1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

LECTURE 1

Symplectic Fixed Points and Morse Theory

1.1. The Arnold conjecture

Let ( M, w) be a compact symplectic manifold. The form w determines an iso-

morphism Iw : T M -t TM and the image of an exact 1-form dH: M -t T M

under this isomorphism is called the Hamiltonian vector field generated by the

Hamiltonian function H: M -t JR. It is denoted by XH: M -t TM and is given

by l(XH )w = dH. Let Ht =Ht+ I : M -t IR be a smooth time dependent 1-periodic

family of Hamiltonian functions and consider the Hamiltonian differential equation

(1) ±(t) = Xt(x(t)),

where Xt = XH, for t E JR. The solutions of (1) generate a family of symplecto-

morphisms 'l/Jt : M -t M via

d

dt 'l/Jt = Xt o 'l/Jt, 'I/Jo =id.

The fixed points of the time-1-map 'l/J = 'lj; 1 are in one-to-one correspondence with

the 1-periodic solutions of (1) and we denote the set of such solutions by

P(H) = {x: IR/Z -t M : (1)}.

A periodic solution x is called nondegenerate if all its Floquet multipliers are not
equal to 1, or equivalently,

(2) det(ll - d'lj; 1 (x(O))) =/= 0.

The Arnold conjecture asserts that, in the nondegenerate case, the number of 1-
periodic solutions should be bounded below by the sum of the Betti numbers of
M.

Conjecture 1.1 (Arnold Conjecture). Let (M,w) be a compact symplectic mani-

fold and Ht = Ht+I : M -t IR be a smooth time dependent l-periodic Hamiltonian

function. Suppose that the l-periodic solutions of (1) are all nondegenerate. Then

2n

#P(H) :'.'.': Ldim Hi(M,Q)

i=O

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