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lect07 - SYMPLECTIC GEOMETRY LECTURE 7 Prof Denis Auroux 1...

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SYMPLECTIC GEOMETRY, LECTURE 7 Prof. Denis Auroux 1. Floer homology For a Hamiltonian diffeomorphism f : ( M, ω ) ( M, ω ) , f = φ H 1 , H t : M R 1-periodic in t , we want to look for fixed points of f , i.e. 1-periodic orbits of X H , x ( t ) = X H t ( x ( t )). We consider the Floer complex CF ( f ), whose basis are 1-periodic orbits; these correspond to critical points of the action functional A H on a covering of the free loop space Ω( M ). The differential ’counts’ solutions of Floer’s equations (1) u : R × S 1 M, ∂u + J ( u ( s, t ))( ∂u ( u )) = 0 ∂s ∂t X H t such that lim s →±∞ u ( s, · ) = x ± (1-periodic orbits). The solutions are formal gradient flow lines of A H between the critical points x ± . Theorem 1 (Arnold’s conjecture) . If the fixed points of f are nondegenerate, then #Fix( f ) dim H i ( M ) , i i.e. #Fix( f ) = rk CF rk HF = rk H ( CF , ∂ ) = rk H ( M ) . 1.1. Lagrangian intersections. There is a notion of Lagrangian Floer homology, which is not always defined (in fact, there are explicit obstructions to its existence). The idea is to count intersections of Lagrangian submanifolds L, L M in a manner which is invariant under Hamiltonian deformations (isotopies). Assume that L and L are transverse (if not, e.g. when L = L , replace the submanifold L by the graph L t of an exact 1-form in T L ). To define Floer homology, one defines a complex CF (
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