Consider a Hamiltonian action of a compact Lie group G on a compact symplectic manifold (M,omega) and let G be a subgroup of the diffeomorphism group Diff M. We develop techniques to decide when the maps on rational homotopy and rational homology induced by the classifying map BG-->BG are injective. For example, we extend Reznikov's result for complex projective space CPn 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 lambda is a Hamiltonian circle action that contracts in G:=Ham(M,omega) then there is an associated nonzero element in pi(3)(G) that deloops to a nonzero element of H-4(BG). This result (as well as many others) extends to c-symplectic manifolds (M,alpha), ie, 2n-manifolds with a class a is an element of H-2(M) such that a(n)not equal0. The proofs are based on calculations of certain characteristic classes and elementary homotopy theory.
|Number of pages||41|
|Journal||Geometry & Topology|
|Publication status||Published - 2005|
- Hamiltonian action
- symplectic characteristic class
- fiber integration