1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

LECTURE 3. FLOER HOMOLOGY 177

detail in Floer-Hofer [12]. The rough idea is to prove first that the moduli spaces

M(x-, x+) are all orientable, and then to choose a system of coherent orientations

under which Floer's gluing maps (discussed below) are orientation preserving. That

such orientations exist is proved in [12]. These orientations are not unique. ·As in

the finite dimensional case they involve one choice of orientation for each critical

point x E P(H). With these orientations in place one can define a number c(u) E

{±1} for u E M^1 (x - , x+; H, J) by comparing this coherent orientation of M^1 with

the obvious flow orientation. The Floer boundary operator is now defined by

(28) c(u)(x)

xE'P( H) (u]ENfl(y,x;H ,J)

μ cz(x;H)=k-l(mod 2N)

for a periodic orbit y E P(H) with μcz(y; H) = k(mod 2N).

Theorem 3.5 (Floer). If (M,w) is monotone and HE 'Hreg then f)F o f)F = 0.

Proof. In the case N 2". 2 the proof is as in the finite dimensional case. The key is

to understand the ends of the 2-dimensional moduli space M^2 (z, x; H , J). If N 2". 2

one proves as before that no bubbling can occur and the ends correspond to pairs

of connecting orbits v E M^1 (z,y) and u E M^1 (y,x). In other words, we can think

of M^2 (z, x) = M^2 (z, x)/IR as a compact 1-manifold whose boundary is given by

(29) 8M^2 (z, x) = LJ M^1 (z, y) x M^1 (y , x).

yEP(H)

This follows from Floer's gluing theorem (see Section 3.3 below). Hence one obtains

(30) c(v)c(u) = 0

vEP(H) [v]ENt1(z, y) [u]EM^1 (y,x)

μ cz(y;H)=k(mod 2N)

for every pair x, z E P(H) with μcz(z ; H) = k + l(mod 2N) and μcz(x; H)

k - l(mod 2N).

In the case of minimal Chern number N = 1 there is an additional subtlety

arising from the presence of J-holomorphic spheres with Chern number

c1(v) = r v*ci = 1.

ls2

The bubbling of such spheres cannot be excluded apriori. However, if such bubbling
does occur, then the remaining limit solution u must have zero energy and hence
be of the form u(s, t) = x(t) = z(t). Now there are two facts which prevent such
bubbles. The first is a more subtle version of the compactness theorem (discussed in

Chapter 4 below) which asserts that the image of the bubbling sphere v: 82 --+ M

must intersect the remaining limit curve u(s, t) = x(t). But now, for a generic
J , the J-holomorphic curves with Chern number 1 form moduli space M(l; J) C

Map(8^2 , M) of dimension 2n + 2. Dividing by the 6-dimensional conformal group

G = PSL(2, C) we obtain a moduli space M(l; J)/G of dimension 2n - 4. Taking
account of the fact that each sphere is 2-dimensional we obtain finally a space
M(l; J) x 82 /G of dimension 2n - 2. Thus the points on J-holomorphic spheres
of Chern number 1 form, for a generic J, a codimension-2 subset of M, and for
a generic H these spheres therefore do not intersect the 1-dimensional periodic
orbits of H. Hence no such bubbling can occur, and we can proceed with the same
argument as above. D