Novikov-symplectic cohomology and exact Lagrangian embeddings

TL;DR

This paper establishes a relationship between the second homotopy groups of an exact Lagrangian submanifold and its ambient manifold under mild conditions, using Novikov homology and symplectic cohomology techniques.

Contribution

It introduces a Novikov homology framework for symplectic cohomology and generalizes Viterbo's transfer map to relate loop space homologies.

Findings

The image of (L) in (N) has finite index under certain homotopy conditions.

No assumptions on Maslov class or orientability of L are needed.

The results apply to manifolds with finitely generated higher homotopy groups.

Abstract

Let L be an exact Lagrangian submanifold inside the cotangent bundle of a closed manifold N. We prove that if N satisfies a mild homotopy assumption then the image of \pi_2(L) in \pi_2(N) has finite index. We make no assumption on the Maslov class of L, and we make no orientability assumptions. The homotopy assumption is either that N is simply connected, or more generally that \pi_m(N) is finitely generated for each m \geq 2. The result is proved by constructing the Novikov homology theory for symplectic cohomology and generalizing Viterbo's construction of a transfer map between the homologies of the free loopspaces of N and L.