Homotopy properties of Hamiltonian group actions

Abstract

Consider a Hamiltonian action of a compact Lie group $G$ on a compact symplectic manifold $\left(M,\omega \right)$ and let $\mathcal{G}$ 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 $BG\to B\mathcal{G}$ are injective. For example, we extend Reznikov’s result for complex projective space ${ℂ\mathbb{P}}^n$ to show that both in this case and the case of generalized flag manifolds the natural map $H_{\ast }\left(BSU\left(n+1\right)\right)\to H_{\ast }\left(B\mathcal{G}\right)$ is injective, where $\mathcal{G}$ 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 $\mathcal{G}:=Ham\left(M,\omega \right)$ then there is an associated nonzero element in ${\pi }_3\left(\mathcal{G}\right)$ that deloops to a nonzero element of $H_4\left(B\mathcal{G}\right)$. 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.