#### Volume 13, issue 2 (2009)

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Novikov-symplectic cohomology and exact Lagrangian embeddings

### Alexander F Ritter

Geometry & Topology 13 (2009) 943–978
##### 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 $\alpha \in {H}^{1}\left({\mathsc{ℒ}}_{0}N\right)$ is the transgression of a nonzero class in ${H}^{2}\left(Ñ\right)$. Combining these results yields the above obstructions to the existence of $L$.

##### Keywords
symplectic homology, Novikov homology, exact Lagrangian
Primary: 57R17
Secondary: 57R58