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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Quantum characteristic classes and the Hofer metric

Yasha Savelyev

Geometry & Topology 12 (2008) 2277–2326
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), ), with its Pontryagin product to QH2n+(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 2000
Primary: 53D45
Secondary: 53D35, 22E67
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
Yasha Savelyev
Stony Brook University
Department of Mathematics
Stony Brook
NY 11790
USA
http://www.math.sunysb.edu/~yasha