1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

LECTURE 3. FLOER HOMOLOGY

In particular, DuR 'T/u + DvR 'T/v = DR*'T/ and hence

llDR'T/llw1.p < llDuR 'T/ullLP + llDvR *'T/vllLP

< C1 (llDuRDUR 'T/ullLP + llDvRDVR 'T/vllLP)

< C1 (llDR(,8RDR*'T/ - ,8R

1

'T/)llLP

+ llDR((l - ,8R)DR*'T/ + ,8R

1

'T/)llLP)

< 2c1 II DRDR 'T/ II LP+ R 2c1 II DR 'T/ - DR'T/lb + 4c1 R ll'T/llLP

< 2c1 llDRDR*'T/lb +

2

~

1

llDR*'T/llLP + ~ ll'T/llw1,p

< 2c1 llDRDR 'T/llLv + 2c1 + R 4coc2 llDR 'T/llLv ·

In the last step we have used (33). With (2c 1 +4c 0 c 2 )/R:::;1/2 we obtain

llDR*'T/llw1,p::::: 4c1 llDRDR*'T/llLP

181

as claimed. This proves the proposition. D

With the uniform estimates for the inverse established, it follows easily from

the implicit function theorem that, for R > 0 sufficiently large, there exists a unique

solution WR E M(z, x; H, J) near v#Ru of the form

WR= expV#Ru(DR*'T/)

for some 'T/ E W^2 ·P (IR x 81 , ( v# RU) *TM). The details are standard and will not be

carried out here. If the original moduli spaces M(z, y; H , J) and M(y, x; H , J) are

zero dimensional, then the images of the gluing maps describe precisely the ends of

the one dimensional moduli space M(z, x; H, J). In other words, the complement

of these images is a compact I-manifold. This justifies the formula (29) in the proof

of Theorem 3.5.

3.4. Invariance of Floer homology

The proof of Theorem 3.6 is based on the following construction. We assume

throughout that J is fixed and discuss the dependence of the Floer homology groups

on the Hamiltonian function. Variations of the almost complex structure can easily

be incorporated by an analogout construction. Given two regular Hamiltonians

H°', Hf3 E Hreg choose a homotopy H°'/3 = { H:,f} from H°' to Hf3 such that

Hs ,t °'/3 - { - HHf' /3

t )

for s :S -1,

for s ~ 1.

Now consider the time-dependent version of equation (7) with Ht replaced by Hs ,t·

This equation has the form

(34) 08 U + J(u)OtU - 'lH~f (u) = 0.

We consider solutions of (34) which satisfy the limit conditions

(35) lim u(s, t) = x°'(t), lim u(s, t) = xf3(t)

S--+- 00 S--+00

for some periodic solutions x°' E P(H°') and xf3 E P(Hf3). The solutions of (34)
and (35) form a moduli space

M(x°',x/3) = M(x°',x^13 ;H°'^13 ) = {u: IR x 81 ----> M: (34), (35)}