Transactions of the American Mathematical Society

Signature of relations in mapping class groups and non-holomorphic Lefschetz fibrations

Author(s): Hisaaki Endo; Seiji Nagami
Journal: Trans. Amer. Math. Soc. 357 (2005), 3179-3199.
MSC (2000): Primary 57R20; Secondary 57N13, 20F05, 14D06
Posted: September 2, 2004
Retrieve article in: PDF DVI PostScript

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

and

which violate slope bounds for non-hyperelliptic fibrations on algebraic surfaces of general type.

1.: J. Amorós, F. Bogomolov, L. Katzarkov and T. Pantev, Symplectic Lefschetz fibrations with arbitrary fundamental groups, J. Diff. Geom. 54 (2000), 489-545. MR 2002g:57051
2.: T. Arakawa and T. Ashikaga, Local splitting families of hyperelliptic pencils, I, Tohoku Math. J. 53 (2001), 369-394. MR 2002h:14041
3.: T. Ashikaga and K. Konno, Global and local properties of pencils of algebraic curves, Algebraic Geometry 2000, Azumino, Advanced Studies in Pure Mathematics 36, 2000, pp. 1-49. MR 2003k:14003
4.: D. Auroux, Fiber sums of genus 2 Lefschetz fibrations, Turkish J. Math. 25 (2001), 1-10. MR 2004b:57033
5.: J. Birman and H. Hilden, On mapping class groups of closed surfaces as covering spaces, Advances in the Theory of Riemann surfaces, Ann. Math. Stud. 66, Princeton Univ. Press, 1971, pp. 81-115. MR 45:1169
6.: K. S. Brown, Cohomology of Groups, Graduate Texts in Mathematics 87, Springer-Verlag, 1982. MR 83k:20002
7.: C. Cadavid, A remarkable set of words in the mapping class group, Dissertation, Univ. of Texas, Austin, 1998.
8.: Z. Chen, On the lower bound of the slope of a non-hyperelliptic fibration of genus , Intern. J. Math. 4 (1993), 367-378. MR 94g:14012
9.: M. Dehn, Die Gruppe der Abbildungsklassen, Acta Math. 69 (1938), 135-206.
10.: S. K. Donaldson, Lefschetz pencils on symplectic manifolds, J. Diff. Geom. 53 (1999), 205-236. MR 2002g:53154
11.: C. J. Earle and J. Eells, A fibre bundle description of Teichmüller theory, J. Diff. Geom. 3 (1969), 19-43. MR 43:2737a
12.: H. Endo, Meyer's signature cocycle and hyperelliptic fibrations (with Appendix written by T. Terasoma), Math. Ann. 316 (2000), 237-257. MR 2001b:57047
13.: H. Endo, M. Korkmaz, D. Kotschick, B. Ozbagci and A. Stipsicz, Commutators, Lefschetz fibrations and the signatures of surface bundles, Topology 41 (2002), 961-977. MR 2003f:57051
14.: R. Fintushel and R. J. Stern, Constructions of smooth 4-manifolds, Documenta Mathematica, Extra Volume ICM 1998, II, pp. 443-452. MR 99g:57033
15.: T. Fuller, Diffeomorphism types of genus 2 Lefschetz fibrations, Math. Ann. 311 (1998), 163-176. MR 99f:57035
16.: S. Gervais, Presentation and central extensions of mapping class groups, Trans. Amer. Math. Soc. 348 (1996), 3097-3132. MR 96j:57016
17.: R. E. Gompf, A topological characterization of symplectic manifolds, preprint, math.SG /0210103.
18.: R. E. Gompf and A. I. Stipsicz, 4-manifolds and Kirby calculus, Graduate Studies in Mathematics 20, American Mathematical Society, 1999.MR 2000h:57038
19.: H. Hopf, Fundamentalgruppe und zweite Bettische Gruppe, Comment. Math. Helv. 14 (1942), 257-309. MR 3:316e
20.: T. Ito, Splitting of singular fibers in certain holomorphic fibrations, J. Math. Sci. Univ. Tokyo 9 (2002), 425-480. MR 2003h:32040
21.: D. Johnson, The structure of the Torelli group I: A finite set of generators for $\mathcal{I}$ , Ann. of Math. 118 (1983), 423-442. MR 85a:57005
22.: A. Kas, On the handlebody decomposition associated to a Lefschetz fibration, Pacific J. Math. 89 (1980), 89-104. MR 82f:57012
23.: K. Konno, A note on surfaces with pencils of non-hyperelliptic curves of genus , Osaka J. Math. 28 (1991), 737-745. MR 93c:14035
24.: K. Konno, Non-hyperelliptic fibrations of small genus and certain irregular canonical surfaces, Ann. Sc. Norm. Pisa Sup. Ser.IV, vol.XX (1993), 575-595. MR 95b:14026
25.: M. Korkmaz, Noncomplex smooth 4-manifolds with Lefschetz fibrations, Internat. Math. Res. Not. 2001, No. 3, 115-128. MR 2001m:57036
26.: F. Luo, A presentation of mapping class groups, Math. Res. Lett. 4 (1997), 735-739. MR 99b:57031
27.: W. Meyer. Die Signatur von Flächenbündeln, Math. Ann. 201 (1973), 239-264. MR 48:9715
28.: Y. Matsumoto, On 4-manifolds fibered by tori, II, Proc. Japan Acad. 59 (1983), 100-103. MR 84j:57010b
29.: Y. Matsumoto, Lefschetz fibrations of genus two - a topological approach -, Proceedings of the 37th Taniguchi Symposium on ``Topology and Teichmüller Spaces'', World Scientific, Singapore, 1996, pp. 123-148.MR 2000h:14038
30.: T. Morifuji, On Meyer's function of hyperelliptic mapping class groups, J. Math. Soc. Japan 55 (2003), 117-129. MR 2003m:57054
31.: S. Morita, Characteristic classes of surface bundles, Invent. Math. 90 (1987), 551-577. MR 89e:57022
32.: S. Nagami, A note on orientations of fixed point sets of spin structure preserving involutions, to appear in Kobe J. Math.
33.: B. Ozbagci, Signatures of Lefschetz fibrations, Pacific J. Math. 202 (2002), 99-118. MR 2002k:57066
34.: B. Ozbagci and A. Stipsicz, Noncomplex smooth 4-manifolds with genus-2 Lefschetz fibrations, Proc. Amer. Math. Soc. 128 (2000), 3125-3128. MR 2000m:57036
35.: B. Siebert and G. Tian, On hyperelliptic $C^{\infty}$ -Lefschetz fibrations of four-manifolds, Commun. Contemp. Math. 1 (1999), 466-488. MR 2001g:57053
36.: I. Smith, Lefschetz fibrations and the Hodge bundle, Geometry $\&$ Topology 3 (1999), 211-233. MR 2000j:57059
37.: I. Smith, Lefschetz pencils and divisors in moduli space, Geometry $\&$ Topology 5 (2001), 579-608.MR 2002f:57056
38.: A. I. Stipsicz, On the number of vanishing cycles in Lefschetz fibrations, Math. Res. Lett. 6 (1999), 449-456. MR 2000g:57046
39.: A. I. Stipsicz, Indecomposability of certain Lefschetz fibrations, Proc. Amer. Math. Soc. 129 (2000), 1499-1502. MR 2001h:57029
40.: A. I. Stipsicz, Spin structures on Lefschetz fibrations, Bull. London Math. Soc. 33 (2001), 466-472. MR 2002a:53062
41.: A. I. Stipsicz, Singular fibers in Lefschetz fibrations on manifolds with , Topology and its Appl. 117 (2002), 9-21. MR 2002j:57048
42.: T. Terasoma, An appendix to Endo's paper, Math. Ann. 316 (2000), 255-256. MR 2001b:57047
43.: V. G. Turaev, First symplectic Chern class and Maslov indices, J. Soviet Math. 37 (1987), 1115-1127.MR 86m:58059
44.: B. Wajnryb, An elementary approach to the mapping class group of a surface, Geometry $\&$ Topology 3 (1999), 405-466. MR 2001a:20059
45.: G. Xiao, Fibered algebraic surfaces with low slope, Math. Ann. 276 (1987), 449-466. MR 88a:14046

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 57R20, 57N13, 20F05, 14D06

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

DOI: 10.1090/S0002-9947-04-03643-8
PII: S 0002-9947(04)03643-8
Keywords: Signature, mapping class group, Lefschetz fibration, relation, signature cocycle, slope
Received by editor(s): November 16, 2003
Posted: September 2, 2004
Dedicated: Dedicated to Professor Yukio Matsumoto for his 60th birthday
Copyright of article: Copyright 2004, American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6850(e) ISSN 0002-9947(p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2008, American Mathematical Society Privacy Statement		Search the AMS