1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

196 D. SALAMON, FLOER HOMOLOGY

(ii) If a.,{3 ET with a.E{3 then (cp~)-^1 o cp~ converges to Zcxf3, uniformly on

compact subsets of S^2 - {z13cx}·

(iii) For i = 1, ... , k, Zi = limv-+oo(cp~,)-^1 (zi).

Note that the first two conditions here summarize the convergence behaviour of se-

quences as discussed in Section 4.1. In that case there are no marked points and the

third condition can be ignored. Definition 4.5 is the natural generalization of this

concept of convergence to sequences of stable maps modelled on more complicated

trees. Here we do not assume that the maps in the sequence are all modelled on the

same tree. Although it is always possible to choose a subsequence modelled on the

same tree, the definition should allow for sequences of stable map which do not all

belong to the same stratum (tree structure) in the space of stable maps. Moreover,

if the limit curve has nontrivial automorphisms, then the tree homomorphisms r

may not be uniquely determined by the sequence (uv,zv) and its limit (u,z).

Fix a spherical homology class A E H2(M, Z) and denote by

Mo,k,A = Mo,k,A(M, J)

the set of stable J-holomorphic curves (u, z) of genus zero in M with k marked

points which represent the class A. The quotient space will be denoted by

Mo,k,A = Mo,k,A(M, J) = Mo,k,A(M, J)j"".

Thus the elements of Mo,k,A(M, J) are equivalence classes of stable maps in M

under the equivalence relation of Definition 4.4.

Definition 4.5 defines a topology on this quotient space, called the Gromov

topology. A sequence of equivalence classes [uv, z v] converges to [u, z] in this

topology if (uv, zv) Gromov converges to (u, z). A subset F C Mo,k,A(M, J)

is called Gromov closed if the limit of every Gromov convergent sequence in

F lies again in F. A subset U C Mo,k,A(M, J) is called Gromov open if its

complement is Gromov closed. That this defines a topology on Mo,k,A(M, J) is

obvious. However, that convergence with respect to this topology is equivalent to

Gromov convergence is not immediately obvious.

Remark 4.6. Let (X, U) b e a topological space in which limits are unique and,

for every x E X and every A C X, we have x E cl(A) if and only if there exists a

sequence Xn EA converging to x. Consider the space

cc xx xf\I

of all pairs (xo, (xn)n) of elements Xo E X and sequences Xn E X such that Xn
converges to xo. Then the collection C of convergent sequences has the following
properties.

(Constant) If Xn = xo for all n EN then (xo, (xn)n) EC.

(Subsequence) If (xo, (x n)n) E C and g : N ~ N is strictly increasing, then

(xo, (xg(n))n) EC.

(Subsubsequence) If, for every strictly increasing function g: N ~ N, there

exists a strictly increasing function f : N ~ N such that (x 0 , (xgof(n))n) EC,

then (xo, (xn)n) E C.

(Diagonal) If (xo, (xk)k) E C and (xk, (xk,n)n) E C for every k then there

exist sequences ki, ni E N such that (xo, (xk,,n,)i) EC.

(Uniqueness of limits) If (xo, (xn)n) EC and (yo, (xn)n) EC then Xo =YD·