Geometry & Topology 13 (2009) 943–978

DOI: 10.2140/gt.2009.13.943

Abstract

Let N be a closed manifold satisfying a mild homotopy assumption. Then for any exact Lagrangian L ⊂ T^*N the map π₂(L) → π₂(N) has finite index. The homotopy assumption is either that N is simply connected, or more generally that π_m(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;L_α), and we show that Viterbo functoriality holds. We prove that the symplectic cohomology SH^*(T^*N;L_α) 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.

Keywords

symplectic homology, Novikov homology, exact Lagrangian

Mathematical Subject Classification

Primary: 57R17

Secondary: 57R58

References

Publication

Received: 13 November 2007
Revised: 24 December 2008
Accepted: 21 December 2008
Published: 8 January 2009
Proposed: Simon Donaldson
Seconded: Jim Bryan, Ron Stern

Authors