#### Volume 12, issue 4 (2008)

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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 $LHam\left(M,\omega \right)$ with values in $Q{H}_{\ast }\left(M\right)$, 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 ${H}_{\ast }\left(\Omega Ham\left(M,\omega \right),ℚ\right)$, with its Pontryagin product to $Q{H}_{2n+\ast }\left(M\right)$ with its quantum product. As an application we prove an extension to higher dimensional geometry of the loop space $LHam\left(M,\omega \right)$ 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