A Lagrangian Piunikhin-salamon-schwarz Morphism

In this article we explore to what extend the techniques of Piunikhin, Salamon and Schwarz in [PSS96] can be carried over to Lagrangian Floer homology. In [PSS96] the authors establish an isomorphism between Hamiltonian Floer homology and singular homology of the underlying symplectic manifold. In general, Lagrangian Floer homology is not isomorphic to the singular homology of the Lagrangian submanifold. Depending on the minimal Maslov number of the Lagrangian submanifold we construct two homomorphisms between Lagrangian Floer homology and singular homology in certain degrees. In degrees where both maps are defined we prove them to be inverse to each other. Examples show that this statement is sharp. We derive various applications.