194 D. SALAMON, FLOER HOMOLOGYFor future reference it is useful to introduce some notation. For every paira, f3 E T with a =/= f3 there exists a unique ordered set of vertices /o, ... , /m E T
such that /iE/i+li /i =/= /i+2, /o = a, and Im = /3. We call this the chain (of
edges) running from a to f3 and denote the set of vertices belonging to this
chain by
[a,/3]=[/3,a]={'Yi: i=O, ... ,m}.
Cutting any edge o:E/3 decomposes the tree Tinto two components. The componentcontaining /3 will be denoted by Taf3 and is given by Taf3 = {! E T : /3 E [a, 1]}.
This set is called a branch of t h e tree T (see Figure 13). Note that T is the
disjoint union of {a} and the branches Taf3 over all /3 E T with o:Ef3. Moreover,
T = Ta/3 U Tf3a whenever o:E/3.
Definition 4.3 (Stable maps). Let ( M, w) be a compact symplectic manifold andJ E J(M,w). A stable J-holomorphic curve of genus zero in M with k
marked points, modelled over a tree (T, E), is a tuple
(u, z) = ({ua }aETi {za/3}aEf3, {ai,zi}1s;i9)
with th e following properties. For each a E T, Uo; : S^2 -+ M is a J -holomorphic
sphere, for all a, /3 E T with o:E /3' Zo;f3 E S^2 ' and ( 0:1, Z1)' ... ' ( O:k' Zk) are finitely
many points in T x S^2 , satisfying th e following conditions (see Figure 14).(i) If a,/3 ET with o:E/3 then u 0 (za{3) = Uf3(Z(3a)·
(ii) If o:E/3, o:E1, and /3 =/= /, then Zaf3 =/= Za'Y· If o:i = O:j with i =/= j then
zi =/= z 1. If o:i = a and o:E /3 then zi =/= Zaf3.
(iii) If u 0 is a constant function th en the setZa = Z 0 (u,z) = {zaf3 : /3 ET, o:E/3} U { zi 1 :':::: i :':::: k , O:i =a}
consists of at least three elements.If ( u, z) is a stable map then the tree T carries natural weights
ma(u) = E(ua ) = r Uo; *w.
Js2
The weight m 0 can only be zero if the a-sphere carries at least three special points.It is useful to introduce the notation
E 0 (u, D) = 1 Ua *w + L ma{3(u), ffia(3(u) = L E(u'Y),
n oEf3 'YET 0 f3
z 0 f3Enfor a, /3 E T with o:E/3 and any open set D c S^2. Then the total energy
E(u) = L E(ua )
a ETof the stable map (u, z) is equal to E 0 (u, S^2 ) for any a ET.
There is a natural equivalence relation on the set of stable maps.^1 The equiva-
lence relation is essentially given by complex diffeomorphisms of the domains of the
curves which identify the maps, the singular points, and the marked points. Now(^1) Strictly spea king, trees do not form a set but a category. So the collection of tuples (u, z) with
the stated properties, which are modelled over trees, is not actually a set. However , if we restrict
our definition of trees with m vertices to me::ming a r elation on the set { 1, ... , m} with the stated
properties then the stable maps in M form a set.