Novikov-symplectic Cohomology and Exact Lagrangian Embeddings

We prove that if N is a closed simply connected manifold and j : L →֒ T ∗N is an exact Lagrangian embedding, then H2(N) → H2(L) is injective and the image of π2(L) → π2(N) has finite index. Viterbo proved that there is a transfer map on free loopspaces H∗(L0N) → H∗(L0L) which commutes under the inclusion of constant loops with the ordinary transfer map H∗(N) → H∗(L). This commutative diagram still holds if one introduces a Novikov bundle of local coefficients induced by the transgression τ(β) ∈ H(L0N) of a non-zero class β ∈ H2(N). By proving the vanishing of the Novikov homology H∗(L0N ; Λτ(β)) we obtain a contradiction to Viterbo functoriality if τ(jβ) ∈ H(L0L) vanished. This will yield the above obstructions to the existence of j.