Novikov-symplectic cohomology and exact Lagrangian embeddings

Ritter, Alexander F

doi:10.2140/gt.2009.13.943

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Novikov-symplectic cohomology and exact Lagrangian embeddings

Alexander F Ritter

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 $L \subset T^{*} N$ the map $π_{2} (L) \to π_{2} (N)$ 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 $m \geq 2$ . 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 ${S H}^{*} (M; {\underline{L}}_{α})$ , and we show that Viterbo functoriality holds. We prove that the symplectic cohomology ${S H}^{*} (T^{*} N; {\underline{L}}_{α})$ is isomorphic to the Novikov homology of the free loopspace. Given the homotopy assumption on $N$ , we show that this Novikov homology vanishes when $α \in H^{1} (ℒ_{0} N)$ is the transgression of a nonzero class in $H^{2} (Ñ)$ . Combining these results yields the above obstructions to the existence of $L$ .

Keywords

symplectic homology, Novikov homology, exact Lagrangian

Mathematical Subject Classification 2000

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

Alexander F Ritter
Department of Mathematics MIT Cambridge, MA 02139 USA

Volume 13, issue 2 (2009)