## Geometry & Topology

### Quantum characteristic classes and the Hofer metric

Yasha Savelyev

#### 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 $QH2n+∗(M)$ 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.

#### Article information

Source
Geom. Topol., Volume 12, Number 4 (2008), 2277-2326.

Dates
Revised: 18 July 2008
Accepted: 5 June 2008
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.gt/1513800127

Digital Object Identifier
doi:10.2140/gt.2008.12.2277

Mathematical Reviews number (MathSciNet)
MR2443967

Zentralblatt MATH identifier
1146.53070

#### Citation

Savelyev, Yasha. Quantum characteristic classes and the Hofer metric. Geom. Topol. 12 (2008), no. 4, 2277--2326. doi:10.2140/gt.2008.12.2277. https://projecteuclid.org/euclid.gt/1513800127

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