Abstract:We prove that any closed connected exact Lagrangian manifold L in a connected cotangent bundle T * N is up to a finite covering space lift a homology equivalence. We prove this by constructing a fibrant parametrized family of ring spectra FL parametrized by the manifold N . The homology of FL will be (twisted) symplectic cohomology of T * L. The fibrancy property will imply that there is a Serre spectral sequence converging to the homology of FL and the product combined with intersection product on N induces a product on this spectral sequence. This product structure and its relation to the intersection product on L is then used to obtain the result. Combining this result with work of Abouzaid we arrive at the conclusion that L → N is always a homotopy equivalence.

Given a closed exact Lagrangian in the cotangent bundle of a closed smooth manifold, we prove that the projection to the base is a simple homotopy equivalence.

Abstract. We show that the transfer map on Floer homotopy types associated to an exact Lagrangian embedding is an equivalence. This provides an obstruction to representing isotopy classes of Lagrangian immersions by Lagrangian embeddings, which, unlike previous obstructions, is sensitive to information that cannot be detected by Floer cochains. We show this by providing a concrete computation in the case of spheres.

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Let L and N be two smooth manifolds of the same dimension.