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Parametrized ring-spectra and the nearby Lagrangian conjecture
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.ORCID iD: 0000-0003-2618-5712
2013 (English)In: Geometry and Topology, ISSN 1465-3060, E-ISSN 1364-0380, Vol. 17, no 2, 639-731 p.Article in journal (Refereed) Published
Abstract [en]

Let L be an embedded closed connected exact Lagrangian sub-manifold 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 (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 -> N is always a homotopy equivalence.

Place, publisher, year, edition, pages
2013. Vol. 17, no 2, 639-731 p.
National Category
Natural Sciences
URN: urn:nbn:se:uu:diva-205411DOI: 10.2140/gt.2013.17.639ISI: 000321304500002OAI: oai:DiVA.org:uu-205411DiVA: diva2:642124
Available from: 2013-08-20 Created: 2013-08-16 Last updated: 2016-04-29

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