184 D. SALAMON, FLOER HOMOLOGY
0 1Figure 10. The parametrized moduli space M^0 (y"'-, yf3; {H~^13 })Proof of Theorem 3.6: Let H"'-f3 be any regular homotopy from Hex to Hf3 and
denote by Hf3cx the inverse homotopy. By Lemma 3.10, these homotopies induce
chain maps cpcxf3 : CF(Hcx) ____, CF(Hf3) and cpf3cx : CF(Hf3) ____, CF(Hcx). By
Lemma 3.11, the composition cpcxf3 o cpf3cx is equal to the map induced by some
homotopy from H a to itself. Now the constant homotopy induces the identity map
on CF*(Hcx). Hence it follows from Lemma 3.12 that cpcxf3 o cpf3a is chain homotopy
equivalent to the identity. Hence cpcxf3 is a chain homotopy equivalence with chain
homotopy inverse cpf3cx. Hence cpaf3 induces an isomorphism on Floer homology. D
3.5. A natural isomorphism
There are essentially two ways to prove that the Floer homology groups of a pair
(H, J) agree with the ordinary homology of M. The first is to use the indepen-
dence of the Floer homology groups of the Hamiltonian, and then to prove that, if
Ht = H is a smooth time independent Morse function with sufficiently small sec-
ond derivatives, then the 1-dimensional moduli spaces of Floer's connecting orbits
consist entirely of gradient fl.ow lines of H. Once this is established, computing
the Floer chain complex reduces to the computation of the Morse complex of H,
and hence the resulting Floer homology groups agree with Morse homology, and
hence, by Theorem 1.12, with the singular homology of M. This method was used
by Floer [11] and also in [47] and [19].
An entirely different approach was found by Piunikhin-Salamon-Schwarz [38].
The idea is to consider perturbed J-holomorphic planes u : <C ____, M which satisfy
the following conditions.
• z f-7 u(z) is a J-holomorphic curve for lzl < 1.
• u( e^2 7r(s+it)) satisfies (34) for s > 0 where Hs,t = 0 for s :::; 0 and Hs, t = Ht
is independent of s for s > l.
• u(e^2 7r(s+it)) converges to a periodic solution x(t) = x(t + 1) of (1) ass____, oo.
• u(O) E wu(y; f) for some Morse function f : M ____,JR and a critical point y.
One can think of these as J-holomorphic spiked disks, where the spike is
the gradient fl.ow line from y to u(O) (see Figure 11). Without the condition
u(O) E Wu(y; f) these perturbed J-holomorphic planes form a manifold of local
dimension 2n - μH(x, u) near u. Hence the condition u(O) E Wu(y; f) gives rise to
a zero dimensional moduli space whenever the index difference is zero , i.e. ind / (y) =