152 D. SALAMON, FLOER HOMOLOGY
the free abelian group generated by the critical points of f. This complex is graded
by the Morse index and the boundary operator 8 = aM: CMk(j) __, CMk-1(!) is
defined by
8M (y) = L s(u)(x)
. xECdt(f) [u)EM(y,x)
md1(x)=k-l
for y E Crit(j) with ind1(Y) = k. Here the sign s(u) is given by Exercise 1.11.
This complex (CM(j),8M) is called the Morse-Smale-Witten complex. The
remarkable observation is that 8M is indeed a boundary operator and that the
homology of this complex agrees with the homology of M.
Theorem 1.12 (Morse,Smale,Witten). Suppose that (5) is a Morse-Smale flow.
Let CM(!) and aM be defined as above. Then aM 0 aM = 0 and there is a natural
isomorphism
ker 8M
HMk(M,f;Z) =. BM __, Hk(M;Z)
lm
where Hk(M;Z) denotes the singular homology of M.
Proof. Here is a sketch of the argument which proves aM 0 aM
equivalent to the formula
(6) L s(v)s(u) = 0
. vECdt(f) [v)EM(z,y) [u)EM(y,x)
md1(y)=k
0. This is
for every pair of critical points x,z E Crit(j) with ind1(x) = k - 1 and ind1(z) =
k + 1. This is proved by studying the ends of the 1-dimensional moduli space
M(z,x). The endpoints of this moduli space are in one-to-one correspondence
with the set of pairs of gradient flow lines ( u, v) running from z to x, via some
intermediate critical point y , necessarily of index k. This assertion follows from a
combination of compactness and gluing arguments. Since every compact 1-manifold
has an even number of boundary points, we conclude that "pairs of connecting orbits
come in pairs" and this proves (6) modulo 2. (See Figure 2.)
Now the manifold M(z, x) carries a natural orientation inherited from the
orientations of wu(z) and wu(x). Using this orientation one can show that the
indices s(u 0 )s(vo) and s(u 1 )s(v 1 ), correponding to the two ends of a component of
M(z, x), cancel out. This proves (6). D
YoFigure 2. The Morse-Smale-Witten complexGeometrically, one can think of the formal sum l:i mi wu (Xi), corresponding
to an element I:i miXi E ker aM' as a cycle representing the image in H*(M; Z)