LECTURE 3
Rational Floer Homology
The goal of this lecture is to explain the definition of the Floer homology
groups of a Hamiltonian H E C^00 (M x IR/Z) for monotone symplectic manifolds.
The first section is devoted to the fundamental compactness result which asserts
that, for a generic Hamiltonian, there are only finitely many connecting orbits of
index 1. Section 3.2 explains the definition of the Floer homology groups and their
main properties. The following three sections provide some details of the proofs.
Section 3.3 explains Floer's gluing construction, Section 3.4 outlines the proof that
Floer homology is an invariant, and Section 3.5 explains the isomorphism between
Floer homology and Morse homology. Sections 3.6 and 3.7 discuss the extension of
Floer homology to symplectic manifolds with vanishing first Chern class.
3.1. Compactness
Denote by M^1 (x-, x+; H, J) = {u E M(x-,x+; H , J) : μ(u; H) = 1} the one di-
mensional part of the moduli space of connecting orbits. The goal of this section is
to prove the following.
Proposition 3.1. If ( M, w) is monotone and H E Hreg then the quotient space
M^1 (x~, x+; H, J) = M^1 (x-, x+; H, J)/IR
is a finite set for every pair x± E P(H).
For v: 82 --+ Mand J E :J(M,w) define
- 1 01 2
8J(v) = 2.(dv +Jo dv o i) E D j (8 , v*TM),
where i denotes the standard complex structure on 82. The function v is called a
J-holomorphic sphere if 8J(v) = 0. The energy of v is the integral
E(v) = { v*w.
Js 2
This is equal to half the L^2 -norm of dv and hence is positive whenever vis noncon-
stant.
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