Parametrized ring-spectra and the nearby Lagrangian conjecture

Abstract

Let $L$ be an embedded closed connected exact Lagrangian submanifold in a connected cotangent bundle $T^{\ast }N$. In this paper we prove that such an embedding is, up to a finite covering space lift of $T^{\ast }N$, a homology equivalence. We prove this by constructing a fibrant parametrized family of ring spectra $\mathcal{ℱ}\mathcal{ℒ}$ parametrized by the manifold $N$. The homology of $\mathcal{ℱ}\mathcal{ℒ}$ will be the (twisted) symplectic cohomology of $T^{\ast }L$. The fibrancy property will imply that there is a Serre spectral sequence converging to the homology of $\mathcal{ℱ}\mathcal{ℒ}$. The fiber-wise ring structure combined with the 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 $L\to N$ is always a homotopy equivalence.