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.

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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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