Geometry & Topology, Vol. 9 (2005) Paper no. 3, pages 121--162.

Homotopy properties of Hamiltonian group actions

Jarek Kedra and Dusa McDuff

Abstract. 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.

Keywords. Symplectomorphism, Hamiltonian action, symplectic characteristic class, fiber integration

AMS subject classification. Primary: 53C15. Secondary: 53D05, 55R40, 57R17.

DOI: 10.2140/gt.2005.9.121

E-print: arXiv:math.SG/0404539

Submitted to GT on 30 April 2004. (Revised 22 December 2004.) Paper accepted 27 December 2004. Paper published 28 December 2004.

Notes on file formats

Jarek Kedra Dusa McDuff
Institute of Mathematics US, Wielkopolska 15, 70-451 Szczecin, Poland
and Mathematical Institute Polish Academy of Sciences
Sniadeckich 8, 00-950 Warszawa, Poland
Department of Mathematics, Stony Brook University
Stony Brook, NY 11794-3651, USA

URL: and

GT home page

EMIS/ELibM Electronic Journals

Outdated Archival Version

These pages are not updated anymore. They reflect the state of 21 Apr 2006. For the current production of this journal, please refer to