Quantum characteristic classes and the Hofer metric

Savelyev, Yasha¹

¹ Stony Brook University, Department of Mathematics, Stony Brook, NY 11790, USA

Geometry & topology, Tome 12 (2008) no. 4, pp. 2277-2326.

Résumé

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 $H_{\ast }\left(\Omega Ham\left(M,\omega \right),\mathbb{Q}\right)$, 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.

Détail

DOI : 10.2140/gt.2008.12.2277

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

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