#### Geometry & Topology, Vol. 9 (2005)
Paper no. 21, pages 935--970.

## 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,
omega) into a disjoint union of an isotropic 2-complex L and a disc
bundle over a symplectic surface Sigma which is Poincare dual to a
multiple of the form omega. We show that then one can recover the
homotopy type of the symplectomorphism group of M from the orbit of
the pair (L, Sigma). This allows us to compute the homotopy type of
certain spaces of Lagrangian submanifolds, for example the space of
Lagrangian RP^2 in CP^2 isotopic to the standard one.
**Keywords**.
Lagrangian, symplectomorphism, homotopy

**AMS subject classification**.
Primary: 57R17.
Secondary: 53D35.

**E-print:** `arXiv:math.SG/0404496`

**E-print:** `arXiv:math.SG/0404496`

**DOI:** 10.2140/gt.2005.9.935

Submitted to GT on 25 June 2004.
(Revised 24 September 2004.)
Paper accepted 18 January 2005.
Paper published 25 May 2005.

Notes on file formats
Joseph Coffey

Courant Institute for the Mathematical Sciences, New York University

251 Mercer Street, New York, NY 10012, USA

Email: coffey@cims.nyu.edu

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