1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

LECTURE 5. MULTI-VALUED PERTURBATIONS 225

for all x, z E P(H). As in the monotone case, this follows by examining the moduli

space M^2 (z, x; H , J, r). For a generic perturbation r and a generic Hamiltonian

H, this space is a regular oriented branched 1-manifold with rational weights. Via

Floer's gluing theorem, it can be compactified by including as boundary points

pairs [v#u] with [u] E M^1 (y,x;H,J,f) and [v] E M^1 (z,y;H,J,r). Associated

to each boundary point is a rational weight p([v#u]). One can show that these

weights are given by

p([v#u]) = p(v)p(u)

and hence the result follows from Lemma 5.11. This proves that 8 o 8 = 0.

The rest of the proof is a longer story, but the details are essentially as in the

standard case. The easiest approch is to fix the perturbation r and the almost

complex structure J, and only vary the Hamiltonian H. Then one can employ
the methods of Section 3.4 to construct the isomorphism between Floer homology

and ordinary homology. It is, however, more elegant to prove first that the Floer

homology groups are independent of H , J , and r. For more details see [23]. D

The above construction can easily be extended to arbitrary compact symplectic
manifold by using Novikov rings as in Section 3.7. Hence we have proved the
following theorem, modulo some analytical details which are carried out in [20, 21,
22, 23]. Other proofs are given in [14, 28].

Theorem 5.15. Suppose that ( M, w) is a compact symplectic manifold. Let Ht =
Ht+l be a smooth time dependent Hamiltonian on M such that all the l-periodic

solutions of (1) are nondegenerate. Then #P(H) 2 dim H*(M, Q).