Geometry & Topology

Symplectomorphism groups and isotropic skeletons

Joseph Coffey

Full-text: Open access

Abstract

The symplectomorphism group of a 2–dimensional surface is homotopy equivalent to the orbit of a filling system of curves. We give a generalization of this statement to dimension 4. The filling system of curves is replaced by a decomposition of the symplectic 4–manifold (M,ω) into a disjoint union of an isotropic 2–complex L and a disc bundle over a symplectic surface Σ which is Poincare dual to a multiple of the form ω. We show that then one can recover the homotopy type of the symplectomorphism group of M from the orbit of the pair (L,Σ). This allows us to compute the homotopy type of certain spaces of Lagrangian submanifolds, for example the space of Lagrangian 22 isotopic to the standard one.

Article information

Source
Geom. Topol., Volume 9, Number 2 (2005), 935-970.

Dates
Received: 25 June 2004
Revised: 24 September 2004
Accepted: 18 January 2005
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513799608

Digital Object Identifier
doi:10.2140/gt.2005.9.935

Mathematical Reviews number (MathSciNet)
MR2140995

Zentralblatt MATH identifier
1083.57034

Subjects
Primary: 57R17: Symplectic and contact topology
Secondary: 53D35: Global theory of symplectic and contact manifolds [See also 57Rxx]

Keywords
Lagrangian symplectomorphism homotopy

Citation

Coffey, Joseph. Symplectomorphism groups and isotropic skeletons. Geom. Topol. 9 (2005), no. 2, 935--970. doi:10.2140/gt.2005.9.935. https://projecteuclid.org/euclid.gt/1513799608


Export citation

References

  • Miguel Abreu, Topology of symplectomorphism groups of $S\sp 2\times S\sp 2$, Invent. Math. 131 (1998) 1–23
  • Miguel Abreu, Dusa McDuff, Topology of symplectomorphism groups of rational ruled surfaces, J. Amer. Math. Soc. 13 (2000) 971–1009
  • P Biran, Lagrangian barriers and symplectic embeddings, Geom. Funct. Anal. 11 (2001) 407–464
  • Joseph Coffey, A Symplectic Alexander Trick and Spaces of Symplectic Sections, Ph.D. thesis, State University of New York, Stony Brook, Stony Brook, New York (2003)
  • S K Donaldson, Symplectic submanifolds and almost-complex geometry, J. Differential Geom. 44 (1996) 666–705
  • Y Eliashberg, L Polterovich, Local Lagrangian $2$–knots are trivial, Ann. of Math. (2) 144 (1996) 61–76
  • M Gromov, Pseudoholomorphic curves in symplectic manifolds, Invent. Math. 82 (1985) 307–347
  • Richard Hind, Lagrangian spheres in $S^2 \times S^2$.
  • François Lalonde, Dusa McDuff, The classification of ruled symplectic $4$–manifolds, Math. Res. Lett. 3 (1996) 769–778
  • Eugene Lerman, Symplectic cuts, Math. Res. Lett. 2 (1995) 247–258
  • J Peter May, Simplicial objects in algebraic topology, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL (1992), reprint of the 1967 original
  • Dusa McDuff, The structure of rational and ruled symplectic $4$–manifolds, J. Amer. Math. Soc. 3 (1990) 679–712
  • Dusa McDuff, Symplectomorphism groups and almost complex structures, from: “Essays on geometry and related topics, Vol. 1, 2”, Monogr. Enseign. Math. 38, Enseignement Math., Geneva (2001) 527–556
  • Edwin H Spanier, Algebraic topology, Springer–Verlag, New York (1981)
  • W P Thurston, Some simple examples of symplectic manifolds, Proc. Amer. Math. Soc. 55 (1976) 467–468