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

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 $\mathbb{Z}$–graded symplectic Floer cohomology is an integral lifting of the usual ${\mathbb{Z}}_{\Sigma \left(L\right)}$–graded Floer–Oh cohomology. We prove the Künneth formula for the spectral sequence and an ring structure on it. The ring structure on the ${\mathbb{Z}}_{\Sigma \left(L\right)}$–graded Floer cohomology is induced from the ring structure of the cohomology of the Lagrangian sub–manifold via the spectral sequence. Using the $\mathbb{Z}$–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 $L$.