Parametrized Ring-Spectra and the Nearby Lagrangian Conjecture

TL;DR

This paper proves that any closed connected exact Lagrangian in a cotangent bundle is homotopy equivalent to the base manifold, using parametrized ring spectra and spectral sequences.

Contribution

It introduces a fibrant parametrized family of ring spectra to establish the homotopy equivalence, advancing the understanding of the nearby Lagrangian conjecture.

Findings

Lagrangian manifolds are homotopy equivalent to base manifolds.

Constructs a fibrant parametrized family of ring spectra.

Uses spectral sequences to relate homology and intersection products.

Abstract

We prove that any closed connected exact Lagrangian manifold L in a connected cotangent bundle T*N is up to a finite covering space lift 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 (twisted) symplectic cohomology of T*L. The fibrancy property will imply that there is a Serre spectral sequence converging to the homology of FL and the product combined with 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.