Geometry & Topology 9 (2005) 121–162

DOI: 10.2140/gt.2005.9.121

Abstract

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 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 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 a in H²(M) such that aⁿ≠0. The proofs are based on calculations of certain characteristic classes and elementary homotopy theory.

Keywords

symplectomorphism, Hamiltonian action, symplectic characteristic class, fiber integration

Mathematical Subject Classification

Primary: 53C15

Secondary: 53D05, 55R40, 57R17

References

Publication

Received: 30 April 2004
Revised: 22 December 2004
Accepted: 27 December 2004
Published: 28 December 2004
Proposed: Ralph Cohen
Seconded: Leonid Polterovich, Frances Kirwan

Authors