Journal of the Mathematical Society of Japan

Geometric decompositions of 4-dimensional orbifold bundles

Jonathan A. HILLMAN

Full-text: Open access

Abstract

We consider geometric decompositions of aspherical 4-manifolds which fibre over 2-orbifolds. We show that no such manifold admits infinitely many fibrations over hyperbolic base orbifolds and that “most” Seifert fibred 4-manifolds over hyperbolic bases have a decomposition induced from a decomposition of the base.

Article information

Source
J. Math. Soc. Japan, Volume 63, Number 3 (2011), 871-886.

Dates
First available in Project Euclid: 1 August 2011

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1312203804

Digital Object Identifier
doi:10.2969/jmsj/06330871

Mathematical Reviews number (MathSciNet)
MR2836748

Zentralblatt MATH identifier
1229.57023

Subjects
Primary: 57N13: Topology of $E^4$ , $4$-manifolds [See also 14Jxx, 32Jxx]

Keywords
cusp decomposition geometry horizontal 4-manifold orbifold vertical

Citation

HILLMAN, Jonathan A. Geometric decompositions of 4-dimensional orbifold bundles. J. Math. Soc. Japan 63 (2011), no. 3, 871--886. doi:10.2969/jmsj/06330871. https://projecteuclid.org/euclid.jmsj/1312203804


Export citation

References

  • W. Barth, C. Peters and A. Van der Ven, Compact Complex Surfaces, Ergebnisse der Math. 3 Folge Bd 4, Springer-Verlag, 1984.
  • B. H. Bowditch, Atoroidal surface bundles over surfaces, Geom. Funct. Anal., 19 (2009), 943–988.
  • J. O. Button, Mapping tori with first Betti number at least two, J. Math. Soc. Japan, 59 (2007), 351–370.
  • J. A. Hillman, Four-Manifolds, Geometries and Knots, Geometry and Topology Monographs 5, Geometry and Topology Publications, 2002, (Revision 2007).
  • J. A. Hillman, On 4-manifolds which admit geometric decompositions, J. Math. Soc. Japan, 50 (1998), 415–431.
  • J. A. Hillman and S. K. Roushon, Surgery on$\widetilde{\mathbf{SL}}\times\mathbf{E}^n$-manifolds, Canad. Math. Bull., 54 (2011), 283–287.
  • F. E. A. Johnson, A group-theoretic analogue of the Parshin-Arakelov Theorem, Archiv. Math., Basel, 63 (1994), 354–361.
  • F. E. A. Johnson, A rigidity theorem for group extensions, Archiv. Math., Basel, 73 (1999), 81–89.
  • M. Kemp, Geometric Seifert 4-manifolds with hyperbolic bases, J. Math. Soc. Japan, 60 (2008), 17–49.
  • C. J. Leininger and A. W. Reid, A combination theorem for Veech subgroups of the mapping class group, Geom. Funct. Anal., 16 (2006), 403–436.
  • M. Ue, Geometric 4-manifolds in the sense of Thurston and Seifert 4-manifolds I, J. Math. Soc. Japan, 42 (1990), 511–540.
  • E. Vogt, Foliations of codimension 2 with all leaves compact on closed 3-, 4- and 5-manifolds, Math. Z., 157 (1977), 201–223.
  • C. T. C. Wall, Geometric structures on compact complex analytic surfaces, Topology, 25 (1986), 119–153.