## Geometry & Topology

### Homotopy properties of Hamiltonian group actions

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

#### Article information

Source
Geom. Topol., Volume 9, Number 1 (2005), 121-162.

Dates
Received: 30 April 2004
Revised: 22 December 2004
Accepted: 27 December 2004
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513799561

Digital Object Identifier
doi:10.2140/gt.2005.9.121

Mathematical Reviews number (MathSciNet)
MR2115670

Zentralblatt MATH identifier
1077.53072

#### Citation

Kędra, Jarek; McDuff, Dusa. Homotopy properties of Hamiltonian group actions. Geom. Topol. 9 (2005), no. 1, 121--162. doi:10.2140/gt.2005.9.121. https://projecteuclid.org/euclid.gt/1513799561

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