Algebraic and Geometric Topology 4 (2004), paper no. 30, pages 647-684.

The Z-graded symplectic Floer cohomology of monotone Lagrangian sub-manifolds

Weiping Li

Abstract. We define an integer graded symplectic Floer cohomology and a Fintushel-Stern type spectral sequence which are new invariants for monotone Lagrangian sub-manifolds and exact isotopes. The Z-graded symplectic Floer cohomology is an integral lifting of the usual Z_Sigma(L)-graded Floer-Oh cohomology. We prove the Kunneth formula for the spectral sequence and an ring structure on it. The ring structure on the Z_Sigma(L)-graded Floer cohomology is induced from the ring structure of the cohomology of the Lagrangian sub-manifold via the spectral sequence. Using the Z-graded symplectic Floer cohomology, we show some intertwining relations among the Hofer energy e_H(L) of the embedded Lagrangian, the minimal symplectic action sigma(L), the minimal Maslov index Sigma(L) and the smallest integer k(L, phi) of the converging spectral sequence of the Lagrangian L.

Keywords. Monotone Lagrangian sub-manifold, Maslov index, Floer cohomology, spectral sequence

AMS subject classification. Primary: 53D40. Secondary: 53D12, 70H05.

DOI: 10.2140/agt.2004.4.647

E-print: arXiv:math.GT/0409332

Submitted: 3 December 2002. Accepted: 9 August 2004. Published: 3 September 2004.

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Weiping Li
Department of Mathematics, Oklahoma State University
Stillwater, Oklahoma 74078-0613

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