### Abstract

Consider a Hamiltonian action of a compact Lie group G on a compact symplectic manifold (M,omega) 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 lambda is a Hamiltonian circle action that contracts in G:=Ham(M,omega) 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,alpha), ie, 2n-manifolds with a class a is an element of H-2(M) such that a(n)not equal0. The proofs are based on calculations of certain characteristic classes and elementary homotopy theory.

Original language | English |
---|---|

Pages (from-to) | 121-162 |

Number of pages | 41 |

Journal | Geometry & Topology |

Volume | 9 |

DOIs | |

Publication status | Published - 2005 |

### Keywords

- Symplectomorphism
- Hamiltonian action
- symplectic characteristic class
- fiber integration

### Cite this

*Geometry & Topology*,

*9*, 121-162. https://doi.org/10.2140/gt.2005.9.121

**Homotopy properties of Hamiltonian group actions.** / Kedra, Jarek; McDuff, Dusa.

Research output: Contribution to journal › Article

*Geometry & Topology*, vol. 9, pp. 121-162. https://doi.org/10.2140/gt.2005.9.121

}

TY - JOUR

T1 - Homotopy properties of Hamiltonian group actions

AU - Kedra, Jarek

AU - McDuff, Dusa

PY - 2005

Y1 - 2005

N2 - Consider a Hamiltonian action of a compact Lie group G on a compact symplectic manifold (M,omega) 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 lambda is a Hamiltonian circle action that contracts in G:=Ham(M,omega) 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,alpha), ie, 2n-manifolds with a class a is an element of H-2(M) such that a(n)not equal0. The proofs are based on calculations of certain characteristic classes and elementary homotopy theory.

AB - Consider a Hamiltonian action of a compact Lie group G on a compact symplectic manifold (M,omega) 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 lambda is a Hamiltonian circle action that contracts in G:=Ham(M,omega) 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,alpha), ie, 2n-manifolds with a class a is an element of H-2(M) such that a(n)not equal0. The proofs are based on calculations of certain characteristic classes and elementary homotopy theory.

KW - Symplectomorphism

KW - Hamiltonian action

KW - symplectic characteristic class

KW - fiber integration

U2 - 10.2140/gt.2005.9.121

DO - 10.2140/gt.2005.9.121

M3 - Article

VL - 9

SP - 121

EP - 162

JO - Geometry & Topology

JF - Geometry & Topology

SN - 1364-0380

ER -