The ℤ–graded symplectic Floer cohomology of monotone Lagrangian sub-manifolds

Li, Weiping

doi:10.2140/agt.2004.4.647

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

Weiping Li

Algebraic & Geometric Topology 4 (2004) 647–684

DOI: 10.2140/agt.2004.4.647

arXiv: math.GT/0409332

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 $ℤ_{Σ (L)}$ –graded Floer–Oh cohomology. We prove the Künneth formula for the spectral sequence and an ring structure on it. The ring structure on the $ℤ_{Σ (L)}$ –graded Floer cohomology is induced from the ring structure of the cohomology of the Lagrangian sub–manifold via the spectral sequence. Using the $ℤ$ –graded symplectic Floer cohomology, we show some intertwining relations among the Hofer energy $e_{H} (L)$ of the embedded Lagrangian, the minimal symplectic action $σ (L)$ , the minimal Maslov index $Σ (L)$ and the smallest integer $k (L, ϕ)$ of the converging spectral sequence of the Lagrangian $L$ .

Keywords

monotone Lagrangian submanifold, Maslov index, Floer cohomology, spectral sequence

Mathematical Subject Classification 2000

Primary: 53D40

Secondary: 53D12, 70H05

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Publication

Received: 3 December 2002

Accepted: 9 August 2004

Published: 3 September 2004

Authors

Weiping Li
Department of Mathematics Oklahoma State University Stillwater OK 74078-0613 USA

Volume 4, issue 2 (2004)