218 D. SALAMON, FLOER HOMOLOGYThe proof is by induction over the number of sets in the open cover M = LJj M(j).
Fix any index j = Ji. By Step 1, the closure of the set M (j 1 ) can be covered by
finitely many such paths /a : [O, l] ---> M. To see this simply choose the /a to
be reparametrizations of the curves 'Pi-l : cl(cpi(Mi )) ---> cl(Mi)· Now under theassumptions of Step 1, one checks easily that the complement M = M - M(j1) is
again a compact branched manifold covered by open sets M(j) = M(j) - M(j1),
j "I-j 1 , which still satisfy the requirements of Step l. This completes the induction
argument and the proof of Step 2.
Step 3: We prove the lemma.Step 2 defines a directed graph with vertices x E V and edges /a· The edges carry
rational weights Aa = Ai(a) > 0. Note that all the boundary points x E 8M are
vertices and that
p( X) = L Aa - L Aa
l'a(l)=x l'a(O)=x
for x E 8M. On the other hand, by Step 2,>.(x) = L Aa = L Aa
la (O)=x l'a(l)=x
for x E V - 8M. HenceL p(x) = L ( L Aa - L Aa) = 0.
xEBM xEV 10 (l)=x 10 (0)= x
This proves the lemma. DLet us now return to the moduli space M = M(A; J, r) of (J, f)-holomorphic
spheres representing the homology class A E H 2 (M, Z). The ''free" axiom guaran-
tees that the action of the reparametrization group G = PSL(2, <C) on M is free.However, the branches Mi = M(A; J,/i) will not, in general, be invariant under
G. Nevertheless, by using local slices (as in the case of principal bundles), one canshow that the quotient M/G is again a branched manifold of dimension
dim M(A; J, r)/G = 2n + 2c 1 (A) - 6.
So far we have not addressed the compactness question. As in the case of J holo-morphic curves, the moduli space M/G will not be compact, in general, but bub-
bling may occur. To obtain bubble trees of (J, f)-holomorphic curves in the limit,
we must ensure the compatibility, under Gromov convergence, of our multi-valued
perturbations correponding to different homotopy classes.
5.5. Perturbations and stable maps
The goal of this section is to obtain the same kind of compactness results for
the perturbed equations ( 47) as were discussed in Section 4 for J-holomorphic
curves. For this it is useful to make sure that the perturbation vanishes in a
neighbourhood of any point near which bubbling occurs. Moreover, we must match
the perturbations on the components of a limiting bubble tree with the perturbation
on the approximating curves. This requires a refinement of the construction in
Section 5.2. Namely, we shall introduce perturbations which not only depend on
the curve u but also on a finite set of marked points, and are required to vanish
near the marked points.