Geometry & Topology 9 (2005) 935–970

DOI: 10.2140/gt.2005.9.935

arXiv: math.SG/0404496

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.

Keywords

Lagrangian, symplectomorphism, homotopy

Mathematical Subject Classification

Primary: 57R17

Secondary: 53D35

References

Forward citations

Publication

Received: 25 June 2004
Revised: 24 September 2004
Accepted: 18 January 2005
Published: 25 May 2005
Proposed: Leonid Polterovich
Seconded: Yasha Eliashberg, Tomasz Mrowka

Authors