1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

LECTURE 3. FLOER HOMOLOGY 185

u

y

u(Oc

()

• )

x

Figure 11. The isomorphism between Morse homology and Floer homology

μH(x, u). In the zero dimensional case counting the solutions with suitable orienta-

tions gives rise to a chain homomorphism CM*(!)--+ CF*(H). In [38] it is shown

that this map induces an isomorphism on homology HM*(!)--+ HF*(M,w;H,J).

This construction can also be used to prove that the corresponding isomorphism

on cohomology identifies the quantum cohomology structure on H*(M) with the

pair-of-pants product on Floer cohomology. For details see [38].

3.6. Calabi-Yau manifolds

Let ( M, w) be a compact symplectic manifold whose first Chern class vanishes over

n2 ( M), i.e.

(36)

for every smooth map v : 8^2 --+ M. It is an immediate consequence of this condition

that the Conley-Zehnder index of a (nondegenerate) periodic solution x E P(H)

gives rise to a well defined integer

μH(x) = n - μcz(1lfx)

(see page 168), and hence Floer homology will be graded over the integers. How-

ever, care must be taken with bubbling of J-holomorphic spheres of Chern number

zero. Such bubbling no longer leads to connecting orbits of strictly lower index.

But the good news is that spheres with Chern number zero form subsets of M of

codimension 4, and hence will generically avoid the spaces of connecting orbits with

index difference 1 or 2, since these form geometrically at most 3-dimensional sets.

More precisely, if c 1 vanishes on n2(M) then, for a generic J E :J(M,w), the

moduli space

M^8 (J) = { v: 82 --+ M : 8J(v) = 0, vis simple}

of J-holomorphic spheres which are not multiply covered is a smooth manifold of di-

mension 2n. Dividing by the action of the reparametrization group G = PSL(2, q

gives a space M^5 (J)/G of dimension 2n - 6. Since each sphere is 2-dimensional
we obtain a space W^8 (J) = M^8 (J) x 8^2 /G of dimension 2n - 4 with the obvi-
ous evaluation map ev : W^8 (J) --+ M. The image of this map is the compact
codimension-4 subset of all points in M which lie on some J-holomorphic sphere
(with Chern number zero). For a generic Hamiltonian H E 'Hreg this set will not

intersect the moduli spaces M^1 (x-, x+; H, J) and M^2 (x-, x+; H, J) of connecting

orbits with index 1 or 2. This shows that no bubbling can occur for sequences of
such connecting orbits.
With this understood, there is an additional difficulty arising from the presence
of possibly infinitely many connecting orbits with index difference 1, with energy