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Parametrized ring-spectra and the nearby Lagrangian conjecture

Thomas Kragh

Appendix: Mohammed Abouzaid

Geometry & Topology 17 (2013) 639–731

Let L be an embedded closed connected exact Lagrangian submanifold in a connected cotangent bundle TN. In this paper we prove that such an embedding is, up to a finite covering space lift of TN, a homology equivalence. We prove this by constructing a fibrant parametrized family of ring spectra parametrized by the manifold N. The homology of will be the (twisted) symplectic cohomology of TL. 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 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.

exact Lagrangian, cotangent bundle, parametrized spectrum, nearby Lagrangian conjecture, Maslov index, Floer homotopy
Mathematical Subject Classification 2010
Primary: 53D12
Secondary: 55R70, 55T10
Received: 6 September 2011
Revised: 19 June 2012
Accepted: 30 July 2012
Published: 18 April 2013
Proposed: Yasha Eliashberg
Seconded: Leonid Polterovich, Ralph Cohen
Thomas Kragh
Department of Mathematics
Uppsala University
PO Box 480
SE-751 06 Uppsala
Mohammed Abouzaid
Department of Mathematics
Columbia University
2990 Broadway
New York, NY 10027
Simons Center for Geometry and Physics
State University of New York
Stony Brook, NY 11794-3636