Geometry & Topology

Quantum characteristic classes and the Hofer metric

Yasha Savelyev

Full-text: Open access

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.

Article information

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

Dates
Received: 9 February 2008
Revised: 18 July 2008
Accepted: 5 June 2008
First available in Project Euclid: 20 December 2017

Permanent link to this document
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

Subjects
Primary: 53D45: Gromov-Witten invariants, quantum cohomology, Frobenius manifolds [See also 14N35]
Secondary: 53D35: Global theory of symplectic and contact manifolds [See also 57Rxx] 22E67: Loop groups and related constructions, group-theoretic treatment [See also 58D05]

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

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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