We give a construction of the Floer homology of the pair of vnon-compact Lagrangian submanifolds, which satisfies natural continuity property under the Hamiltonian isotopy that moves the infinity but leaves the intersection set of the pair compact. This construction uses the concept of Lagrangian cobordism and certain singular Lagrangian submanifolds. We apply this construction to conormal bundles (or varieties) in the cotangent bundle, and relate it to a conjecture made by MacPherson on the intersection theory of the characteristic Lagrangian cycles associated to the perverse sheaves constructible to a complex stratification on the complex algebraic manifold.
OH, YONG-GEUM (2001) "Floer homology and its continuity for non-compact Lagrangian submanifolds," Turkish Journal of Mathematics: Vol. 25: No. 1, Article 6. Available at: https://journals.tubitak.gov.tr/math/vol25/iss1/6