#### 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

and

Department of Mathematics, Stony Brook University

Stony Brook, NY 11794-3651, USA

Email: kedra@univ.szczecin.pl, dusa@math.sunysb.edu

URL: http://www.univ.szczecin.pl/~kedra and http://www.math.sunysb.edu/~dusa

GT home page

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