#### Volume 4, issue 2 (2004)

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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
##### 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 ${ℤ}_{\Sigma \left(L\right)}$–graded Floer–Oh cohomology. We prove the Künneth formula for the spectral sequence and an ring structure on it. The ring structure on the ${ℤ}_{\Sigma \left(L\right)}$–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 ${e}_{H}\left(L\right)$ of the embedded Lagrangian, the minimal symplectic action $\sigma \left(L\right)$, the minimal Maslov index $\Sigma \left(L\right)$ and the smallest integer $k\left(L,\varphi \right)$ of the converging spectral sequence of the Lagrangian $L$.

##### Keywords
monotone Lagrangian submanifold, Maslov index, Floer cohomology, spectral sequence
##### Mathematical Subject Classification 2000
Primary: 53D40
Secondary: 53D12, 70H05