158 D. SALAMON, FLOER HOMOLOGYThe meaning of this result is not only that, by accident, M(x-, x+) is a smooth
manifold, but that the natural Fredholm operator, obtained by linearizing (7), is
surjective for every connecting orbit. The proof will be outlined in the next lecture.
Remark 1.25. We shall see below that for every nondegenerate 1-periodic solution
x E P(H) and every smooth map u: B ___, M with u(e^2 7rit) = x(t) there is a well-
defined Conley-Zehnder index μH(x, u) which satisfies
μH(x,A#u) = μH(x,u) - 2c1(A)
for every A E 7r 2 (M). Since
aH(x, A#u) = aH(x, u) - w(A),it follows that the difference
(11) r/H(x) = μH(x, u) - 2rnH(x, u)is independent of the choice of the function u : B ___, M used to define it. That this
difference satisfies the requirements of Theorem 1.24 will follow from Exercise 1.23
and the fact that μ(u;H) = μH(x-,u-) - μH(x+,u-#u). Note that, without
specifying the map u : B ___, M, the Conley-Zehnder index of a periodic solution
x E P(H) is only well defined modulo 2N. D