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Parametrized ring-spectra
and the nearby Lagrangian conjecture
Thomas KraghAppendix: Mohammed Abouzaid |
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Geometry & Topology 17 (2013) 639–731
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Abstract |
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Let be an embedded closed connected exact Lagrangian submanifold in a connected cotangent bundle . In this paper we prove that such an embedding is, up to a finite covering space lift of , a homology equivalence. We prove this by constructing a fibrant parametrized family of ring spectra parametrized by the manifold . The homology of will be the (twisted) symplectic cohomology of . The fibrancy property will imply that there is a Serre spectral sequence converging to the homology of . The fiber-wise ring structure combined with the intersection product on induces a product on this spectral sequence. This product structure and its relation to the intersection product on is then used to obtain the result. Combining this result with work of Abouzaid we arrive at the conclusion that is always a homotopy equivalence. |
Keywords
exact Lagrangian, cotangent bundle, parametrized spectrum,
nearby Lagrangian conjecture, Maslov index, Floer homotopy
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Mathematical Subject Classification 2010
Primary: 53D12
Secondary: 55R70, 55T10
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References |
Publication
Received: 6 September 2011
Revised: 19 June 2012
Accepted: 30 July 2012
Published: 18 April 2013
Proposed: Yasha Eliashberg
Seconded: Leonid Polterovich, Ralph Cohen
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Authors |
| Thomas Kragh | |
| Department of Mathematics Uppsala University PO Box 480 SE-751 06 Uppsala Sweden |
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| Mohammed Abouzaid | |
| Department of Mathematics Columbia University 2990 Broadway New York, NY 10027 USA and Simons Center for Geometry and Physics State University of New York Stony Brook, NY 11794-3636 USA |
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