Geometry & Topology

Virtual Betti numbers of genus 2 bundles

Joseph D Masters

Full-text: Open access

Abstract

We show that if M is a surface bundle over S1 with fiber of genus 2, then for any integer n, M has a finite cover M˜ with b1(M˜)>n. A corollary is that M can be geometrized using only the “non-fiber" case of Thurston’s Geometrization Theorem for Haken manifolds.

Article information

Source
Geom. Topol., Volume 6, Number 2 (2002), 541-562.

Dates
Received: 15 January 2002
Revised: 9 August 2002
Accepted: 19 November 2002
First available in Project Euclid: 21 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513882934

Digital Object Identifier
doi:10.2140/gt.2002.6.541

Mathematical Reviews number (MathSciNet)
MR1941723

Zentralblatt MATH identifier
1009.57023

Subjects
Primary: 57M10: Covering spaces
Secondary: 57R10: Smoothing

Keywords
3–manifold geometrization virtual Betti number genus 2 surface bundle

Citation

Masters, Joseph D. Virtual Betti numbers of genus 2 bundles. Geom. Topol. 6 (2002), no. 2, 541--562. doi:10.2140/gt.2002.6.541. https://projecteuclid.org/euclid.gt/1513882934


Export citation

References

  • M Baker, Covers of Dehn fillings on once-punctured torus bundles, Proc. Amer. Math. Soc. 105 (1989) 747–754
  • M Baker, Covers of Dehn fillings on once-punctured torus bundles II, Proc. Amer. Math. Soc. 110 (1990) 1099–1108
  • D Gabai, On 3–manifolds finitely covered by surface bundles, from: “Low-dimensional Topology and Kleinian Groups (Coventry/Durham, 1984)”, LMS Lecture Note Series 112, Cambridge University Press (1986)
  • S P Humphries, Generators for the mapping class group, from: “Topology of Low-Dimensional Manifolds”, Proceedings of the Second Sussex Conference, 1977, Lecture Notes in Mathematics 722, Springer–Verlag, Berlin (1979)
  • T Kanenobu, The augmentation subgroup of a pretzel link, Mathematics Seminar Notes, Kobe University, 7 (1979) 363–384
  • J D Masters, Virtual homology of surgered torus bundles", to appear in Pacific J. Math.
  • W D Neumann, A W Reid, Arithmetic of hyperbolic 3–manifolds, Topology '90, de Gruyter (1992) 273–309
  • J-P Otal, Thurston's hyperbolization of Haken manifolds, from: “Surveys in Differential Geometry, Vol. III”, (Cambridge MA 1996), Int. Press, Boston MA (1998) 77–194
  • A W Reid, Arithmeticity of knot complements, J. London Math. Soc. 43 (1991) 171–184
  • A W Reid, Isospectrality and commensurability of arithmetic hyperbolic 2– and 3–manifolds, Duke Math. J. 65 (1992), no. 2, 215–228
  • R Riley, Parabolic representations and symmetries of the knot $9_{32}$, from: “Computers and Geometry and Topology”, (M C Tangora, editor), Lecture Notes in Pure and Applied Math. 114, Dekker (1988) 297–313
  • W Thurston, A norm for the homology of 3–manifolds, Mem. Amer. Math. Soc. 59 (1986) 99–130
  • F Waldhausen, On irreducible 3–manifolds which are sufficiently large, Ann. of Math. 87 (1968) 195–203