Volume 9, issue 2 (2005)

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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Symplectomorphism groups and isotropic skeletons

Joseph Coffey

Geometry & Topology 9 (2005) 935–970

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 2 2 isotopic to the standard one.

Keywords
Lagrangian, symplectomorphism, homotopy
Mathematical Subject Classification 2000
Primary: 57R17
Secondary: 53D35
References
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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