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The $\mathbb{Z}$–graded symplectic Floer cohomology of monotone Lagrangian sub-manifolds

Weiping Li

Algebraic & Geometric Topology 4 (2004) 647–684

arXiv: math.GT/0409332


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 –graded symplectic Floer cohomology is an integral lifting of the usual Σ(L)–graded Floer–Oh cohomology. We prove the Künneth formula for the spectral sequence and an ring structure on it. The ring structure on the Σ(L)–graded Floer cohomology is induced from the ring structure of the cohomology of the Lagrangian sub–manifold via the spectral sequence. Using the –graded symplectic Floer cohomology, we show some intertwining relations among the Hofer energy eH(L) of the embedded Lagrangian, the minimal symplectic action σ(L), the minimal Maslov index Σ(L) and the smallest integer k(L,ϕ) of the converging spectral sequence of the Lagrangian L.

monotone Lagrangian submanifold, Maslov index, Floer cohomology, spectral sequence
Mathematical Subject Classification 2000
Primary: 53D40
Secondary: 53D12, 70H05
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Received: 3 December 2002
Accepted: 9 August 2004
Published: 3 September 2004
Weiping Li
Department of Mathematics
Oklahoma State University
Stillwater OK 74078-0613