Novikov-symplectic cohomology and exact Lagrangian embeddings

Abstract

Let $N$ be a closed manifold satisfying a mild homotopy assumption. Then for any exact Lagrangian $L\subset T^{\ast }N$ the map ${\pi }_2\left(L\right)\to {\pi }_2\left(N\right)$ has finite index. The homotopy assumption is either that $N$ is simply connected, or more generally that ${\pi }_m\left(N\right)$ is finitely generated for each $m\ge 2$. The manifolds need not be orientable, and we make no assumption on the Maslov class of $L$.

We construct the Novikov homology theory for symplectic cohomology, denoted ${SH}^{\ast }\left(M;{\underset{¯}{L}}_{\alpha }\right)$, and we show that Viterbo functoriality holds. We prove that the symplectic cohomology ${SH}^{\ast }\left(T^{\ast }N;{\underset{¯}{L}}_{\alpha }\right)$ is isomorphic to the Novikov homology of the free loopspace. Given the homotopy assumption on $N$, we show that this Novikov homology vanishes when $\alpha \in H^1\left({\mathcal{ℒ}}_{0}N\right)$ is the transgression of a nonzero class in $H^2\left(Ñ\right)$. Combining these results yields the above obstructions to the existence of $L$.