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Gauged Hamiltonian Floer homology I: Definition of the Floer homology groups


Author: Guangbo Xu
Journal: Trans. Amer. Math. Soc. 368 (2016), 2967-3015
MSC (2010): Primary 53D20, 53D40; Secondary 37J05
DOI: https://doi.org/10.1090/tran/6643
Published electronically: October 20, 2015
MathSciNet review: 3449264
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Abstract: We construct the vortex Floer homology group $ VHF\left ( M, \mu ; H\right )$ for an aspherical Hamiltonian $ G$-manifold $ (M, \omega , \mu )$ and a class of $ G$-invariant Hamiltonian loops $ H_t$, following a proposal of Cieliebak, Gaio, and Salamon (2000). This is a substitute for the ordinary Hamiltonian Floer homology of the symplectic quotient of $ M$. The equation for connecting orbits is a perturbed symplectic vortex equation on the cylinder $ \mathbb{R} \times S^1$. We achieve the transversality of the moduli space by the classical perturbation argument instead of the virtual technique, so the homology can be defined over $ \mathbb{Z}$.


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Additional Information

Guangbo Xu
Affiliation: Department of Mathematics, 410N Rowland Hall, University of California Irvine, Irvine, California 92697
Address at time of publication: Department of Mathematics, Princeton University, Fine Hall, Washington Rd., Princeton, New Jersey 08544
Email: guangbox@math.uci.edu, guangbox@math.princeton.edu

DOI: https://doi.org/10.1090/tran/6643
Received by editor(s): August 28, 2014
Received by editor(s) in revised form: August 31, 2014, September 26, 2014, and December 31, 2014
Published electronically: October 20, 2015
Article copyright: © Copyright 2015 American Mathematical Society