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

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)$andthesmallestintegerk(L,phi$ of the converging spectral sequence of the Lagrangian L.

Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-30.abs.html

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