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arXiv · 1107.4674

Parametrized Ring-Spectra and the Nearby Lagrangian Conjecture

Thomas Kragh
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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.

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Thomas Kragh. 2011-07-23. Parametrized Ring-Spectra and the Nearby Lagrangian Conjecture. https://doi.org/10.2140/gt.2013.17.639

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