Symplectomorphism groups and isotropic skeletons

Coffey, Joseph

doi:10.2140/gt.2005.9.935

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Symplectomorphism groups and isotropic skeletons

Joseph Coffey

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 ${ℝ ℙ}^{2} \subset {ℂ ℙ}^{2}$ isotopic to the standard one.

Keywords

Lagrangian, symplectomorphism, homotopy

Mathematical Subject Classification 2000

Primary: 57R17

Secondary: 53D35

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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

Joseph Coffey
Courant Institute for the Mathematical Sciences New York University 251 Mercer Street New York New York 10012 USA

Volume 9, issue 2 (2005)