178 D. SALAMON, FLOER HOMOLOGY
The Floer homology groups of a regular pair (H, J) are defined as the homology
of the chain complex (CF*(H), aF):
ker8F
HF*(M,w,H,J;F) = ~·
imu
The key observation is that these homology groups are an invariant, i.e. they do not
depend on the almost complex structure or the Hamiltonian. Moreover, they are
naturally isomorphic to the ordinary homology groups of M. This is the content of
the following two theorems.
Theorem 3.6 (Floer). Let (M,w) be a monotone symplectic manifold. For two
pairs (H°', J°') and (Hf3, Jf3), which satisfy th e regularity requirements for the def-
inition of Floer cohomology, there exists a natural isomorphism
q>f3°': HF(M,w,H°',J°') ---t HF(M,w,Hf3,Jf3).
If ( H'"Y, J'Y) is another such pair then°'°' =id.
Theorem 3.7 (Floer). Let (M,w) be a monotone symplectic manifold. Then, for
every regular pair (H°',J°'), there exists an isomorphism
j=k(mod 2N)
These maps are natural in the sense that
q>f3 o q>f3cx = <P°'.The Arnold conjecture for monotone symplectic manifolds (and general coeffi-
cient rings) follows immediately from Theorem 3.7. The proofs will be explained in
the next three sections.
Exercise 3.8. Prove the naturality of Floer's gradient fl.ow equation (7). More
precisely, suppose that u : JR x JR/Z ---t M is a solution of (7) and 'Pt = 'Pt+l is a
loop of Hamiltonian symplectomorphisms, generated by the Hamiltonian functions
K t = Kt+1 : M ---t JR via
'Po= id.Prove that the function u(s,t) = 'Pt-^1 (u(s,t)) satisfies
8su + 1t(ii)8tii - 'lHt(ii) = 0,
where
jt ='Pt* Jt, H t = (Ht - Kt) 0 'Pt,and the gradient is computed with respect to the metric induced by Jt. Deduce
that the Hamiltonian loop 'Pt induces an isomorphism of Floer homologies.
HF*(M,w,Ht, Jt) ---t HF*(M,w, (Ht - Kt) o 'Pt,'Pt* Jt) ·This isomorphism will not, in general, agree with the one of Theorem 3.6. One
can think of this as an action of the fundamental group of the group Ham( M, w) of
Hamiltonian symplectomorphisms on Floer homology. This action has recently been
used by Seidel and Lalonde-McDuff-Polterovich to derive nontrivial information
about the fundamental group of Ham\M,w). D