Geometry & Topology

Homotopy properties of Hamiltonian group actions

Jarek Kędra and Dusa McDuff

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Consider a Hamiltonian action of a compact Lie group G on a compact symplectic manifold (M,ω) and let G be a subgroup of the diffeomorphism group DiffM. We develop techniques to decide when the maps on rational homotopy and rational homology induced by the classifying map BGBG are injective. For example, we extend Reznikov’s result for complex projective space n to show that both in this case and the case of generalized flag manifolds the natural map H(BSU(n+1))H(BG) is injective, where G denotes the group of all diffeomorphisms that act trivially on cohomology. We also show that if λ is a Hamiltonian circle action that contracts in G:= Ham(M,ω) then there is an associated nonzero element in π3(G) that deloops to a nonzero element of H4(BG). This result (as well as many others) extends to c-symplectic manifolds (M,a), ie, 2n–manifolds with a class aH2(M) such that an0. The proofs are based on calculations of certain characteristic classes and elementary homotopy theory.

Article information

Geom. Topol., Volume 9, Number 1 (2005), 121-162.

Received: 30 April 2004
Revised: 22 December 2004
Accepted: 27 December 2004
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53C15: General geometric structures on manifolds (almost complex, almost product structures, etc.)
Secondary: 53D05: Symplectic manifolds, general 55R40: Homology of classifying spaces, characteristic classes [See also 57Txx, 57R20] 57R17: Symplectic and contact topology

symplectomorphism Hamiltonian action symplectic characteristic class fiber integration


Kędra, Jarek; McDuff, Dusa. Homotopy properties of Hamiltonian group actions. Geom. Topol. 9 (2005), no. 1, 121--162. doi:10.2140/gt.2005.9.121.

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