Volume 9, issue 1 (2005)

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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
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 (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 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 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 an0. 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
References
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Publication
Received: 30 April 2004
Revised: 22 December 2004
Accepted: 27 December 2004
Published: 28 December 2004
Proposed: Ralph Cohen
Seconded: Leonid Polterovich, Frances Kirwan
Authors
Jarek Kędra
Institute of Mathematics US
Wielkopolska 15
70-451 Szczecin
Poland
http://www.univ.szczecin.pl/~kedra/
Dusa McDuff
Department of Mathematics
Stony Brook University
Stony Brook
New York 11794-3651
USA
http://www.math.sunysb.edu/~dusa/