Floer homology for magnetic fields with at most linear growth on the universal cover

Details Floer homology for magnetic fields with at most linear growth on the universal cover Floer homology for magnetic fields with at most linear growth on the universal cover

2011-08-15

arxiv.org/abs/1108.3044v2

Mathematics

Symplectic Geometry

24 pages, V2 - minor corrections, final version to appear in JFA

Scientific paper

The Floer homology of a cotangent bundle is isomorphic to loop space homology of the underlying manifold, as proved by Abbondandolo-Schwarz, Salamon-Weber, and Viterbo. In this paper we show that in the presence of a Dirac magnetic monopole which admits a primitive with sublinear growth on the universal cover, the Floer homology in atoroidal free homotopy classes is again isomorphic to loop space homology. As a consequence we prove that for any atoroidal free homotopy class and any sufficiently small T>0, any magnetic flow associated to the Dirac magnetic monopole has a closed orbit of period T belonging to the given free homotopy class. In the case where the Dirac magnetic monopole admits a bounded primitive on the universal cover we also prove the Conley conjecture for Hamiltonians that are quadratic at infinity, i.e., we show that such Hamiltonians have infinitely many periodic orbits.

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