200 D. SALAMON, FLOER HOMOLOGYopen stratum in Mo,n·^4 We shall use cross ratios to construct an embedding ofMo,n into (5^2 )N for N = (l). Our embedding is reminiscent of a construction by
Fulton and MacPherson in [15] for higher dimensional varieties. That paper also
contain more references about the moduli space Mo,n·
The basic observation is that the relative position of four distinct points on the
2-sphere is, up to complex isomorphisms, determined by the cross ratio. To see thisnote that, for any three distinct points z 0 , z 1 , z 2 E 52 = C U { oo} there is a unique
fractional linear transformation rp E G which sends zo to 0, z1 to 1, and z2 to oo.
This transformation is given byThe cross ratio is well defined for (zo,z 1 ,z2,z3) E (5^2 )^4 - ~3 and satisfies{
oo,w(zo,z1,z2,z3) = 1,
0,(40)
if zo = Z1 or Z2 = Z3'
if Zo = Z2 or Z3 = Z1'
if zo = Z3 or Z1 = Z2.Here ~3 = {(zo,z1,z2,z3) : :3i < j < k 3 Zi = Zj = zk} denotes the set of
quadruples with three equal points. Moreover, the cross ratio is invariant under thediagonal action of G, meaning that w( rp(zo), rp(z 1 ), rp(z2), rp(z3)) = w(zo, z 1 , z2, z3)
for rp E G.
Let us now introduce the maps wijke: Mo,n--+ 52 given by(41) Wijke(z) = w(zi,Zj,Zk,ze)
for any four distinct integers i, j, k, f, E {l, ... , n }. We claim that these maps extend
continuously to Mo,n, that they collectively form an injection of Mo,n into (5^2 )N,
that this injection is a homeomorphism onto its image, and that this image is a
smooth (in fact algebraic) submanifold of (5^2 )N.
Exercise 4.17. Prove that the maps Wijke: Mo,n --+ 52 defined by (41) satisfy(42) Wjike = Wijek = 1 - Wijkf.,(43) Wjkem = Wijkm -^1
Wijkm - Wijkf.for any five distinct integers i, j, k, f, m E {l,. .. , n }.
It is useful to introduce the notation
In = { ( i, j, k, f) E N^4 : i, j, k, f, are pairwise distinct and ~ n} ,
and write the elements of (5^2 fn in the form w = {w!}IEin·
DProposition 4.18. The maps Wijke : Mo,n --+ 32 defined by (41) extend to con-
tinuous functions Mo,n --+ 32 , still denoted by Wijke. The resulting map
Mo,n--+ (5^2 )In: Z f--) {w1(z)}JEin
is injective and its image is the set Mn C (5^2 )In of all tuples w = { WJ} IEin which
satisfy (42) and (43).
(^4) ·we denote by Mo,n what others denote by Mo,n and by Mo,n what others denote by Mo,n·