LECTURE 3. FLOER HOMOLOGY 1793.3. Floer's gluing theorem
The goal of this section is to provide more details for the proof of Theorem 3.5
and to prepare the background for the proofs of Theorems 3.6 and 3.7. The basic
construction is very simple. Given two connecting orbitsv E M(z, y ; H, J), u E M(y,x;H,J),
with surjective Fredholm operators Dv and Du , one constructs a one parameter
family of approximate solution
WR= v#Ru
of (7) running directly from z to x. One then proves that the linearized operator
D-wn is surjective for large Rand has a right inverse which satisfies a uniform bound,independent of R. It then follows from the infinite dimensional implicit function
theorem that near WR there is a true solution WR E M(z , x; H, J) of (7). This
construction gives rise to a gluing map
(31) lz,y,x: M(z, y; H, J) x (Ro, oo) x M(y, x; H, J) ---t M(z, x; H, J).
This is not quite accurate, unless the index differences are 1 and so the moduli spaces
M(z, y; H , J) and M(y, x; H , J) are compact. In general, this map is only defined
on any given compact subset of M(z, y; H, J) x M(y, x; H, J) and the constants in
the estimtes will depend on this subset. With this understood, the gluing map (31)
is a diffeomorphism onto an open subset of M(z, x; H, J).
v(s+R,t)I
I
I
-R -R/2 0u(s-R,t)s
R/2 R
Figure 8. Floer's gluing constructionHere are some more details of Floer's gluing construction. The approximate
solution is illustrated in Figure 8. Explicitly, it can be defined by
l
v(s+R,t),
expy(t) (/3( -s - R/2)TJ( s + R, t)),
v#Ru(s, t) = y(t),
expy(t) (/3( s - R/2)~( s - R, t)),
u(s - R, t),s ~ -R/2 - 1,
-R/2 - 1 ~ s ~ -R/2,
-R/2 ~ s ~ R/2,
R/2 ~ s ~ R/2 + 1,
s ?". R/2 + 1,
where ~( s, t), TJ( s, t) E Ty(t) M are chosen such tha t u( s, t) = expy(t) (~( s, t)) for all t
and large negatives and v(s, t) = expy(t) (TJ(s, t)) for all t and large positives. Here
f3 : JR. __, [O, 1] is a cutoff function equal to 1 for s ?". 1 and equal to 0 for s ~ 0. Let
us fix the two solutions u and v and assume that Du and Dv are surjective. The
next proposition shows that there is a uniformly bounded family of right inverses
for the operators DR = Dv# nu for R sufficiently large.