Parametrized Ring-Spectra and the Nearby Lagrangian Conjecture

By Thomas J. Kragh. 2011

69 citations recorded by the source corpus.

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 FL parametrized by the manifold N . The homology of FL 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 FL. 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. 53D12; 55R70, 55T10

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/89512dfe642c5b26791f0eb5a83de711cd4f007c

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

1. D. Auroux
2. P. Seidel
3. I. Smith
4. L. Katzarkov
5. Nick Sheridan
6. P. Biran
7. O. Cornea
8. M. Abouzaid

In Mathematical Physics

1. D. Auroux
2. P. Seidel
3. I. Smith
4. L. Katzarkov
5. Nick Sheridan
6. O. Cornea
7. P. Biran
8. Alexander F. Ritter

Citations in the corpus

• A Beginner’s Introduction to Fukaya Categories (2013, background): “ standard Liouville form) is Hamiltonian isotopic to the zero section. A BEGINNER’S INTRODUCTION TO FUKAYA CATEGORIES 37 Theorem 4.6 (Fukaya-Seidel-Smith [21], Nadler-Zaslow [31], Abouzaid [7], Kragh [26]). Let Lbe a compact connected exact Lagrangian submanifold of T∗N. Then as an object of W(T∗N),”