Sections of surface bundles and Lefschetz fibrations
HTML articles powered by AMS MathViewer
- by R. İnanç Baykur, Mustafa Korkmaz and Naoyuki Monden PDF
- Trans. Amer. Math. Soc. 365 (2013), 5999-6016 Request permission
Abstract:
We investigate the possible self-intersection numbers for sections of surface bundles and Lefschetz fibrations over surfaces. When the fiber genus $g$ and the base genus $h$ are positive, we prove that the adjunction bound $2h-2$ is the only universal bound on the self-intersection number of a section of any such genus $g$ bundle and fibration. As a side result, in the mapping class group of a surface with boundary, we calculate the precise value of the commutator lengths of all powers of a Dehn twist about a boundary component, concluding that the stable commutator length of such a Dehn twist is $1/2$. We furthermore prove that there is no upper bound on the number of critical points of genus–$g$ Lefschetz fibrations over surfaces with positive genera admitting sections of maximal self-intersection, for $g \geq 2$.References
- Denis Auroux, Mapping class group factorizations and symplectic 4-manifolds: some open problems, Problems on mapping class groups and related topics, Proc. Sympos. Pure Math., vol. 74, Amer. Math. Soc., Providence, RI, 2006, pp. 123–132. MR 2264537, DOI 10.1090/pspum/074/2264537
- Christophe Bavard, Longueur stable des commutateurs, Enseign. Math. (2) 37 (1991), no. 1-2, 109–150 (French). MR 1115747
- Jonathan Bowden, On closed leaves of foliations, multisections and stable commutator lengths, J. Topol. Anal. 3 (2011), no. 4, 491–509. MR 2887673, DOI 10.1142/S1793525311000696
- R. Inanc Baykur and Seiichi Kamada; “Classification of broken Lefschetz fibrations with small fiber genera”, preprint; http://arxiv.org/abs/1010.5814.
- V. Braungardt and D. Kotschick, Clustering of critical points in Lefschetz fibrations and the symplectic Szpiro inequality, Trans. Amer. Math. Soc. 355 (2003), no. 8, 3217–3226. MR 1974683, DOI 10.1090/S0002-9947-03-03290-2
- Danny Calegari, scl, MSJ Memoirs, vol. 20, Mathematical Society of Japan, Tokyo, 2009. MR 2527432, DOI 10.1142/e018
- H. Endo and D. Kotschick, Bounded cohomology and non-uniform perfection of mapping class groups, Invent. Math. 144 (2001), no. 1, 169–175. MR 1821147, DOI 10.1007/s002220100128
- 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
- John L. Harer, Stability of the homology of the mapping class groups of orientable surfaces, Ann. of Math. (2) 121 (1985), no. 2, 215–249. MR 786348, DOI 10.2307/1971172
- Mustafa Korkmaz, Low-dimensional homology groups of mapping class groups: a survey, Turkish J. Math. 26 (2002), no. 1, 101–114. MR 1892804
- Mustafa Korkmaz, Stable commutator length of a Dehn twist, Michigan Math. J. 52 (2004), no. 1, 23–31. MR 2043394, DOI 10.1307/mmj/1080837732
- Mustafa Korkmaz, Problems on homomorphisms of mapping class groups, Problems on mapping class groups and related topics, Proc. Sympos. Pure Math., vol. 74, Amer. Math. Soc., Providence, RI, 2006, pp. 81–89. MR 2264533, DOI 10.1090/pspum/074/2264533
- Mustafa Korkmaz and Burak Ozbagci, Minimal number of singular fibers in a Lefschetz fibration, Proc. Amer. Math. Soc. 129 (2001), no. 5, 1545–1549. MR 1713513, DOI 10.1090/S0002-9939-00-05676-8
- Dieter Kotschick, Stable length in stable groups, Groups of diffeomorphisms, Adv. Stud. Pure Math., vol. 52, Math. Soc. Japan, Tokyo, 2008, pp. 401–413. MR 2509718, DOI 10.2969/aspm/05210401
- D. Kotschick, Quasi-homomorphisms and stable lengths in mapping class groups, Proc. Amer. Math. Soc. 132 (2004), no. 11, 3167–3175. MR 2073290, DOI 10.1090/S0002-9939-04-07508-2
- Ai-Ko Liu, Some new applications of general wall crossing formula, Gompf’s conjecture and its applications, Math. Res. Lett. 3 (1996), no. 5, 569–585. MR 1418572, DOI 10.4310/MRL.1996.v3.n5.a1
- S. Morita, Characteristic classes of surface bundles and bounded cohomology, A fête of topology, Academic Press, Boston, MA, 1988, pp. 233–257. MR 928403
- Olga Plamenevskaya and Jeremy Van Horn-Morris, Planar open books, monodromy factorizations and symplectic fillings, Geom. Topol. 14 (2010), no. 4, 2077–2101. MR 2740642, DOI 10.2140/gt.2010.14.2077
- Yoshihisa Sato, 2-spheres of square $-1$ and the geography of genus-2 Lefschetz fibrations, J. Math. Sci. Univ. Tokyo 15 (2008), no. 4, 461–491 (2009). MR 2546906
- Ivan Smith, Geometric monodromy and the hyperbolic disc, Q. J. Math. 52 (2001), no. 2, 217–228. MR 1838364, DOI 10.1093/qjmath/52.2.217
- 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, Chern numbers of certain Lefschetz fibrations, Proc. Amer. Math. Soc. 128 (2000), no. 6, 1845–1851. MR 1641113, DOI 10.1090/S0002-9939-99-05172-2
- 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
Additional Information
- R. İnanç Baykur
- Affiliation: Max Planck Institut für Mathematik, Bonn, Germany – and – Department of Mathematics, Brandeis University, Waltham, Massachusetts 02453
- MR Author ID: 794751
- Email: baykur@mpim-bonn.mpg.de, baykur@brandeis.edu
- Mustafa Korkmaz
- Affiliation: Department of Mathematics, Middle East Technical University, Ankara, Turkey
- Email: korkmaz@metu.edu.tr
- Naoyuki Monden
- Affiliation: Department of Mathematics, Osaka University, Osaka, Japan
- Email: n-monden@cr.math.sci.osaka-u.ac.jp
- Received by editor(s): October 26, 2011
- Received by editor(s) in revised form: February 10, 2012, March 6, 2012, and March 11, 2012
- Published electronically: August 2, 2013
- © Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 365 (2013), 5999-6016
- MSC (2010): Primary 57R22, 57R17, 20F65
- DOI: https://doi.org/10.1090/S0002-9947-2013-05840-0
- MathSciNet review: 3091273