#### Volume 9, issue 1 (2005)

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Homotopy properties of Hamiltonian group actions

### Jarek Kędra and Dusa McDuff

Geometry & Topology 9 (2005) 121–162
 arXiv: math.SG/0404539
##### Abstract

Consider a Hamiltonian action of a compact Lie group $G$ on a compact symplectic manifold $\left(M,\omega \right)$ and let $\mathsc{G}$ be a subgroup of the diffeomorphism group $DiffM$. We develop techniques to decide when the maps on rational homotopy and rational homology induced by the classifying map $BG\to B\mathsc{G}$ 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 ${H}_{\ast }\left(BSU\left(n+1\right)\right)\to {H}_{\ast }\left(B\mathsc{G}\right)$ is injective, where $\mathsc{G}$ 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 $\mathsc{G}:=Ham\left(M,\omega \right)$ then there is an associated nonzero element in ${\pi }_{3}\left(\mathsc{G}\right)$ that deloops to a nonzero element of ${H}_{4}\left(B\mathsc{G}\right)$. This result (as well as many others) extends to c-symplectic manifolds $\left(M,a\right)$, ie, $2n$–manifolds with a class $a\in {H}^{2}\left(M\right)$ such that ${a}^{n}\ne 0$. 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 2000
Primary: 53C15
Secondary: 53D05, 55R40, 57R17