Homotopy properties of Hamiltonian group actions

Details Homotopy properties of Hamiltonian group actions Homotopy properties of Hamiltonian group actions

2004-04-29

arxiv.org/abs/math/0404539v3

Geom. Topol. 9(2005) 121-162

Mathematics

Symplectic Geometry

Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol9/paper3.abs.html

Scientific paper

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.

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