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 T*N t h e m a p π2(L) π2(N) has finite index. T he homotopy assumption is either that N is simply connected, or more generally that πm(N) is finitely generated for each m ≥ 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* (M;Lα), and we show that Viterbo functoriality holds. We prove that the symplectic cohomology SH* (T*N;Lα) is isomorphic to the Novikov homology of the free loopspace. Given the homotopy assumption on N, we show that this Novikov homology vanishes when α in H1(L0N) is the transgression of a nonzero class in H2(Ñ). Combining these results yields the above obstructions to the existence of L. © Copyright 2009 Mathematical Sciences Publishers. All rights reserved.