212 D. SALAMON, FLOER HOMOLOGYsemigroup property of the class of perturbations considered above. The family of
perturbations takes the form of a pair of maps
H x B---> 2£ : (h, v) 1-t fh(v), H x £---> Q: (h,v,ry) 1-+ Ah(v,ry),
such that, for each h, the pair (rh, Ah) satisfies the ''finiteness'', "conformality",
"energy'', and "local structure" axioms, In addition, the branches H x Ui ---> £ :
( h , v) 1-+ /i,h ( v) are required to be linear in h for each v E Ui, The crucial condition
is that the linear operator
TvBk,p x H---> £~-l,p: (~, h) 1-t D[)j(v)~ - D/i,h(v)~ - %,r,,(v)is surjective whenever 8J(v) = /i,h(v) and llhll < c (for a sufficiently small number
c; > 0). Under these assumptions the universal moduli spaces
are all smooth Hilbert manifolds. Now choose a common regular value h of the
projections
Mi,e:(J, {fh}h) ---> Hwith llhll < c; to obtain a perturbation (fh, Ah) which satisfies the "transversality"
axiom. That such a regular value exists follows from the Sard-Smale theorem.That a family of perturbations {fhh with the above properties exists, follows
from Corollary 5.5 via superposition.
To construct a perturbation which satisfies the "free" axiom we use a similar
argument. We note that, if v o <p = v for some <p E G - {id} then, for every point
z E 82 , there exists a point z' E S^2 - {z} with v(z) = v(z'). The goal is to show
that this does not happen generically for the solutions of ( 4 7). To see this considerthe universal spaces Xi consisting of tuples ( v, z 1 , ... , Zm, z~, ... , z~, J, h) such that
v E Ui, J E .J(M,w), h EH, Zj, zj E S^2 with Zj =f. zj, and
v(z1) = v(z~), ... , v(zm) = v(z~).
One can use the techniques of [31] to prove that the spaces Xi are Banach manifolds
and the projections
Xi---> .J(M,w) x Hare Fredholm maps of index 2n + 2c 1 (Ai) + m(4 - 2n), where A E H 2 (M,Z) is
the homology class of v E Ui. If n 2". 3 and m = mi is chosen sufficiently large,
then this index is negative. Now let (J, h) be a common regular value of these
projections. Then every solution v E Ui of DJ(v) E fh(v) has less than mi double
points. This implies that v o <p =f. v for every <p E G - {id}. Hence, for any such
common regular value (J, h) , the pair (J, rh) satisfies the ''free" axiom. Full details
of these arguments are given in [22].
5.4. Branched moduli spaces
Let (f, >.)be a weighted multi-valued perturbation with branches /i : Ui --->£which
satisfies all the axioms in Section 5.2. Fix a spherical homology class A E H 2 (M, 'IL)
and consider the moduli space