Signature of relations in mapping class groups and non-holomorphic Lefschetz fibrations
HTML articles powered by AMS MathViewer
- by Hisaaki Endo and Seiji Nagami PDF
- Trans. Amer. Math. Soc. 357 (2005), 3179-3199 Request permission
Abstract:
We introduce the notion of signature for relations in mapping class groups and show that the signature of a Lefschetz fibration over the 2-sphere is the sum of the signatures for basic relations contained in its monodromy. Combining explicit calculations of the signature cocycle with a technique of substituting positive relations, we give some new examples of non-holomorphic Lefschetz fibrations of genus $3, 4$ and $5$ which violate slope bounds for non-hyperelliptic fibrations on algebraic surfaces of general type.References
- J. Amorós, F. Bogomolov, L. Katzarkov, and T. Pantev, Symplectic Lefschetz fibrations with arbitrary fundamental groups, J. Differential Geom. 54 (2000), no. 3, 489–545. With an appendix by Ivan Smith. MR 1823313, DOI 10.4310/jdg/1214339791
- Tatsuya Arakawa and Tadashi Ashikaga, Local splitting families of hyperelliptic pencils. I, Tohoku Math. J. (2) 53 (2001), no. 3, 369–394. MR 1844374, DOI 10.2748/tmj/1178207417
- Sampei Usui, Mark Green, Luc Illusie, Kazuya Kato, Eduard Looijenga, Shigeru Mukai, and Shuji Saito (eds.), Algebraic geometry 2000, Azumino, Advanced Studies in Pure Mathematics, vol. 36, Mathematical Society of Japan, Tokyo, 2002. MR 1971510, DOI 10.2969/aspm/03610000
- Denis Auroux, Fiber sums of genus 2 Lefschetz fibrations, Turkish J. Math. 27 (2003), no. 1, 1–10. MR 1975329
- Joan S. Birman and Hugh M. Hilden, On the mapping class groups of closed surfaces as covering spaces, Advances in the Theory of Riemann Surfaces (Proc. Conf., Stony Brook, N.Y., 1969) Ann. of Math. Studies, No. 66, Princeton Univ. Press, Princeton, N.J., 1971, pp. 81–115. MR 0292082
- Kenneth S. Brown, Cohomology of groups, Graduate Texts in Mathematics, vol. 87, Springer-Verlag, New York-Berlin, 1982. MR 672956, DOI 10.1007/978-1-4684-9327-6
- C. Cadavid, A remarkable set of words in the mapping class group, Dissertation, Univ. of Texas, Austin, 1998.
- Zhi Jie Chen, On the lower bound of the slope of a nonhyperelliptic fibration of genus $4$, Internat. J. Math. 4 (1993), no. 3, 367–378. MR 1228579, DOI 10.1142/S0129167X93000194
- 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), no. 2, 205–236. MR 1802722, DOI 10.4310/jdg/1214425535
- Clifford J. Earle and James Eells, A fibre bundle description of Teichmüller theory, J. Differential Geometry 3 (1969), 19–43. MR 276999
- Hisaaki Endo, Meyer’s signature cocycle and hyperelliptic fibrations, Math. Ann. 316 (2000), no. 2, 237–257. MR 1741270, DOI 10.1007/s002080050012
- H. Endo, M. Korkmaz, D. Kotschick, B. Ozbagci, and A. Stipsicz, Commutators, Lefschetz fibrations and the signatures of surface bundles, Topology 41 (2002), no. 5, 961–977. MR 1923994, DOI 10.1016/S0040-9383(01)00011-8
- Ronald Fintushel and Ronald J. Stern, Constructions of smooth $4$-manifolds, Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998), 1998, pp. 443–452. MR 1648094
- Terry Fuller, Diffeomorphism types of genus $2$ Lefschetz fibrations, Math. Ann. 311 (1998), no. 1, 163–176. MR 1624287, DOI 10.1007/s002080050182
- Sylvain Gervais, Presentation and central extensions of mapping class groups, Trans. Amer. Math. Soc. 348 (1996), no. 8, 3097–3132. MR 1327256, DOI 10.1090/S0002-9947-96-01509-7
- R. E. Gompf, A topological characterization of symplectic manifolds, preprint, math.SG /0210103.
- Robert E. Gompf and András I. Stipsicz, $4$-manifolds and Kirby calculus, Graduate Studies in Mathematics, vol. 20, American Mathematical Society, Providence, RI, 1999. MR 1707327, DOI 10.1090/gsm/020
- Sergio Sispanov, Generalización del teorema de Laguerre, Bol. Mat. 12 (1939), 113–117 (Spanish). MR 3
- Toshio Ito, Splitting of singular fibers in certain holomorphic fibrations, J. Math. Sci. Univ. Tokyo 9 (2002), no. 3, 425–480. MR 1930415
- Dennis Johnson, The structure of the Torelli group. I. A finite set of generators for ${\cal I}$, Ann. of Math. (2) 118 (1983), no. 3, 423–442. MR 727699, DOI 10.2307/2006977
- A. Kas, On the handlebody decomposition associated to a Lefschetz fibration, Pacific J. Math. 89 (1980), no. 1, 89–104. MR 596919, DOI 10.2140/pjm.1980.89.89
- Kazuhiro Konno, A note on surfaces with pencils of nonhyperelliptic curves of genus $3$, Osaka J. Math. 28 (1991), no. 3, 737–745. MR 1144482
- Kazuhiro Konno, Nonhyperelliptic fibrations of small genus and certain irregular canonical surfaces, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 20 (1993), no. 4, 575–595. MR 1267600
- Mustafa Korkmaz, Noncomplex smooth 4-manifolds with Lefschetz fibrations, Internat. Math. Res. Notices 3 (2001), 115–128. MR 1810689, DOI 10.1155/S107379280100006X
- Feng Luo, A presentation of the mapping class groups, Math. Res. Lett. 4 (1997), no. 5, 735–739. MR 1484704, DOI 10.4310/MRL.1997.v4.n5.a11
- Werner Meyer, Die Signatur von Flächenbündeln, Math. Ann. 201 (1973), 239–264 (German). MR 331382, DOI 10.1007/BF01427946
- Yukio Matsumoto, On $4$-manifolds fibered by tori, Proc. Japan Acad. Ser. A Math. Sci. 58 (1982), no. 7, 298–301. MR 682687
- Yukio Matsumoto, Lefschetz fibrations of genus two—a topological approach, Topology and Teichmüller spaces (Katinkulta, 1995) World Sci. Publ., River Edge, NJ, 1996, pp. 123–148. MR 1659687
- Takayuki Morifuji, On Meyer’s function of hyperelliptic mapping class groups, J. Math. Soc. Japan 55 (2003), no. 1, 117–129. MR 1939188, DOI 10.2969/jmsj/1196890845
- Shigeyuki Morita, Characteristic classes of surface bundles, Invent. Math. 90 (1987), no. 3, 551–577. MR 914849, DOI 10.1007/BF01389178
- S. Nagami, A note on orientations of fixed point sets of spin structure preserving involutions, to appear in Kobe J. Math.
- Burak Ozbagci, Signatures of Lefschetz fibrations, Pacific J. Math. 202 (2002), no. 1, 99–118. MR 1883972, DOI 10.2140/pjm.2002.202.99
- Burak Ozbagci and András I. Stipsicz, Noncomplex smooth $4$-manifolds with genus-$2$ Lefschetz fibrations, Proc. Amer. Math. Soc. 128 (2000), no. 10, 3125–3128. MR 1670411, DOI 10.1090/S0002-9939-00-05390-9
- Bernd Siebert and Gang Tian, On hyperelliptic $C^\infty$-Lefschetz fibrations of four-manifolds, Commun. Contemp. Math. 1 (1999), no. 2, 255–280. MR 1696101, DOI 10.1142/S0219199799000110
- Ivan Smith, Lefschetz fibrations and the Hodge bundle, Geom. Topol. 3 (1999), 211–233. MR 1701812, DOI 10.2140/gt.1999.3.211
- Ivan Smith, Lefschetz pencils and divisors in moduli space, Geom. Topol. 5 (2001), 579–608. MR 1833754, DOI 10.2140/gt.2001.5.579
- András I. Stipsicz, On the number of vanishing cycles in Lefschetz fibrations, Math. Res. Lett. 6 (1999), no. 3-4, 449–456. MR 1713143, DOI 10.4310/MRL.1999.v6.n4.a7
- András I. Stipsicz, Indecomposability of certain Lefschetz fibrations, Proc. Amer. Math. Soc. 129 (2001), no. 5, 1499–1502. MR 1712877, DOI 10.1090/S0002-9939-00-05681-1
- András I. Stipsicz, Spin structures on Lefschetz fibrations, Bull. London Math. Soc. 33 (2001), no. 4, 466–472. MR 1832559, DOI 10.1017/S0024609301008232
- András I. Stipsicz, Singular fibers in Lefschetz fibrations on manifolds with $b_2^+=1$, Topology Appl. 117 (2002), no. 1, 9–21. MR 1874001, DOI 10.1016/S0166-8641(00)00105-X
- Hisaaki Endo, Meyer’s signature cocycle and hyperelliptic fibrations, Math. Ann. 316 (2000), no. 2, 237–257. MR 1741270, DOI 10.1007/s002080050012
- V. G. Turaev, The first symplectic Chern class and Maslov indices, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 143 (1985), 110–129, 178 (Russian, with English summary). Studies in topology, V. MR 806563
- Bronislaw Wajnryb, An elementary approach to the mapping class group of a surface, Geom. Topol. 3 (1999), 405–466. MR 1726532, DOI 10.2140/gt.1999.3.405
- Gang Xiao, Fibered algebraic surfaces with low slope, Math. Ann. 276 (1987), no. 3, 449–466. MR 875340, DOI 10.1007/BF01450841
Additional Information
- Hisaaki Endo
- Affiliation: Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan
- Email: endo@math.wani.osaka-u.ac.jp
- Seiji Nagami
- Affiliation: 3-6-3-10 Sakuranchou, Toyonaka, Osaka 560-0054, Japan
- Email: nagami-s@est.hi-ho.ne.jp
- Received by editor(s): November 16, 2003
- Published electronically: September 2, 2004
- © Copyright 2004 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 357 (2005), 3179-3199
- MSC (2000): Primary 57R20; Secondary 57N13, 20F05, 14D06
- DOI: https://doi.org/10.1090/S0002-9947-04-03643-8
- MathSciNet review: 2135741
Dedicated: Dedicated to Professor Yukio Matsumoto for his 60th birthday