Geometry & Topology

The Seiberg–Witten equations and the Weinstein conjecture II: More closed integral curves of the Reeb vector field

Clifford Henry Taubes

Full-text: Open access

Abstract

Let M denote a compact, orientable 3–dimensional manifold and let a denote a contact 1–form on M; thus a da is nowhere zero. This article explains how the Seiberg–Witten Floer homology groups as defined for any given Spin structure on M give closed, integral curves of the vector field that generates the kernel of da.

Article information

Source
Geom. Topol., Volume 13, Number 3 (2009), 1337-1417.

Dates
Received: 22 April 2007
Revised: 24 November 2008
Accepted: 1 December 2008
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513800249

Digital Object Identifier
doi:10.2140/gt.2009.13.1337

Mathematical Reviews number (MathSciNet)
MR2496048

Zentralblatt MATH identifier
1200.57018

Subjects
Primary: 57R17: Symplectic and contact topology 57R57: Applications of global analysis to structures on manifolds, Donaldson and Seiberg-Witten invariants [See also 58-XX]

Citation

Taubes, Clifford Henry. The Seiberg–Witten equations and the Weinstein conjecture II: More closed integral curves of the Reeb vector field. Geom. Topol. 13 (2009), no. 3, 1337--1417. doi:10.2140/gt.2009.13.1337. https://projecteuclid.org/euclid.gt/1513800249


Export citation

References

  • C Abbas, K Cieliebak, H Hofer, The Weinstein conjecture for planar contact structures in dimension three, Comment. Math. Helv. 80 (2005) 771–793
  • S K Donaldson, Floer homology groups in Yang–Mills theory, Cambridge Tracts in Mathematics 147, Cambridge University Press, Cambridge (2002) With the assistance of M Furuta and D Kotschick
  • Y Eliashberg, Contact $3$–manifolds twenty years since J Martinet's work, Ann. Inst. Fourier $($Grenoble$)$ 42 (1992) 165–192
  • H Hofer, Pseudoholomorphic curves in symplectizations with applications to the Weinstein conjecture in dimension three, Invent. Math. 114 (1993) 515–563
  • T Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer New York, New York (1966)
  • P B Kronheimer, T S Mrowka, Monopoles and contact structures, Invent. Math. 130 (1997) 209–255
  • P B Kronheimer, T S Mrowka, Monopoles and three-manifolds, New Mathematical Monographs 10, Cambridge University Press, Cambridge (2007)
  • G Meng, C H Taubes, $\underline{\rm SW}=$ Milnor torsion, Math. Res. Lett. 3 (1996) 661–674
  • T Mrowka, Y Rollin, Legendrian knots and monopoles, Algebr. Geom. Topol. 6 (2006) 1–69
  • C H Taubes, $SW \Rightarrow Gr$: From Seiberg–Witten equations to pseudo-holomorphic curves in Seiberg Witten and Gromov invariants for symplectic 4–manifolds, International Press, Somerville MA (2005)
  • C H Taubes, Asymptotic spectral flow for Dirac operators, Comm. Anal. Geom. 15 (2007) 569–587
  • C H Taubes, The Seiberg-Witten equations and the Weinstein conjecture, Geom. Topol. 11 (2007) 2117–2202