166 D. SALAMON, FLOER HOMOLOGY
where sign S is the signature (the number of positive minus the number of
negative eigenvalues).(Determinant) (-l)n-μcz(llt) = signdet(Il - '11(1)).
(Inverse) μcz(w-^1 ) = μcz(wT) = -μcz(W).
Here is an alternative definition of the Conley-Zehnder index in terms of cross-
ing numbers, as in [40]. Any path W E SP(n) can be expressed as a solution of an
ordinary differential equation
~(t) = JoS(t)w(t), w(O) = Il,
where t r-+ S(t) = S(t)T is a smooth path of symmetric matrices. A number
t E [O, 1] is called a crossing if det(Il - w(t)) = 0. If t is a crossing then the
crossing form is the quadratic form r(w, t) : ker(Il - w(t)) --+JR. defined by
r(w, t)fo = wo(fo, ~(t)fo) = (fo, S(t)fo)for fo E ker ( n - w ( t)). A crossing t is called regular if the crossing form is
nondegenerate. Regular crossings are isolated. For a path W E SP(n) with only
regular crossings the Conley-Zehnder index is given by the formula
μcz(W) = ~sign(S(O))+ 2...:signr(w,t)
t > Owhere the sum runs over all crossings t > 0. That both definitions of the Conley-
Zehnder index agree is proved in Robbin-Salamon [40].2.5. The spectral flow
The operator D =as + Joat +Scan be written in the form
a
D = as+ A(s)
where A(s): W^1 •^2 (S^1 ,JR.^2 n )--+ L^2 (S^1 ,JR.^2 n) is given by
a
(21) A(s) =Jo at+ S(s, ·).
This is a smooth family of unbounded self-adjoint operators on the Hilbert space
H = L^2 (S1,JR.^2 n). It turns out that the limit operators
A±= lim A(s)
s->±ooare both invertible. In this case the Fredholm index of D is given by the spectral
flow of A (see Atiyah-Patodi-Singer [1]). In our discussion we follow the exposition
in [41].
Intuitively, the spectral flow is the number of eigenvalues of A(s) crossing zero
from negative to positive ass moves from -oo to +oo (see Figure 6). More formally,
the spectral flow can be defined as follows. A number s E JR. is called a crossing
if ker A(s) i-{O}. Ifs is a crossing then the crossing form is the quadratic form
I'(A, s) : ker A(s) --+JR. defined by
r(A, s)~ = (~, A.(s)~)H
for ~ E ker A(s). A crossing s is called regular if the crossing form is nondegen-
erate. Regular crossings are isolated. For a smooth family s r-+ A(s) with only