## Algebraic & Geometric Topology

### The $\mathbb{Z}$–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 $ℤΣ(L)$–graded Floer–Oh cohomology. We prove the Künneth formula for the spectral sequence and an ring structure on it. The ring structure on the $ℤΣ(L)$–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 $eH(L)$ 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 $L$.

#### Article information

Source
Algebr. Geom. Topol., Volume 4, Number 2 (2004), 647-684.

Dates
Received: 3 December 2002
Accepted: 9 August 2004
First available in Project Euclid: 21 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513882527

Digital Object Identifier
doi:10.2140/agt.2004.4.647

Mathematical Reviews number (MathSciNet)
MR2100676

#### Citation

Li, Weiping. The $\mathbb{Z}$–graded symplectic Floer cohomology of monotone Lagrangian sub-manifolds. Algebr. Geom. Topol. 4 (2004), no. 2, 647--684. doi:10.2140/agt.2004.4.647. https://projecteuclid.org/euclid.agt/1513882527

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