#### Volume 17, issue 2 (2013)

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Parametrized ring-spectra and the nearby Lagrangian conjecture

### Appendix: Mohammed Abouzaid

Geometry & Topology 17 (2013) 639–731
##### 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 $\mathsc{ℱ}\mathsc{ℒ}$ parametrized by the manifold $N$. The homology of $\mathsc{ℱ}\mathsc{ℒ}$ 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 $\mathsc{ℱ}\mathsc{ℒ}$. 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.

##### Keywords
exact Lagrangian, cotangent bundle, parametrized spectrum, nearby Lagrangian conjecture, Maslov index, Floer homotopy
##### Mathematical Subject Classification 2010
Primary: 53D12
Secondary: 55R70, 55T10