1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

176 D. SALAMON, FLOER HOMOLOGY

+

---X 0 =X

Figure 7. Limit behaviour for Floer's connecting orbits with bounded derivatives

Proof of Proposition 3.1: The formula (10) shows that

r1H(x-) - 'T/H(x+) + 2TE(u) = μ(u; H) = 1

for u E M^1 (x-,x+;H,J) and hence the energy inequality (25) is automatically

satisfied for sequences uv E M^1 (x-,x+;H,J). Moreover, the limit solutions u 1 E

M(x 1 , x 1 _ 1 ; H, J) in Corollary 3 .4 satisfy

t, μ(u1; H) t, ( 'T/H(x1) - 'TJH(x1_i) + 2TE(u1))

m

j=l

< 'T/H(x-) - 'TJH(x+) + 2TE(uv) - 2Tf.n

1-2£.N.

Here R. is the number of bubbles, the first equation follows from (10), the third

inequality from (27), and the last equation from (10) and the fact that Tn = N.

If H E Hreg then Theorem 1.24 guarantees that μ( u 1 ; H) 2". 1 for every j. This

implies that m = 1 and R. = 0, i.e. there is no bubbling and there is a single limit

solution u = u 1 E M(x-, x+; H, J). In this case one can show that uv(s + s!, t)

converges to u(s, t) in the W^1 ·P-norm on the noncompact domain IR x S^1. This

shows that, in the monotone case, the moduli space M^1 ( x-, x+; H, J) is compact

for every pair x± E P(H). Since this space is also a zero dimensional manifold it

must be a finite set. D

3.2. Floer homology

Continue with the monotone case and, for H E Hreg, consider the chain complex

IF (x).

xE"P(H)

μcz(x;H)=k(mod 2N)

Here IF is a principal ideal domain (which we shall choose to be either of Z 2 , Z,

or Q). In the case IF = Z2 the boundary operator is defined by counting the

connecting orbits u running from x - to x+ which satisfy μ( u; H) = l. For other

coefficient rings one has to take account of orientations. The latter is discussed in