Parametrized ring-spectra and the nearby Lagrangian conjecture

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

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

Authors

Thomas Kragh
Department of Mathematics Uppsala University PO Box 480 SE-751 06 Uppsala Sweden

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

Volume 17, issue 2 (2013)

Thomas Kragh

Appendix: Mohammed Abouzaid

Abstract

Keywords

Mathematical Subject Classification 2010

References

Publication

Authors