Algebraic & Geometric Topology

On sections of hyperelliptic Lefschetz fibrations

Shunsuke Tanaka

Full-text: Open access

Abstract

We construct a relation among right-handed Dehn twists in the mapping class group of a compact oriented surface of genus g with 4g+4 boundary components. This relation gives an explicit topological description of 4g+4 disjoint (1)–sections of a hyperelliptic Lefschetz fibration of genus g on the manifold 2#(4g+5)¯2.

Article information

Source
Algebr. Geom. Topol., Volume 12, Number 4 (2012), 2259-2286.

Dates
Received: 6 March 2012
Revised: 10 August 2012
Accepted: 20 August 2012
First available in Project Euclid: 19 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513715456

Digital Object Identifier
doi:10.2140/agt.2012.12.2259

Mathematical Reviews number (MathSciNet)
MR3020206

Zentralblatt MATH identifier
1268.57010

Subjects
Primary: 57N13: Topology of $E^4$ , $4$-manifolds [See also 14Jxx, 32Jxx]
Secondary: 20F34: Fundamental groups and their automorphisms [See also 57M05, 57Sxx]

Keywords
4–manifold mapping class group Lefschetz fibration relation section Dehn twist monodromy hyperelliptic rational surface

Citation

Tanaka, Shunsuke. On sections of hyperelliptic Lefschetz fibrations. Algebr. Geom. Topol. 12 (2012), no. 4, 2259--2286. doi:10.2140/agt.2012.12.2259. https://projecteuclid.org/euclid.agt/1513715456


Export citation

References

  • J Amorós, F Bogomolov, L Katzarkov, T Pantev, Symplectic Lefschetz fibrations with arbitrary fundamental groups, J. Differential Geom. 54 (2000) 489–545
  • H Endo, M Korkmaz, D Kotschick, B Ozbagci, A Stipsicz, Commutators, Lefschetz fibrations and the signatures of surface bundles, Topology 41 (2002) 961–977
  • B Farb, D Margalit, A primer on mapping class groups, Princeton Math. Series 49, Princeton Univ. Press (2012)
  • R E Gompf, A I Stipsicz, $4$-manifolds and Kirby calculus, Graduate Studies in Mathematics 20, American Mathematical Society (1999)
  • T Ito, Splitting of singular fibers in certain holomorphic fibrations, J. Math. Sci. Univ. Tokyo 9 (2002) 425–480
  • S Kitagawa, K Konno, Fibred rational surfaces with extremal Mordell–Weil lattices, Math. Z. 251 (2005) 179–204
  • M Korkmaz, B Ozbagci, On sections of elliptic fibrations, Michigan Math. J. 56 (2008) 77–87
  • Y Matsumoto, Lefschetz fibrations of genus two–-a topological approach, from: “Topology and Teichmüller spaces”, (S Kojima, Y Matsumoto, K Saito, M Seppälä, editors), World Sci. Publ., River Edge, NJ (1996) 123–148
  • S Ç Onaran, On sections of genus two Lefschetz fibrations, Pacific J. Math. 248 (2010) 203–216
  • M-H Saitō, K-I Sakakibara, On Mordell–Weil lattices of higher genus fibrations on rational surfaces, J. Math. Kyoto Univ. 34 (1994) 859–871
  • I Smith, Geometric monodromy and the hyperbolic disc, Q. J. Math. 52 (2001) 217–228