LECTURE l. SYMPLECTIC FIXED POINTS AND MORSE THEORY 151of points in M whose gradient lines connect y to x (the space of connecting orbits)
is a smooth submanifold of M whose dimension is given by the difference of the
Morse indices:
dim M(y, x; !) = ind1(Y) - ind1(x).
One can think of M(y, x; !) as the space of gradient flow lines u: JR----; M running
from y = lim 8 , 00 u( s) to x = lims->+oo u( s). The group JR acts on M (y, x; f) by
translations and the quotient M(y, x; f) = M(y, x; !)/JR is a manifold of dimension
ind f (y) - ind f ( x) - 1 (whenever it is nonempty). Hence the Morse-Smale condition
implies that ind1(x) < ind1(Y) whenever there is a connecting orbit from y to x.
In short, the index decreases strictly along flow lines.
Exercise 1.8. Prove the dimension formula. D
Exercise 1.9. Prove that for every sequence uv E M(y, x; f) there exists a subse-
quence (still denoted by uv), finitely many critical points x 0 = x, x 1 , ... , Xm = y,
finitely many gradient flow lines u 1 E M(x 1 ,x 1 _ 1 ;!), and sequences sj E JR, such
that, for every j, uv(s + sj) converges to u 1 (s), uniformly on compact subsets of
R This limit behaviour is illustrated in Figure l. (See [45] if you get stuck.) D
Figure 1. Limit behaviour for connecting orbitsExercise 1.10. Prove that the quotient space M (y, x; !) = M (y, x; !) /JR of gradi-
ent flow lines from y to x is a finite set whenever the difference of the Morse indices
is equal to l. D
Exercise 1.11. Fix an orientation Ox of the unstable manifold wu ( x) for every
critical point x of f. Show how this gives rise to a natural orientation for each
connecting orbit (with index difference 1). Hint: For z E M (y, x; f) the differential
of the gradient flow d<r/(z) determines, for large t, a vector space isomorphism
Tz wu(y) n V' f(z)J_ ___,Tc wu(x).Define c:(z) = ±1, depending on whether this isomorphism is orientation preserving
or orientation reversing. This works even if the manifold M is not orientable. D
Let us now assume that the gradient flow of f is a Morse-Smale system and fix
an orientation of wu(x) for every critical point x. Denote by
CM*(!)= EB Z(x)
df(x)=O