Volume 13, issue 2 (2009)

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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
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 TN the map π2(L) π2(N) has finite index. The 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;L¯α), and we show that Viterbo functoriality holds. We prove that the symplectic cohomology SH(TN;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 H1(0N) 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 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