Quantum characteristic classes and the Hofer metric

Savelyev, Yasha

doi:10.2140/gt.2008.12.2277

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Quantum characteristic classes and the Hofer metric

Yasha Savelyev

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 $L Ham (M, ω)$ with values in $Q H_{*} (M)$ , and their $S^{1}$ 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 $H_{*} (Ω Ham (M, ω), ℚ)$ , with its Pontryagin product to $Q H_{2 n + *} (M)$ with its quantum product. As an application we prove an extension to higher dimensional geometry of the loop space $L Ham (M, ω)$ 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

Volume 12, issue 4 (2008)