1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

154 D. SALAMON, FLOER HOMOLOGY

Figure 3. A Morse-Smale gradient flow on the 2-torus

Figure 4. A Morse-Smale gradient flow on the real projective plane

Exercise 1.18. Consider the function f : cpn --+JR given by

n

f([zo: z1: · ··:Zn])= Lilzjl^2.

j=l

Find the critical points and compute their Morse indices. Compute the homology

groups of cpn. 0

1.4. Symplectic action

Let us now return to Hamiltonian differential equations in the monotone case.
In this section we show how the contractible 1-periodic solutions of (1) can be
interpreted as the critical points of the (circle valued) symplectic action functional
on the space CM of contractible loops in M. Here is how this works.
Throughout we think of a loop in M as a smooth map x : JR --+ M which

satisfies x(t + 1) = x(t) for t E R A tangent vector to .CM at such a loop x is

a vector field ~ along x. Explicitly, we think of ~ as a smooth map ~ : JR --+ TM

which satisfies ~(t) E Tx(t)M and ~(t + 1) = ~(t) fort ER We denote the space of

such vector fields by C^00 (JR/'1L,x*TM) = Tx.CM. For each 1-periodic Hamiltonian

Ht= Ht+l as above the loop space .CM carries a natural 1-form WH: T.CM--+ JR,

defined by

WH(x;~) = fo

1

w(i:(t)-Xt(x(t)),~(t))dt

for~ E Tx.CM. The zeros of this 1-form are precisely the 1-periodic solutions of (1).