220 D. SALAMON, FLOER HOMOLOGY
carries an action of the group G(T) of automorphisms of the bubble tree. The
elements of G(T) are tuples (!, {'Pa} a ET) where f : T ---+ T is a tree automorphism
and 'Pa E G for a ET. The group operation is given by
(g, {'l/!a}aET) O (!, {<p}aET) = (g^0 f, {'tfJJ(a)^0 'Pa}aET),
and the action on B(T, k) by (!, <p)*(u , z) = (ii, z), where
Uf(a) = Ua o 'Pa -l, ZJ(a)f({3) = 'Pa(Zaf3), cli = f(ai), Zi ='Pai (zi)
(see Definition 4.4). It follows from (49) that the moduli space of solutions of (51)
is invariant under the action of G(T). The ''free" axiom can be extended to ensure
that the action of G(T) on this moduli space is free. For details see [22].
Compatibility
The crucial compatibility condition for the perturbations in different homology
classes is continuity with respect to the Gromov topology. More precisely
this means the following.
(52)
(53)
(Compatibility) If a sequence ( u", zI, ... , zk) Gromov converges to a stable
map (u, z) E B(T, k), with corresponding sequences <p~ E G (see Defini-
tion 4.5), then
and
/\ ( U v , Z1, v ... , Zk, v. 'T) " )
77ver(uv ,zr , ... ,zk>
II ( "°::; l "11" -111 1 r,= «:
for E: > 0 sufficiently small. In (52) the convergence is uniform on S^2. That
u " o<p~ only converges uniformly on compact subsets of S^2 - Za is immaterial
because the perturbation vanishes in a neighbourhood of Z a.
The existence of an almost complex structure J and a family of perturbations
(r A, AA), one for each homology class A E H 2 (M, Z) (and each number k of marked
points), which satisfy the axioms of Section 5.2 and the "compatibility" axiom, is
proved by induction over w(A). For w(A) < 2/i all J-holomorphic curves are simple.
Hence transversality can be achieved by a generic choice of J. Now suppose that
the perturbations have been constructed for w(A) < m!i for some integer m ;::: 2,
and fix a homology class B with w(B) < (m + l)li. Then one can use the gluing
construction, of Appendix A in [31], and the induction hypothesis, to define the
perturbation in a Gromov neighbourhood of any given bubble tree, representing
the class B. The estimate for the inverse in [31], Appendix A, then shows that
the glued perturbations automatically satisfy the crucial "transversality" axiom.
Away from the bubble trees one can use the methods of Section 5.3 to construct
the perturbations. For more details of the induction argument see [22].
5.6. Rational Gromov-Witten invariants
The techniques explained so far sufficP-to define the Gromov-Witten invariants for
general symplectic manifolds via solutions of the perturbed equations (50), following
