Geometry & Topology 12 (2008) 2277–2326

DOI: 10.2140/gt.2008.12.2277

Abstract

Given a closed monotone symplectic manifold M, we define certain characteristic cohomology classes of the free loop space LHam(M,ω) with values in QH_*(M), and their S¹ equivariant version. These classes generalize the Seidel representation and satisfy versions of the axioms for Chern classes. In particular there is a Whitney sum formula, which gives rise to a graded ring homomorphism from the ring H_*(ΩHam(M,ω), Q), with its Pontryagin product to QH_2n+*(M) with its quantum product. As an application we prove an extension to higher dimensional geometry of the loop space LHam(M,ω) of a theorem of McDuff and Slimowitz on minimality in the Hofer metric of a semifree Hamiltonian circle action.

Keywords

quantum homology, Hamiltonian group, energy flow, loop group, Hamiltonian symplectomorphism, Hofer metric

Mathematical Subject Classification

Primary: 53D45

Secondary: 22E67, 53D35

References

Publication

Received: 9 February 2008
Revised: 18 July 2008
Accepted: 5 June 2008
Published: 2 September 2008
Proposed: Leonid Polterovich
Seconded: Yasha Eliashberg, Simon Donaldson

Authors