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Symplectomorphism groups and isotropic skeletons
Joseph Coffey
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Geometry & Topology 9 (2005)
935–970
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Abstract
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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.
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Keywords
Lagrangian, symplectomorphism,
homotopy
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Mathematical Subject Classification
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
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