Homotopy properties of Hamiltonian group actions

Abstract

Consider a Hamiltonian action of a compact Lie group H on a compact symplectic manifold (M,w) 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 BH --> BG are injective. For example, we extend Reznikov's result for complex projective space CP^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 lambda is a Hamiltonian circle action that contracts in G = Ham(M,w) 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,a), ie, 2n-manifolds with a class a in H^2(M) such that a^n is nonzero. The proofs are based on calculations of certain characteristic classes and elementary homotopy theory.Comment: Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol9/paper3.abs.htm