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A Lagrangian Piunikhin-Salamon-Schwarz morphism and two comparison homomorphisms in Floer homology

In this article we address two issues. First, we explore to what extent the techniques of Piunikhin, Salamon and Schwarz in [PSS96] can be carried over to Lagrangian Floer homology. In [PSS96] an isomorphism between Hamiltonian Floer homology and the singular homology is established. In contrast, Lagrangian Floer homology is not isomorphic to the singular homology of the Lagrangian submanifold, in general. Depending on the minimal Maslov number, we construct for certain degrees two homomorphisms between Lagrangian Floer homology and singular homology. In degrees where both maps are defined we prove them to be isomorphisms. Examples show that this statement is sharp. Second, we construct two comparison homomorphisms between Lagrangian and Hamiltonian Floer homology. They underly no degree restrictions and are proven to be the natural analogs to the homomorphisms in singular homology induced by the inclusion map of the Lagrangian submanifold into the ambient symplectic manifold.