Geometry & Topology

Symplectomorphism groups and isotropic skeletons

Joseph Coffey

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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

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

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

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Lagrangian symplectomorphism homotopy


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

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