#### 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.

Notes on file formats
Weiping Li

Department of Mathematics, Oklahoma State University

Stillwater, Oklahoma 74078-0613

Email: wli@math.okstate.edu

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