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

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