A Note on Local Floer Homology

A. In general, Lagrangian Floer homology HF∗(L, φH(L)) – if well-defined – is not isomorphic to the singular homology of the Lagrangian submanifold L. For arbitrary closed Lagrangian submanifolds a local version of Floer homology HF ∗ (L, φH(L)) is defined in [Flo89, Oh96] which is isomorphic to singular homology. This construction assumes that the Hamiltonian function H is sufficiently C2-small and the almost complex structure involved is sufficiently standard. In this note we develop a new construction of local Floer homology which works for any (compatible) almost complex structure and all Hamiltonian function with Hofer norm less than the minimal (symplectic) area of a holomorphic disk or sphere. The example S 1 ⊂ C shows that this is sharp. If the Lagrangian submanifold is monotone, the grading of local Floer homology can be improved to a Z-grading.