222 D. SALAMON, FLOER HOMOLOGYinjective for each solution. The injective condition is redundant, by the "free" ax-
iom. One can argue, as in [31], that if there were infinitely many solutions, then
there would be a sequence of solutions which Gromov converges to a stable map
modelled over a tree with more than one vertex. This limit curve would then belong
to a moduli space of strictly lower dimension. But since we have started with a
zero dimensional moduli space the limit moduli space has negative dimension and
therefore must be empty. The key point is that for the perturbed equations the ac-
tual dimension always agrees with the virtual dimension, and hence the difficulties
discussed in Section 5.1 do not arise. D
Other approaches to the construction of the Gromov-Witten invariants for
general symplectic manifolds were developed by Li-Tian [26], Ruan [42], and
Siebert [51]. These authors construct, for every J E .:J(M, w), every A E H 2 (M, Z),
and every integer k 2 0, a rational fundamental classes
[Mo,k,A(M,w, J)] E H2n+2ci(A)+2k-6(Mo,k,A(M,w, J); Q)
on the compact ified moduli space Mo,k,A (M, w, J) of stable J-holomorphic curves
in the class A with k marked points. The Gromov-Witten invariants can then be de-
fined by evaluating the cohomology class e 1 a 1 ., · · · ., ek O'.k on the fundamental
class:
A(a1, ... ,ak)= 1 e1a1'--'···'--'ekak.
[Mo,k,A (M,w, J)]
Here the ei: Mo,k,A(M,w, J)--> M denote the obvious evaluation maps.
The standard gluing techniques, as in [31, 27, 43], can be used to prove that
the rational Gromov-Witten invariants satisfy the usual gluing rules. One version
of these rules asserts the associativity of quantum cohomology and the WDVV
equation. Another version involves the Chern classes of certain line bundles over
the moduli space and plays a crucial role in the recent work of Givental on mirror
symmetry [16, 17].
5. 7. Rational Floer homology
The goal of this section is to explain how the above ideas can be used to define
Floer homology groups for symplectic manifolds which satisfy ( 4) with T < 0. Fix
a time dependent Hamiltonian Ht = Ht+ 1 with nondegenerate 1-periodic solutions
x E P(H) and, for each pair x± E P(H), consider the space Z(x-, x+) of smooth
functions u : JR x S^1 --> M which satisfy (8). As in Exercise 2.10, abbreviate
Z = Z(H) =
x±ET-'(H)and consider the vector bundle E --> Z with fibers
Eu= C 0 (JR x S^1 ,u*TM).
For the purpose of this discussion the subscript zero can either denote compact
support or some exponential decay condition. The precise notation is immaterial
because, when it comes to the analysis, we will have to work with suitable Sobolev
completions. To extend the above ideas to Floer homology we must choose a
weighted multi-valued section
.A: E _, Q,