetz fibrations, Internat. Math. Res. Notices, (2001), 115--128.
• K. Konno: A note on surfaces with pencils of nonhyperelliptic curves of genus $3$, Osaka J. Math. 28 (1991), 737--745.
• K. Konno: Nonhyperelliptic fibrations of small genus and certain irregular canonical surfaces, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 20 (1993), 575--595.
• W.B.R. Lickorish: A representation of orientable combinatorial $3$-manifolds, Ann. of Math. (2) 76 (1962), 531--540.
• Y. Matsumoto: Lefschetz fibrations of genus two---a topological approach; in Topology and Teichmüller Spaces (Katinkulta, 1995), World Sci. Publ., River Edge, NJ., 123--148, 1996.
• J. Milnor: Singular Points of Complex Hypersurfaces, Ann. of Math. Stud. 61, Princeton Univ. Press, Princeton, N.J., 1968.
• J. Nielsen: Die Struktur periodischer Transformationen von Flächen, Math. -fys. Medd. Danske Vid. Selsk. 15, (1937) (English transl. in Jakob Nielsen collected works, Vol. 2'', 65--102).
• P.A. Smith: Abelian actions on $2$-manifolds, Michigan Math. J. 14 (1967), 257--275.
• S. Takamura: Towards the classification of atoms of degenerations. I, Splitting criteria via configurations of singular fibers, J. Math. Soc. Japan 56 (2004), 115--145.
• S. Takamura: Toward the classification of atoms of degenerations, II, Linearization of degenerations of complex curves, RIMS Preprint 1344 (2001).
• S. Takamura: Splitting Deformations of Degenerations of Complex Curves, Towards the classification of atoms of degenerations, III, Lecture Notes in Mathematics 1886, Springer, Berlin, 2006.
• S. Takamura: Toward the classification of atoms of degenerations, IV, In preparation.
• S. Takamura: Toward the classification of atoms of degenerations, V, In preparation.
• K. Yokoyama: Classification of periodic maps on compact surfaces. I, Tokyo J. Math. 6 (1983), 75--94.