Geometry & Topology
- Geom. Topol.
- Volume 9, Number 1 (2005), 121-162.
Homotopy properties of Hamiltonian group actions
Consider a Hamiltonian action of a compact Lie group on a compact symplectic manifold and let be a subgroup of the diffeomorphism group . We develop techniques to decide when the maps on rational homotopy and rational homology induced by the classifying map are injective. For example, we extend Reznikov’s result for complex projective space to show that both in this case and the case of generalized flag manifolds the natural map is injective, where denotes the group of all diffeomorphisms that act trivially on cohomology. We also show that if is a Hamiltonian circle action that contracts in then there is an associated nonzero element in that deloops to a nonzero element of . This result (as well as many others) extends to c-symplectic manifolds , ie, –manifolds with a class such that . The proofs are based on calculations of certain characteristic classes and elementary homotopy theory.
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
Permanent link to this document
Digital Object Identifier
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
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. https://projecteuclid.org/euclid.gt/1513799561