## Geometry & Topology

### Symplectomorphism groups and isotropic skeletons

Joseph Coffey

#### 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 $ℝℙ2⊂ℂℙ2$ isotopic to the standard one.

#### Article information

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

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

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

#### 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

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