The \(\mathbb Z\)-graded symplectic Floer cohomology of monotone Lagrangian submanifolds. (Q1880647)

The author defines an integer graded symplectic Floer cohomology and a Fintushel-Stern type spectral sequence which are new invariants for monotone Lagrangian submanifolds and exact isotopes. The \(\mathbb Z\)-graded symplectic Floer cohomology is an integral lifting of the usual \({\mathbb Z}_{\Sigma(L)}\)-graded Floer-Oh cohomology. The Künneth formula for the spectral sequence and a ring structure on it is proven. The ring structure on the \({\mathbb Z}_{\Sigma(L)}\)-graded Floer cohomology is induced from the ring structure of the cohomology of the Lagrangian submanifold via the spectral sequence. Using the \(\mathbb Z\)-graded symplectic Floer cohomology, 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 \(L\) are investigated.