Algebraic & Geometric Topology

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

Weiping Li

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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 –graded symplectic Floer cohomology is an integral lifting of the usual Σ(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 Σ(L)–graded Floer cohomology is induced from the ring structure of the cohomology of the Lagrangian sub–manifold via the spectral sequence. Using the –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

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

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

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Primary: 53D40: Floer homology and cohomology, symplectic aspects
Secondary: 53D12: Lagrangian submanifolds; Maslov index 70H05: Hamilton's equations

monotone Lagrangian submanifold Maslov index Floer cohomology spectral sequence


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.

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