168 D. SALAMON, FLOER HOMOLOGY
Proof of Theorem 2.2 (the index formula): Suppose, without loss of
generality, that w(s, t) = w-(t) for s ~ -T and W(s, t) = w+(t) for s ?': T, and
that the paths r-+ W(s, 1) has only regular crossings. The symplectic loop around
the boundary of the square [-T, T] x [O, 1] is obviously contractible. Hence the
difference of the Conley-Zehnder indices can be expressed in the form
μcz(w+) - μcz(w- ) = μ( {W(s, 1)}-r::;s::;r) = I:signf(W(·, 1), s).
8
By Lemma 2.6, the term on the right agrees with the spectral flow of the operator
family s r-+ A(s), and hence we have
μcz(w+) - μcz(w-) = μspec(A) = indexD.
The last identity is proved in [4 1 ]. This proves the theorem. D
2.6. Transversality
Let us now return to the manifold situation of Theorem 1.24. In order to ap-
ply Theorem 2.2 we must assign a Conley-Zehnder index to every nondegen-
erate periodic solution x E P(H). This can be done as follows. Lineariz-
ing the differential equation (1) along x(t) we obtain linear symplectomorphisms
d1/it(x(O)) : Tx(o)M--+ Tx(t)M. In order to obtain a symplectic path Wx E SP(n)
we must trivialize the tangent bundle x*T M over x:
d,P,(x(O))
Tx(o)M ____,, Tx(t)M
T TSuch a trivialization can be obtained by specifying a disc u : B --+ M such that
u(e^2 rrit) = x(t) and then trivializing u*TM. This gives rise to a Conley-Zehnder
index
μH(x, u) = n - μ.cz(Wx)·
This index satisfies the following.
Corollary 2. 7. Let x± E P(H) be two nondegenerate periodic solutions of (1).
Moreover, let u : JR x JR/Z --+ M be a smooth map which satisfies the limit condi-
tion (8). Let u± : B--+ M satisfy u±(e^2 rrit) = x±(t) and u+ = u-#u. Then
Du: W^1 'P(u*TM)--+ LP(u*TM)is a Fredholm operator and its Fredholm index is given by
(23) indexDu = μ(u,H) = μH(x-,u-) - μH(x+, u+).
Proof. Theorem 2.2. D
Exercise 2.8. Suppose that H 1 = H is a Morse function with sufficiently small
second derivatives. Let x(t) = x be a critical point of H and define u : B --+ M as
the constant disc u(z) = x for z E B. Prove that
μH(x,u) = ind_H(x),
i.e. the Maslov index of the pair (x , u) is equal to the Morse index of x as a critical