LECTURE 2. FREDHOLM THEORY 165Here D* = -8 8 + Jo8t + 8 denotes the formal adjoint operator and the last equality
holds since D*T/ = 0. Taking the q-th power of this inequality and summing over
k we find that 'fJ E W^1 ,q (IR x 8^1 , IR^2 n) and D*T/ = 0. Thus the cokernel of D :wl,p--) L P agrees with the kernel of D* : wl,q--) Lq and therefore is also.finite
dimensional. D2.4. The Conley-Zehnder index
Denote by Sp(2n) the group of symplectic 2n x 2n-matrices. In [2] Conley and
Zehnder introduced a Maslov type index for paths of symplectic matrices. Their
index assigns an integer μcz(W) to every path W : [O, 1] --> Sp(2n) such that w(O) =n and det(TI-w(l)) =I-0. Other expositions are given in Salamon-Zehnder [47] and
Robbin-Salamon [40].
Denote by Sp*(2n) the open and dense set of all symplectic matrices which do
not have 1 as an eigenvalue. This set has two components, distinguished by the sign
of det(TI-w). Its complement is called the Maslov cycle. It is an algebraic variety
of codimension 1 and admits a natural coorientation. The intersection number of
a loop : 8^1 --> Sp(2n) with the Maslov cycle is always even and the Maslov
index μ() is half this intersection number. Alternatively, the Maslov index can be
defined as the degree
μ() = deg(p o )
where p : Sp(2n) --> 81 is a continuous extension of the determinant map det
U(n) = Sp(2n) n 0(2n) --> 81. The map p is not a homomorphism but can be
chosen to be multiplicative with respect to direct sums, invariant under similarity,
and taking the value ±1 for symplectic matrices with no eigenvalues on the unit
circle. These properties determine p uniquely (cf. [4 7]).
Now denote by SP(n) the space of paths w : [O, 1] -+ Sp(2n) with w(O) = n and
w(l) E Sp(2n). Any such path admits an extension W: [O, 2]--> Sp(2n), unique up
to homotopy, such that W ( s) E Sp ( 2n) for s ~ 1 and W ( 2) is one of the matrices
w+ = -TI and w-= diag(2, -1, ... , -1, 1/2, -1, ... , -1). Since p(W±) = ±1 it
follows that p^2 o W : [O, 2] --> 81 is a loop and the Conley-Zehnder index of W is
defined as its degree
μcz(w) = deg(p^2 ow).
The Conley-Zehnder index has the following properties. It is uniquely determined
by the homotopy, loop, and signature properties [47].
(Naturality) For any path <I>: [O, 1]--> Sp(2n), μcz(<I>w<I>-^1 ) = μcz(W).
(Homotopy) The Conley-Zehnder index is constant on the components of
SP(n)(Zero) If W(s) has no eigenvalue on the unit circle for s > 0 then μcz(W) = 0.
(Product) If n' +n" = n identify Sp(2n') EBSp(2n") with a subgroup of Sp(2n)
in the obvious way. Then μcz(W' EB w") = μcz(W') + μcz(w").
(Loop) If : [O, 1]-+ Sp(2n,IR) is a loop with (O) = (l) = n then
μcz(w) = μcz(w) + 2μ().
(Signature) If 8 = 5 T E 1R^2 nx^2 n is a symmetric matrix with 11811 < 27r and
w(t) = exp(Jo8t) thenμcz(W) = ~sign(8)