LECTURE 2. FREDHOLM THEORY 169Exercise 2.9. Let x E P(H) and u: B ____, M be as above. Prove that
μH(x,A#u) = μH(x, u) - 2c1(A)
for every A E 7r2(M), where c1 (A) E Z denotes the integral of the first Chern classc 1 E H^2 (M, Z) of the tangent bundle over A and A#u denotes the disc obtained
by taking the connected sum of a representative of A with u. DExercise 2.10. Given a Hamiltonian H with only nondegenerate 1-periodic so-lutions and x± E P(H), consider the space Z(x-, x+) of all smooth maps
u: JR x 81 ____, M which satisfy (8). Abbreviate
Z(H) = LJ Z(x-, x+).
x±EP(H)If uo1 E Z(xo,x1) and u12 E Z(x1,x2) such that uo1(s,t) = x 1 (t) for s ~ 0 and
u12(s, t) = x1(t) for s:::; 0 define the catenation uo1 #u12 E Z(xo, x2) by
# ( ) {
uo1(s,t), s::;O,Uo1 U12 S, t = U12 ( S, t) , S > _ Q.
Think of the Maslov index as a function Z(H) ____, Z: u r--+ μ(u, H) defined by (23).
Prove that this function has the following properties.
(Homotopy) The Maslov index is constant on the components of Z ( H).(Zero) If x-= x+ = x and u(s, t) = x(t) then
μ(u, H) = 0.
(Catenation)μ(uo1 #u12, H) = μ(uo1, H) + μ(u12, H).
(Chern class) If v : 82 ____, M then
μ(u#v,H)=μ(u,H)+2 { v*c 1.
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(Morse index) Assume Ht = H : M ____, JR is a Morse function with suffi-
ciently small second derivatives. Then the 1-periodic solutions x E P(H)
are the critical points of Hand, for every u E Z(x-, x+) with u(s, t) = u(s),μ(u, H) = indH(x+) - indH(x-).
(Fixed point index) For u E Z(x-, x+),
(-l)μ(u,H) = signdet(ll - dcpH(x-(0))) det(ll - dcpH(x+(o)))
where 'PH = 'ljJ 1 : M ____,Mis the time-1-map of (1). D
Exercise 2.9 shows that, in the case where all 1-periodic solutions x E P(H) are
nondegenerate and w satisfies (4), there exists a well defined function T/H: P(H) ____,
JR which satisfies
rJH(x) = μH(x, u) - 2rnH(x, u)
for every u: B ____, M with u(e^2 7fit) = x(t). By Exercise 1.23,
μ(u, H) = T/H(x-) -TJH(x+) + 2TE(u)
for every u E M(x-, x+; H, J). This proves the index formula in Theorem 1.24.
Let us now define